Exploring Neutrino Mass Orderings through Supernova Neutrino Detection

Abstract

Core-collapse supernovae (SNe) are one of the most powerful cosmic sources of neutrinos, with energies of several MeV. The emission of neutrinos and antineutrinos of all flavors carries away the gravitational binding energy of the compact remnant and drives its evolution from the hot initial to the cold final states. Detecting these neutrinos from Earth and analyzing the emitted signals present a unique opportunity to explore the neutrino mass ordering problem. This research outlines the detection of neutrinos from SNe and their relevance in understanding the neutrino mass ordering. The focus is on developing a model-independent analysis strategy, achieved by comparing distinct detection channels in large underground detectors. The objective is to identify potential indicators of mass ordering within the neutrino sector. Additionally, a thorough statistical analysis is performed on the anticipated neutrino signals for both mass orderings. Despite uncertainties in supernova explosion parameters, an exploration of the parameter space reveals an extensive array of models with significant sensitivity to differentiate between mass orderings. The assessment of various observables and their combinations underscores the potential of forthcoming supernova observations in addressing the neutrino mass ordering problem.

1. Introduction

Core-collapse supernovae (SNe) are the final evolutionary stage of stars with masses $M\gtrsim 8M_{\odot }$ and represent a long-awaited observation target for neutrino telescopes. To explain these events, interdisciplinary research that combines nuclear physics, particle physics, and astrophysics is needed. Neutrinos are important for studying the energy balance involved in SN collapses, because only about $1\%$ of the gravitational binding energy is released as kinetic energy in the compact object formation, while the remaining $99\%$ is carried out by neutrinos of all flavors, with energies of several MeV [1]. The mechanisms leading to neutrino production in the SN core are, mainly, electron capture by nucleons $e^{-}+p\to n+{\nu }_e$, pair annihilation $e^{+}+e^{-}\to {\nu }_e+\overline{{\nu }_e}$, flavor conversion ${\nu }_e+\overline{{\nu }_e}\to {\nu }_{\tau ,\mu }+{\overline{\nu }}_{\tau ,\mu }$, and nucleon bremsstrahlung $N+N\to N^{\prime }+N^{\prime }+\nu +\overline{\nu }$ [2]. Once they have traveled through the stellar material and space, neutrinos reaching the Earth can be detected, providing precious information on the stellar core [3].

Studying the signals that the neutrinos leave in the detectors, with an effective neutrino flavor discrimination, it is possible to infer properties on their physics, because the structure of the neutrino mass spectrum and lepton mixing is imprinted into the detected signal. Neutrinos were already observed for the 1987A SN in the Large Magellanic Cloud. During this event, two water Cherenkov detectors, Kamiokande-II and the Irvine–Michigan–Brookhaven (IMB) experiment, observed neutrino interaction events at a time consistent with the estimated time of the collapse [4,5]. Two scintillator detectors, Baksan and LSD [6,7], also reported observations; the latter report was controversial because the events were recorded several hours early. At present, several detectors are ready and waiting for the detection of SN neutrinos from the next galactic explosion. SN neutrinos can be detected via electroweakly or strongly interacting products of weak charged-current (CC) and neutral-current (NC) interactions with electrons and nuclei. Some of the relevant interaction channels for current (and future) detectors are the inverse beta decay and neutrino–proton elastic scattering for scintillation detectors, the inverse beta decay and neutrino–electron elastic scattering for water Cherenkov detectors, and the absorption interaction on ${}^{40}$Ar in liquid Argon time projection chambers, among others.

Analytical models exist for the SN neutrino energy spectra that are useful for studying and predicting SN neutrino signals. The spectral distribution is often parametrized by a three-parameter fit that allows for deviations from a strictly thermal spectrum (motivated by analytic simplicity) [8,9]. These three parameters are the total neutrino energy ${ϵ}_{\alpha }$, the neutrino mean energy $\langle E_{\alpha }\rangle$, and the spectral index or shape parameter ${\beta }_{\alpha }$, where $\alpha =({\nu }_e,{\overline{\nu }}_e,{\nu }_x)$. SN simulations provide the indicative values of the mentioned parameters. However, they depend on many details of the neutrino transport inside the star, the properties of the incompletely known neutron-star equation of state, the properties of the collapsing star, and time [10,11]. During the accretion phase, which takes place a few tens to hundreds of milliseconds after the bounce, the expected neutrino energy spectrum would exhibit a flavor hierarchy $\langle E_{{\nu }_e}\rangle <\langle E_{{\overline{\nu }}_e}\rangle <\langle E_{{\nu }_x}\rangle$ [2]. In recent years, several works have presented sensitivity studies of the energy spectrum of neutrinos concerning spectral parameters and developed strategies to minimize dependence on spectral models [12,13]. An interesting approach is through a combination of observables, for example, comparing CC- and NC-induced events [14,15,16], to achieve model-independent signatures. Also, the ratio of events in these detection channels can distinguish between different neutrino mixing effects and processes. In Reference [16], the authors study the possibility of disentangling a pure matter effect from a complete flavor equalization on the fluxes due to the neutrino–neutrino interactions.

The analysis and reconstruction of SN neutrino fluxes is an interesting tool to clarify the role of neutrinos in stellar explosion events and nucleosynthesis, as well as for studying physics at high densities and, in particular, the neutrino oscillation phenomena [3,17]. The neutrino oscillation phenomenon consists of a quantum process in which a flavor neutrino is described as a superposition of mass eigenstates, allowing it to change families between its emission and detection. Different detectors on Earth have measured neutrino fluxes from the sun, atmosphere, and reactors and reported the presence of this phenomenon consistently [18,19,20]. The neutrino mass ordering is one of the major open issues in this regard, being one of the experimental priorities in the area of particle physics. This is called normal (inverted) if $\Delta m_{31}^2=m_3^2-m_1^2\ge 0(\le 0)$. Finding out which of the two cases occurs in nature has profound implications for cosmology, searches for neutrino mass, and studies of double-beta decays. In particular, neutrinos from SNe can be used to study this problem, because studying the neutronization burst, the time profile on the early signal, and the Earth matter effect can provide evidence of the neutrino mass ordering [21,22,23].

Given the aforementioned information, it is expected that a possible difficulty is that both core-collapse physics and neutrino physics affect the nature of the SN neutrino burst, and it may not be trivial to disentangle the two. The more experimental data we can gather in as many detectors around the globe as possible, and as far as we can find model-independent signatures, the better our chances will be of disentangling the various effects.

This research investigates whether the neutrino counts from SNe within the first second after the burst, detected across various channels and detectors, can elucidate the correct neutrino mass ordering. The main goal is to formulate a model-independent analysis approach by comparing diverse detection channels in large underground detectors, aiming to identify potential indicators of mass ordering in the neutrino sector. Furthermore, a thorough statistical analysis is conducted on the expected signals for both mass orderings, aiming to ascertain its discernibility.

2. Supernova Neutrino Fluxes

The time-integrated neutrino flux streaming off the SN can be expressed as (for the un-oscillated case):

$$F_{\alpha }^0\left(E\right)=\frac{{ϵ}_{\alpha }}{4\pi d^2\langle E_{\alpha }\rangle }{f}_{{\nu }_{\alpha }}\left(E\right),$$

(1)

where $\langle E_{\alpha }\rangle$ are the neutrino mean energies, ${ϵ}_{\alpha }$ is the total neutrino energy (equivalent to the time-integrated luminosity) for which we assume equipartition among all the neutrino flavors, d is the SN distance, and $f_{{\nu }_{\alpha }}\left(E\right)$ is the time-averaged energy spectrum [9] given by

$$f_{{\nu }_{\alpha }}\left(E\right)=\frac{{(1+{\beta }_{\alpha })}^{1+{\beta }_{\alpha }}}{\Gamma (1+{\beta }_{\alpha })}\frac{E^{{\beta }_{\alpha }}}{{\langle E_{\alpha }\rangle }^{{\beta }_{\alpha }+1}}exp\left[-({\beta }_{\alpha }+1)\frac{E}{\langle E_{\alpha }\rangle }\right]$$

(2)

where $\beta$ is the pinching parameter of the spectral distribution. Given that the values of the spectral parameters depend on the SN simulation or model, the variation within specific ranges will be studied.

To calculate the neutrino fluxes including oscillation effects ($F_{\alpha }\left(E\right)$), it is necessary to determine how the neutrino spectral distributions from Equation (2) are altered by the oscillations, by solving the flavor evolution equations. Calling $\rho$ ($\overline{\rho }$) to the neutrino (antineutrino)-distribution function in its matrix form and $\mathcal{H}$ ($\overline{\mathcal{H}}$) the neutrino (antineutrino) Hamiltonian in the flavor basis, the differential equations that give the dependence of the neutrino (antineutrino)-distribution functions upon the radius are [17,24]

$$\begin{array}{c}\hfill i\frac{\partial \rho }{\partial r}=\left[\mathcal{H},\rho \right],i\frac{\partial \overline{\rho }}{\partial r}=\left[\overline{\mathcal{H}},\overline{\rho }\right].\end{array}$$

The Hamiltonian can be written as

$$\begin{array}{ccc}\hfill \mathcal{H}& =& {\mathcal{H}}^{vac}+{\mathcal{H}}^m+{\mathcal{H}}^{\nu -\nu },\hfill \end{array}$$

(3)

where ${\mathcal{H}}^{vac}$ describes neutrino oscillations in vacuum, ${\mathcal{H}}^m$ represents the neutrino–matter interactions, and ${\mathcal{H}}^{\nu -\nu }$ takes into account the neutrino–neutrino interactions.

Three-Active Scheme: The Two Flavor Approximations

This work focuses on the first milliseconds until the first second of the SN event. In this stage, the effects of matter are dominant, and the self-induced effects can be neglected. Furthermore, during the whole accretion phase the matter potential is also expected to dominate over the neutrino–neutrino potential [25,26] (see Appendix C for details and discussion). Operating within the three-active scheme, the calculation of SN neutrino fluxes incorporates both vacuum and matter effects [3]. The mixing mechanism is assumed to remain unaffected by CP violations. Additionally, a rotation can be performed in this subspace to diagonalize the submatrix $\mu ,\tau$ of Equation (3) [17,27]. The ${\nu }_{\mu }$ and ${\nu }_{\tau }$ fluxes in an SN are very similar and these two flavors play symmetric roles. Therefore, it is useful to define a linear combination ${\nu }_x$ that is essentially identical with the $m_3$ mass eigenstate and mixes with ${\nu }_e$ by means of the ${\theta }_{13}$ mixing angle [17,28]. Because the matter density (and therefore the potential) decreases with the star radius, active neutrinos exhibit two Mikheyev–Smirnov–Wolfenstein (MSW) resonances called H (high density) and L (low density) where the flavor conversion mechanism is amplified [27]. Instead of using the adiabatic approximation, we solved the complete equations numerically, considering the ${\mathcal{H}}^{vac}$ and ${\mathcal{H}}^m$ terms in the Hamiltonian (see Appendix A for details).

For this work, the mixing parameters considered are from PDG 2020 [29]: ${sin}^2\left(2{\theta }_{13}\right)=0.0241$ and $\Delta m_{31}^2=2.52\times {10}^{-3}{\mathrm{eV}}^2$ ($\Delta m_{31}^2=-2.24\times {10}^{-3}{\mathrm{eV}}^2$) for the normal (inverted) mass ordering.

In Figure 1, the fluxes obtained following the above description for an SN event at 10 kpc of distance are presented. The non-oscillation case is shown in the first column, while in the second and third columns we show the fluxes for the case in which the neutrino oscillations for NO (2nd column) and IO (3rd column) are considered.

3. Selected Detectors and Interactions Channels

To calculate the expected signal on Earth, the interactions of SN neutrinos in various channels have been computed. This involved selecting detectors of different technology and incorporating channels sensitive to different flavors. In

Table 1. Selected detectors to perform the calculations of the expected signal. The selected interaction channels (type of current and processes) are shown in the last two columns.

Detector	Location	Tot. Mass (Fid. Mass)	Current	Process
SNO+	SNOLAB	∼780 t (∼0.45 kt)	CC	$\overline{{\nu }_e}+p\to n+e^{+}$
			NC	$\nu +p\to {\nu }^{\prime }+p$
			NC	$\nu {+}^{12}\mathrm{C}{\to }^{12}{\mathrm{C}}^{*}\left(15.11\mathrm{MeV}\right)$$+{\nu }^{\prime }$
HALO	SNOLAB	∼79 t	CC	${\nu }_e{+}^{208}\mathrm{Pb}{\to }^{207}\mathrm{Bi}+\mathrm{n}+{\mathrm{e}}^{-}$
			CC	${\nu }_e{+}^{208}\mathrm{Pb}{\to }^{206}\mathrm{Bi}+2\mathrm{n}+{\mathrm{e}}^{-}$
			NC	${\nu }_x{+}^{208}\mathrm{Pb}{\to }^{207}\mathrm{Pb}+\mathrm{n}$
			NC	${\nu }_x{+}^{206}\mathrm{Pb}{\to }^{207}\mathrm{Pb}+2\mathrm{n}$
DUNE	Fermilab	∼70 kt (∼40 kt)	CC	${\nu }_e{+}^{40}\mathrm{Ar}\to {\mathrm{e}}^{-}{+}^{40}{\mathrm{K}}^{*}$
			CC	${\overline{\nu }}_e{+}^{40}\mathrm{Ar}\to {\mathrm{e}}^{+}{+}^{40}{\mathrm{Cl}}^{*}$
			NC	$\nu {+}^{40}\mathrm{Ar}\to$$\nu {+}^{40}{\mathrm{Ar}}^{*}$
			NC + CC	$\nu +e^{-}\to \nu +e^{-}$

, the chosen detectors, their host laboratory (location), the type of channel (neutral or charged current) they can detect, and the considered processes are presented.

3.1. Interactions in SNO+

The SNO+ experiment, located 2 km underground at SNOLAB in Sudbury, Canada, is a liquid scintillator detector sensitive to neutrinos emitted from an SN in the Milky Way through the interaction channels mentioned in Table 1. Its detection efficiency is assumed to be perfect above the 200 keV threshold. Here, it is assumed that $N_p=3.32\times {10}^{31}$ free protons in a fiducial mass of 0.45 kt and an energy resolution of $5\%/\sqrt{E_{vis}}$ [31]. Neutrino interactions with free protons are considered through inverse beta decay (IBD) $\left({\overline{\nu }}_e+p\to n+e^{+}\right)$, neutrino–proton elastic scattering (pES) $\nu +p\to {\nu }^{\prime }+p$, and neutrino–nucleus reactions in ${}^{12}$C ($\nu -C^{12}$), given by $\nu +{}^{12}\mathrm{C}\to {{}^{12}\mathrm{C}}^{*}\left(15.11\mathrm{MeV}\right)$$+{\nu }^{\prime }$. The details of the calculation of the event rate and cross section are given in the Appendix B. The cross sections for the reactions considered for the SNO+ detector as a function of the neutrino energy are shown in the left panel of Figure 2, while the right panel shows the expected signal for all the mentioned channels.

3.2. Interactions in HALO

The HALO detector is dedicated to the study of SN neutrinos and it is able to observe the neutrons emitted from electron neutrino scattering on lead from charged current (CC) and NC events. Measuring one- and two-neutron events on lead is particularly attractive to extract information about the SN neutrino temperatures [32]. For the listed reactions in Table 1, neutrons are detected using ${}^{3}He$ counters as performed for the SNO experiment. HALO will measure both NC and CC events without distinguishing them because the outgoing electrons are not detected. Because the outgoing lepton is not identified, the total event rate $N^{tot}$ is given by the sum of both the NC and CC expected rates

$$N_{1n\left(2n\right)}^{tot}=N_{1n\left(2n\right)}^{NC}+N_{1n\left(2n\right)}^{CC}$$

(4)

where $N_{1n\left(2n\right)}$ refers to 1-neutron (1n) or 2-neutron (2n) event rates. The CC and NC event rates are given by

$$\begin{array}{ccc}\hfill N_{1n\left(2n\right)}^{CC}& =& N_{Pb}\int \frac{d{\sigma }_{1n\left(2n\right)}^{CC}}{dE}{F}_{{\nu }_e}dE\hfill \\ \hfill N_{1n\left(2n\right)}^{NC}& =& N_{Pb}\int \sum _{\alpha =e,\mu ,\tau }\left[\frac{d{\sigma }_{1n\left(2n\right)}^{NC,\nu }}{dE}{F}_{{\nu }_{\alpha }}+\frac{d{\sigma }_{1n\left(2n\right)}^{NC,\overline{\nu }}}{dE}{F}_{{\overline{\nu }}_{\alpha }}\right]dE\hfill \end{array}$$

(5)

The cross sections for the CC process were taken from Ref. [33], while for the NC channels, the cross sections were extracted from SNOwGlobes [34]. The cross sections as a function of the neutrino energy are plotted in the left panel of Figure 3. The right panel shows the signal expected in the detector in both 1n and 2n cases by assuming a detector efficiency of 36% [32].

3.3. Interactions in DUNE

The Deep Underground Neutrino Experiment (DUNE) will be made up of four 10 kton liquid argon time projection chambers (LArTPCs). DUNE’s dynamic range is such that it is also sensitive to neutrinos with energies down to about 5 MeV. CC interactions of neutrinos from around 5 MeV to several tens of MeV create short electron tracks in liquid argon, potentially accompanied by gamma ray and other secondary particle signatures. This regime is of particular interest for the detection of neutrinos from a galactic SN [35]. DUNE will have a 70 kton liquid argon mass in total [36], of which 40 kton will be fiducial mass (10 kton fiducial mass per module). For each module, a number of argon nuclei ${\mathrm{N}}_{\mathrm{Ar}}=1.5\times {10}^{32}$ and of electrons ${\mathrm{N}}_{\mathrm{e}}=2.7\times {10}^{33}$ is assumed [37]. A detector efficiency, an energy bin distribution, and an energy resolution extracted from [35] have also been considered. In this case, the MARLEY [38] cross sections implemented in SNOwGLoBES [34] are used, as shown in the left panel of Figure 4. The right panel displays the expected signal in the detector.

4. Combined Signals

The signatures of the neutrino mixing effects in the detected signals are closely related to the difference between the electronic and non-electronic fluxes. However, as mentioned before, the fluxes are model-dependent, posing a challenge in the analysis. To address this issue, two approaches have been employed: On the one hand, several works performed a global fit of the data, simultaneously determining the oscillation parameters and the parameters of the original fluxes [39,40,41]. On the other hand, other research has employed a combination of observables to reduce the reliance on specific models and the number of free parameters [15,16,33]. By utilizing this strategy, the analysis becomes less model dependent, providing a more robust characterization of the neutrino mixing phenomena.

Here, a hybrid approach is adopted. In addition to the regular counts expected for each interaction channel, combinations between them are studied. Specifically, the following ratios are analyzed:

• The ratio of 1n and 2n events expected in the HALO detector ($\frac{{\mathrm{N}}_{1\mathrm{n}}}{{\mathrm{N}}_{2\mathrm{n}}}$).
• The ratio between the pES and IBD events expected in the SNO+ detector ($\frac{{\mathrm{N}}_{\mathrm{pES}}}{{\mathrm{N}}_{\mathrm{IBD}}}$).
• The ratio of CC and NC events on Argon in DUNE ($\frac{{\mathrm{N}}_{{\mathrm{CC}}_{\mathrm{Ar}}}}{{\mathrm{N}}_{{\mathrm{NC}}_{\mathrm{Ar}}}}$).
• The ratio of ${\nu }_e$ and $\overline{\nu }e$ events on Argon in the DUNE detector ($\frac{\mathrm{N}{\nu }_e-Ar}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}$).

By considering these combinations, the aim is to extract more information and enhance our sensitivity to neutrino mixing effects and the neutrino mass ordering in the supernova context.

These ratios are attractive because they do not depend on normalization factors (like the SN distance or the time-integrated luminosity $ϵ$) and have already been used in the past to study the ${\theta }_{13}$ mixing angle and for disentangling between different neutrino flavor transformation scenarios [15,16]. In this work, the extension of using these observables to study the neutrino mass ordering is explored. Additionally, variations in the parameters describing the initial neutrino spectrum within the ranges of 8 MeV $<\langle E_{\alpha }\rangle <$ 28 MeV and $2\le \beta \le 4.5$ are examined. Furthermore, the constraint $\langle E_{{\nu }_e}\rangle \le \langle E_{{\overline{\nu }}_e}\rangle \le \langle E{\nu }_x,{\overline{\nu }}_x\rangle$ on the mean energies is imposed.

In Figure 5, the ranges obtained for each ratio are displayed, considering both mass orderings and variations in the spectral parameters within the mentioned ranges.

Regarding the HALO detector ratio, it is observed that the ${\nu }_e{+}^{208}\mathrm{Pb}$ reaction dominates, being prevalent in both 1n and 2n emission channels. Furthermore, a greater number of events are generated in the NO scenario compared to the IO scenario, and this difference is more pronounced in the 2n channel. Consequently, when analyzing the $\frac{\mathrm{N}1n}{\mathrm{N}2n}$ ratio, the IO scenario exhibits higher values than the NO case.

For the SNO+ ratio, IBD events in the IO scenario surpass those in the NO scenario. Consequently, the ratio $\frac{\mathrm{N}pES}{\mathrm{N}IBD}$ peaks in the NO scenario, as the $N_{pES}$ events remain unaffected by neutrino oscillations.

Shifting focus to the DUNE ratio $\frac{{\mathrm{N}}_{CCAr}}{{\mathrm{N}}_{NCAr}}$, the dominant channel for CC interactions is ${\nu }_e{+}^{40}\mathrm{Ar}$. In this case, the NO scenario yields a higher event count than the IO scenario. This behavior is clearly depicted in the ratios presented in the bottom-left panel of Figure 5.

In the case of the DUNE ratio $\frac{\mathrm{N}{\nu }_e-Ar}{\mathrm{N}\overline{\nu }e-Ar}$ (illustrated in the bottom-right panel of Figure 5), this discrepancy is even more pronounced, given that ${\mathrm{N}}_{\overline{\nu }e-Ar}$ is greater for the IO scenario compared to the NO one.

Regarding the behavior of the ratios with respect to the pinching parameter, it is observed that as $\beta$ increases, the individual counts in the studied channels $N_i$ decrease. This is related to the fact that smaller values of $\beta$ increase the number of neutrinos at higher energies and reduce the number of neutrinos at lower energies.

For the ratios $\frac{{\mathrm{N}}_{1\mathrm{n}}}{{\mathrm{N}}_{2\mathrm{n}}}$, $\frac{{\mathrm{N}}_{{\mathrm{CC}}_{\mathrm{Ar}}}}{{\mathrm{N}}_{{\mathrm{NC}}_{\mathrm{Ar}}}}$, and $\frac{\mathrm{N}{\nu }_e-Ar}{\mathrm{N}{\overline{\nu }}_e-Ar}$, the decrease in counts is more pronounced for the denominator, resulting in higher values for the ratio as $\beta$ increases.

On the other hand, for the ratio $\frac{{\mathrm{N}}_{\mathrm{pES}}}{{\mathrm{N}}_{\mathrm{IBD}}}$, ${\mathrm{N}}_{\mathrm{pES}}$ decreases more significantly compared to ${\mathrm{N}}_{\mathrm{IBD}}$, leading to a lower ratio at higher values of $\beta$.

With experimental knowledge of the event ratios, determination of the allowed parameter ranges for each mass ordering would become feasible. In addition, by examining the overlaps between these ranges for the different studied ratios, it would be possible to determine the mass ordering. Combining studies on different observables is crucial to avoid parameter degeneracies and obtain more robust results.

From Figure 5, it is seen that $\frac{N_{1n}}{N_{2n}}>15.76$, $\frac{N_{C{C}_{Ar}}}{N_{N{C}_{Ar}}}<1.41$, and $\frac{{\mathrm{N}}_{{\nu }_e-Ar}}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}<17.14$ exclude NO. Meanwhile, $\frac{N_{1n}}{N_{2n}}<1.33$, $\frac{N_{pES}}{N_{IBD}}>9.21$, $\frac{N_{C{C}_{Ar}}}{N_{N{C}_{Ar}}}>6.57$, and $\frac{{\mathrm{N}}_{{\nu }_e-Ar}}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}>198.29$ exclude IO. Additionally, calculations have been conducted to determine the values of these ratios for three distinct SN models. These are the Dasgupta model [30] and two test models: Test Model 1 and Test Model 2. Model 1 falls into non-overlapping regions for the HALO and SNO+ ratios and is given by $\beta =2.5$, $\langle E_{{\nu }_e}\rangle =8$, $\langle E_{\overline{{\nu }_e}}\rangle =9$, and $\langle E_{{\nu }_x}\rangle =25$, while the second falls in the non-overlapping region of the DUNE ratios and has a small dispersion for SNO+, and it is given by $\beta =3.5$, $\langle E_{{\nu }_e}\rangle =8$, $\langle E_{\overline{{\nu }_e}}\rangle =9$, and $\langle E_{{\nu }_x}\rangle =11$. The three models have been superimposed on the figures along with their respective errors.

In Figure 6, the relative statistical error for the ratios depicted in Figure 5 is presented by solid lines1. The shaded areas illustrate the counts that can be generated within the parameter space considered in this study. The gray or beige color corresponds to the counts generated by NO or IO, respectively. For this plot, the overall factors, d and $ϵ$, were fixed at 10 kpc and 25 foe, respectively. In most of the studied cases, the statistics are sufficiently large, making the statistical error not a limiting factor. Furthermore, the arrows indicate the growth of the counts in each channel for NO (represented by dashed lines) and IO (represented by dotted lines). These arrows have been constructed with $\beta =3$, $\langle E_{{\nu }_e}\rangle =12\mathrm{MeV}$, and $\langle E_{\overline{{\nu }_e}}\rangle =15\mathrm{MeV}$, while the direction of the arrows indicates the increase in $\langle E_{{\nu }_x}\rangle$ from 15 MeV to 28 MeV.

5. Statistical Analysis

A statistical analysis was conducted to compare the expected event counts at each detector for both the NO and IO scenarios. The aim was to investigate the possibility of replicating an expected signal for a particular mass ordering using the opposite ordering while allowing for variations in the spectral SN parameters within a realistic range. The analysis employed a chi-square test to determine the feasibility of such a reproduction. Additionally, the study aimed to identify any preferential channels, detectors, or combinations thereof that could be instrumental in detecting signatures of the neutrino mass ordering.

5.1. Individual Counts Analysis

First, an assessment of the sensitivity of individual channel counts ($N_i$) within each detector was conducted using a chi-square minimization procedure. Subsequently, the overall sensitivity was calculated by considering all the analyzed channels and detectors and minimizing a global chi-square. For the benchmark model, the NO scenario was considered. The indicator is defined by

$${\chi }_{\alpha }^2=\sum _i\frac{{\left[N_i^{\alpha }({\widehat{\theta }}^{SN}, NO)-N_i^{\alpha }({\theta }^{SN}, IO)\right]}^2}{{\sigma }^2(N_i^{\alpha }({\widehat{\theta }}^{SN}, NO))},$$

(6)

where $\alpha$ refers to the considered interaction channel, $N_i^{\alpha }$ are the number of events in the $\alpha$ channel, calculated by integrating the spectra $dN/dE$ over the visible energy, as described in Section 3, and the sum runs over the i-energy bins. For both SNO+ and DUNE, equally spaced bins of 0.5 MeV were considered [35,42]. In the case of HALO, the analysis involved total counts because the detector operates based on neutron counting and does not entail a spectral analysis [32]. For each data point, a statistical error of ${\sigma }_i=\sqrt{n_i}$ is considered. ${\widehat{\theta }}^{SN}$ and ${\theta }^{SN}$ represent the set of parameters $\{\beta ,\langle E_{{\nu }_e}\rangle ,\langle E_{\overline{\nu }e}\rangle ,\langle E{\nu }_x\rangle ,ϵ\}$ that define the initial SN spectra for the NO and IO models, respectively. Furthermore, the sensitivity is calculated by incorporating all the studied channels using a global chi-square, denoted as

$${\chi }_{\mathrm{global}}^2=\sum _{\alpha }{\chi }_{\alpha }^2.$$

(7)

The exploration encompasses parameter ranges for both ${\widehat{\theta }}^{SN}$ and ${\theta }^{SN}$, involving $8\mathrm{MeV}\le \langle E_{{\nu }_j}\rangle \le 28\mathrm{MeV}$, $2<\beta <4.5$, and $1/6\times {10}^{53}\mathrm{erg}<ϵ<1\times {10}^{53}\mathrm{erg}$ (corresponding to $E_{{\nu }_{\mathrm{tot}}}=[1-6]\times {10}^{53}$ erg). Additionally, the constraint $\langle E_{{\nu }_e}\rangle \le \langle E_{{\overline{\nu }}_e}\rangle \le \langle E{\nu }_x,{\overline{\nu }}_x\rangle$ is imposed on the average energies at the neutrinosphere.

The minimization of the statistical indicators was performed using IMINUIT2 [43], a Python interface of the MINUIT2 C++ package (standard tool at CERN) that minimizes the multi-variate function with constraints [44].

As a first step, we did not perform marginalization on $ϵ$ but instead focused on studying the effects of its variation along with the SN distance. To accomplish this, we introduced an overall factor $A=\frac{ϵ\left[\mathrm{foe}\right]}{d^2\left[\mathrm{kpc}\right]}$. It is important to note that the obtained results are applicable to any combination of $ϵ$ and d that yields the same value of A3.

Concerning the analysis of the counts for the individual channels ($N_i$), it was observed that there was a lack of sensitivity to changes in the mass ordering. This was indicated by the fact that ${\chi }_{{\alpha }_{min}}^2<0.6$ for all channels, meaning that the counts produced by the NO could be reproduced using IO by adjusting the spectral parameters accordingly. In contrast, more significant sensitivities were identified when considering the global observable. In Figure 7, the results of the minimization process for the global chi-square are presented for various values of A. The contours illustrate the minimum chi-square as a function of the benchmark model parameters, taking into account the allowed variation in the IO model parameters as well. These provide insights into the parameter regions that enhance sensitivity.

It was observed that for the global observable, the sensitivity increased as the value of $\beta$ decreased and as the value of $\langle E_{{\nu }_x}\rangle$ increased. Qualitatively, this can be attributed to the extension of neutrino fluxes to higher energies in these conditions. Consequently, interaction channels with higher thresholds (such as the 2n channel for HALO and ${\overline{\nu }}_e-Ar$ for DUNE) become relevant and begin to substantially influence the fitting process. In turn, it was observed that the 1n and 2n channels in HALO, as well as the CCAr channels in DUNE, posed the most significant challenges for simultaneous minimization. As expected, higher values of A produce higher sensitivities, because they are associated either with shorter distances or higher luminosities.

Next, parameter A was incorporated into the minimization process. In Figure 8, contours for the global analysis are presented, considering the variation in the luminosity and distance between the models. It was observed that the minimization of the chi-square resulted in smaller values. This outcome was expected due to the incorporation of an additional marginalized parameter. A consistent observation was made that at higher non-electronic energies, there was a greater sensitivity in distinguishing between different models. Additionally, when considering higher luminosities or shorter distances, the discrimination between models with NO or IO becomes more feasible.

5.2. Ratio Analysis

Next, a similar analysis was performed, focusing on the event ratios as defined in Section 4. In this case:

$${\chi }_{{\frac{c{h}_1}{c{h}_2}}_{min}}^2=\frac{{\left[\frac{N_{c{h}_1}}{N_{c{h}_2}}({\widehat{\theta }}^{SN}, NO)-\frac{N_{c{h}_1}}{N_{c{h}_2}}({\theta }^{SN}, IO)\right]}^2}{{\sigma }^2(\frac{N_{c{h}_1}}{N_{c{h}_2}}({\widehat{\theta }}^{SN}, NO))},$$

(8)

where $c{h}_1$ and $c{h}_2$ stand for the two involved channels in the ratio calculation. The errors were calculated as described in Section 4. Unlike the previous case, the ratios no longer depend on $ϵ$ or distance. Thus, the results are not marginalized over epsilon. However, because epsilon contributes to the calculation of the errors, the results are presented for different values of A as defined in the previous subsection. The ranges for all the other parameters remain consistent with those mentioned above. In Figure 9, we show the contour plots of the minimum chi-square for the four studied ratios as a function of $\langle E_{{\nu }_x}\rangle$, because we noted that varying the values of mean energies for electron-type neutrinos has a small effect. Once again, we display the results for different values of the parameter A. Our findings indicate that there is an increased sensitivity with respect to the mass ordering as $E_{{\nu }_x}$ increases, with the exception of the $\frac{N_{C{C}_{Ar}}}{N_{N{C}_{Ar}}}$ ratio, which exhibits the opposite behavior. Additionally, we observed that the sensitivity of the ratios associated with HALO and SNO increases at lower values of $\beta$, while those associated with DUNE show enhanced sensitivity at higher values of $\beta$.

The ratio associated with HALO exhibits the highest sensitivity. This can be attributed to the fact that, for large values of $E_{{\nu }_x}$ (≳18 MeV), the fluxes in the NO scenario extend to higher neutrino energies. Consequently, the ${\nu }_e{+}^{208}\mathrm{Pb}{\to }^{206}\mathrm{Bi}+2\mathrm{n}+{\mathrm{e}}^{-}$ channel becomes significant for the NO ordering, resulting in larger event counts. As a result, there are substantial differences between the $\frac{{\mathrm{N}}_{1\mathrm{n}}}{{\mathrm{N}}_{2\mathrm{n}}}$ ratios for NO and IO, leading to an amplified chi-square numerator. Additionally, for these particular $E_{{\nu }_x}$ values, the event counts for the ${\nu }_e{+}^{208}\mathrm{Pb}{\to }^{207}\mathrm{Bi}+\mathrm{n}+{\mathrm{e}}^{-}$ channel increase for both mass orderings. The combined effect of these factors results in a large number of counts on both channels 1n and 2n, which are associated with smaller statistical errors, reducing the denominator of the chi-square (as depicted in Figure 6) and thereby enhancing the overall sensitivity.

The second dominant ratio associated with SNO+ ($\frac{{\mathrm{N}}_{\mathrm{pES}}}{{\mathrm{N}}_{\mathrm{IBD}}}$) exhibits significant growth at high $\langle E_{{\nu }_x}\rangle$. This growth can be primarily attributed to a substantial increase in counts from both the pES channel and the IBD channel for IO, indicating a strong response at non-electronic energies. In contrast, the counts in the IBD channel for the NO scenario remain relatively unchanged. Consequently, as $\langle E_{{\nu }_x}\rangle$ increases, the numerator of the corresponding chi-square becomes larger. Moreover, the increase in counts in the pES channel leads to a smaller sigma error, ultimately resulting in a higher final chi-square value.

6. Conclusions

The neutrino mass ordering problem remains unsolved; thus, it is interesting to investigate if a future observation of a galactic SN could help to discriminate among possible mass ordering scenarios.

In this work, we have presented predictions for the expected neutrino events in three different detectors with distinct technologies: the scintillator SNO+, the helium and lead detector HALO, and the liquid argon detector DUNE.

First, we presented a study regarding the ratios of events, which let us eliminate the dependence with the distance and with the integrated luminosity. This allowed us to focus specifically on how these ratios vary with respect to the SN model spectral parameters and the mass ordering.

We showed that the ratios $\frac{N_{1n}}{N_{2n}}$, $\frac{N_{pES}}{N_{IBD}}$, $\frac{N_{C{C}_{Ar}}}{N_{N{C}_{Ar}}}$, and $\frac{{\mathrm{N}}_{{\nu }_e-Ar}}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}$ can potentially exclude either normal or inverted mass ordering for the neutrino sector without requiring the assumption of a specific SN model.

We were able to find regions that exclude one ordering or another, finding the following:

• $\frac{N_{1n}}{N_{2n}}>15.76$ excludes NO, while $\frac{N_{1n}}{N_{2n}}<1.33$ excludes IO;
• $\frac{N_{pES}}{N_{IBD}}>9.21$ excludes IO;
• $\frac{N_{C{C}_{Ar}}}{N_{N{C}_{Ar}}}<1.41$ excludes NO, while $\frac{N_{C{C}_{Ar}}}{N_{N{C}_{Ar}}}>6.57$ excludes IO;
• $\frac{{\mathrm{N}}_{{\nu }_e-Ar}}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}<17.14$, excludes NO, while $\frac{{\mathrm{N}}_{{\nu }_e-Ar}}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}>198.29$ excludes IO.

Then, we performed a statistical analysis to study the sensitivities of different observables associated with the change in the neutrino mass ordering. We had observed that analyzing combinations of events across different channels yielded more meaningful insights than examining the isolated counts for individual channels ($N_i$). In particular, we observed that the global observable, encompassing all channels collectively, displayed a heightened sensitivity in discerning between the mixing scenarios. Specifically, the 1n and 2n channels in HALO, along with the CCAr channels in DUNE, posed significant challenges during simultaneous minimization, identifying them as crucial channels.

Through statistical analysis and minimization, we found that the most sensitive ratios to the mass ordering are $\frac{{\mathrm{N}}_{1\mathrm{n}}}{{\mathrm{N}}_{2\mathrm{n}}}$ for HALO and $\frac{{\mathrm{N}}_{\mathrm{pES}}}{{\mathrm{N}}_{\mathrm{IBD}}}$ for SNO+, being two complementary detectors when studying these effects. The fact that they have the same location makes this even more interesting because there will be no effects on the signal due to geographical differences, such as the Earth matter effects. Additionally, the ratio $\frac{{\mathrm{N}}_{{\nu }_e-Ar}}{{\mathrm{N}}_{{\overline{\nu }}_e-Ar}}$ for DUNE is the one that generates larger numerical differences when comparing both mass orderings, but it is also the one with the greatest associated statistical errors.

Although the specific parameters governing future supernova explosions remain unknown, exploring the parameter space within the ranges studied reveals a wide range of models that could exhibit high sensitivity. By examining the sensitivities of various observables, we can determine which combination of experiments would be sensitive in different regions of the parameter space. Through the analysis of multiple signals, considering different channels and ratios, we have the potential to differentiate between different mass orderings. The probability of achieving higher sensitivities for this discrimination increases if future observations occur in the regions of Figure 5 that do not overlap. In general, we observe that higher non-electronic energies offer greater sensitivity in distinguishing between different models, while higher luminosities or shorter distances enhance the discrimination between models with NO or IO, making it more feasible to identify the ordering of neutrino masses.

The data collected from a network of detectors with diverse energy thresholds and detection channels offers a unique and powerful tool for imposing constraints on the parameters of the initial SN neutrino fluxes and determining the neutrino mass ordering. This approach provides valuable insights into the influence of SN spectral parameters and mass ordering on the observed data, leading to a more comprehensive understanding of the underlying physics involved in these phenomena.