Flavor Triangle of the Diffuse Supernova Neutrino Background

For the calculation of the DSNB fluxes we employ two approaches. In the first, we adopt the predictions of Ref. [70], hereafter H18, where the authors performed a detailed study combining multiple simulations of both core collapse of massive stars to neutron stars and black holes. This approach is an attempt to account for the progenitor-dependence of core-collapse neutrino emission. In the second approach, we adopt a simple Fermi-Dirac (FD) model described by a thermal temperature.

In H18, suites of axisymmetric two-dimensional hydrodynamical simulation of core collapse to neutron stars were combined with half a dozen spherically symmetric one-dimensional simulations of core collapse to black holes. For the collapse to neutron stars, the authors considered two sets of simulations: ($i$) a large suite of over 100 progenitors [71] using a leakage approximation for heavy lepton neutrino transport and a more sophisticated ray-by-ray neutrino transport for $\nu_{e}$ and $\bar{\nu}_{e}$, and ($ii$) a smaller suite of 18 progenitors [72] using a ray-by-ray neutrino transport for all neutrino flavors. Both were also augmented by a simulation of core collapse of an ONeMg core star [73]. For the purposes of this study, we adopt the predictions based on the latter set, since we need accurate spectral information for all neutrino flavors.

Here, we briefly summarize the procedure and refer the reader to H18 for details. For each progenitor, the neutrino’s spectral parameters (i.e., the total energy, mean energy, and spectral shape or pinching parameter) are obtained directly from the simulations. Since simulations cover only the first $\sim 1$ second after core bounce, a time extrapolation is necessary to cover the long-term neutrino emission. Such an extrapolation is not needed for collapse to black holes, since the simulations cover the time until black hole formation. The time-summed neutrino spectrum is then fit to a pinched FD functional form as [74, 75],

$\displaystyle F(E)=\frac{(1+\langle\alpha\rangle)^{(1+\langle\alpha\rangle)}}{\Gamma(1+\langle\alpha\rangle)}\frac{E_{\nu}^{\rm{tot}}E^{\langle\alpha\rangle}}{\langle E_{\nu}\rangle^{2+\langle\alpha\rangle}}\exp\Big{[}-(1+\langle\alpha\rangle)\frac{E}{\langle E_{\nu}\rangle}\Big{]},$

(2.1)

where $E_{\nu}^{\rm{tot}}$ is the total neutrino energy emitted from the SNe, $\langle E_{\nu}\rangle$ is the mean neutrino energy and $\langle\alpha\rangle$ is a spectral shape parameter. Next, the mean neutrino spectrum $\frac{dN}{dE}$ for a population of progenitors is computed by weighting each progenitor by the initial mass function (IMF),

$\displaystyle\frac{dN}{dE}=\sum_{i}\frac{\int_{\Delta M_{i}}\psi(M)dM}{\int_{8}^{100}\psi(M)dM}F_{i}(E),$

(2.2)

where $\psi(M)=dn/dM$ is the IMF of stars and $\Delta M_{i}$ is the mass range assigned to the $i$th progenitor. We use a Salpeter IMF which is $\psi(M)\propto M^{\eta}$, where $\eta=-2.35$. The integration is performed using the solar-metallicity progenitors of Ref. [76] augmented by an ONeMg core star [77, 78] to represent the mass range of core-collapse progenitors taken to be between $(8-100)~{}M_{\odot}$.

The contribution from core collapse to black holes is uncertain due to the unknown fraction of progenitors that collapse to black holes. Theoretically, this is a complex prediction impacted by the progenitor structure, the physics of hot dense nuclear matter, as well as unsettled aspects of the SN explosion mechanism (e.g., [79, 80, 81, 82, 83]). However, recent observations suggest the fraction may be fairly large. For example, searches for the disappearance of massive stars [84] lead to a fraction of failed explosions of 4–43% percent at 90% C.L. [85, 86, 87], which is not only consistent with other observables but can also explain puzzles related to SNe and their remnants [88, 89, 90, 91, 92]. A simple procedure to include the uncertain contribution from black holes is to parametrize the fraction of stars that collapse to black holes (e.g., [93, 94]). In this work, we opt instead to adopt a fixed black hole fraction of $17$% (using the parameterization of H18, this corresponds to a “critical compactness” for progenitors to collapse to black holes of $\xi_{2.5}=0.2$) and attribute a generous uncertainty to the DSNB flux when performing our statistical forecasts.

In the second approach, we describe the mean supernova neutrino spectrum effectively by using a thermal FD distribution [64],

$\displaystyle\frac{dN}{dE}=\frac{E_{\nu}^{\rm{tot}}}{6}\frac{120}{7\pi^{4}}\frac{E^{2}}{T_{\nu}^{4}}\frac{1}{e^{E/T\nu}+1},$

(2.3)

where $T_{\nu}$ is the temperature of each neutrino flavor, $E_{\nu}^{\rm{tot}}=3\times 10^{53}$ erg is the total energy liberated, and the factor $1/6$ represents equipartition into the 6 neutrino and anti-neutrinos. The temperatures can be obtained by fitting the neutrino emissions predicted by core-collapse simulations, and generally show the following hierarchy: $T_{\nu_{e}}<T_{\bar{\nu}_{e}}<T_{\nu_{x}}$. Compilations generally cluster around $T_{\nu_{e}}\approx 3$–5 MeV, $T_{\bar{\nu}_{e}}\approx 4$–6 MeV, and $T_{\nu_{x}}\approx 4$–7 MeV [95, 96]. While this approach has often been used in the past literature, it does not take into account the progenitor dependence of the neutrino emission which is now known to be substantial [70, 97]. The benefit however is its simplicity and ease of parameterization. In the following, we therefore consider both approaches.

The DSNB differential flux is the integrated neutrino flux over redshift, appropriately weighted by the core-collapse rate, given by

$\displaystyle\frac{d\phi^{0}}{dE}=c\int_{0}^{z_{max}}R_{CC}(z)\frac{dN}{dE^{\prime}}(1+z)\Big{|}\frac{dt}{dz}\Big{|}dz,$

(2.4)

where $c$ is the speed of light, $z$ is the redshift, $\frac{dN}{dE^{\prime}}$ is the mean neutrino spectrum per core collapse, $E^{\prime}=E(1+z)$, and $|{dt}/{dz}|=H_{0}(1+z)[\Omega_{m}(1+z)^{3}+\Omega_{\Lambda}]^{1/2}$, where $H_{0}=70$ ${\rm{kms}}^{-1}{\rm{Mpc}}^{-1}$ is the Hubble parameter, while $\Omega_{m}=0.3$ and $\Omega_{\Lambda}=0.7$ are the matter and vacuum energy densities, respectively. The upper index in $\phi^{0}$ refers to fluxes which are calculated at the source (pre-oscillation). We take $z_{max}\equiv 5$, which is large enough to incorporate the majority of the DSNB flux. We model the cosmic history of the comoving core-collapse rate, $R_{CC}(z)$, by

$\displaystyle R_{CC}(z)=\dot{\rho}^{*}(z)\frac{\int_{8}^{100}\psi(M)dM}{\int_{0.1}^{100}M\psi(M)dM}$

(2.5)

where $\psi(M)$ is once again the IMF and $\rho^{*}(z)$ is the rate of the cosmic star formation. The cosmic star-formation rate is derived from the observed luminosity density of various star-forming priors, which are converted through use of appropriate conversion factors [98, 99]. There are various compilations of the cosmic star-formation rate and functional fits through subsets of the available data (see, e.g., [100, 101]). We adopt the smoothed piece-wise form of Ref. [102] (see Refs. [100, 102, 95] for numerical values). While the cosmic star-formation rate has significant systematic uncertainty, one of the most dominant—the IMF—fortunately mostly cancels by performing the product with the integration ratio in Eq. (2.5). The remaining uncertainty is dominated by the scatter between the measurements and is generally in the range of $\sim 20$% over redshifts of importance for the DSNB (see discussions in, e.g., Refs. [100, 95, 101, 96]).

Figure 1 shows the differential DSNB flux as a function of the neutrino energy for $\nu_{e}$, $\bar{\nu}_{e}$ and $\nu_{x}$ in orange, blue and purple, respectively. The predictions of H18 containing a black hole fraction of 17% are shown with solid curves. For comparison we also show the fluxes we have calculated using the thermal FD flux with the dashed curve, where we have chosen $T_{\nu_{e}}=5$ MeV, $T_{\bar{\nu}_{e}}=6$ MeV and $T_{\nu_{x}}=7$ MeV, respectively. These thermal FD fluxes are smaller than those of H18 at a few MeV but dominate above $\sim 12$ MeV, giving significantly higher neutrino rates in that region.

Once neutrinos are emitted from the neutrinospheres they will oscillate during their propagation before reaching detectors on Earth. The fluxes after oscillation are therefore a combination of the initial fluxes emitted from the neutrinospheres. During propagation through the progenitor envelope, the Mikheev-Smirnov-Wolfenstein (MSW) effect becomes important due to the coherent scattering of neutrinos on electrons. Due to the matter potential, electron neutrinos $\nu_{e}$ will exit as $\nu_{3}$ while the other states $\nu_{x}$ will exit as $\nu_{1}$ and $\nu_{2}$, in the normal mass hierarchy (NH). In the anti-neutrino sector, electron anti-neutrinos $\bar{\nu}_{e}$ exit as $\bar{\nu}_{1}$, while $\bar{\nu}_{x}$ exit as $\bar{\nu}_{2}$ and $\bar{\nu}_{3}$. Therefore, we may assume the following relations between the temperatures of the flavor and mass eigenstates:

$\displaystyle T_{\nu_{3}}=T_{\nu_{e}},\quad T_{\nu_{1}}=T_{\nu_{2}}=T_{\nu_{x}},\quad T_{\bar{\nu}_{1}}=T_{\bar{\nu}_{e}},\quad T_{\bar{\nu}_{2}}=T_{\bar{\nu}_{3}}=T_{\nu_{x}},$

(2.6)

again for NH. If the matter density along the neutrino trajectory varies slowly (the adiabatic propagation) then neutrinos and anti-neutrinos arrive at terrestrial detectors with the same mass eigenstate as they were in SNe, and we can write $\phi_{\alpha}=\sum_{i}|U_{\alpha i}|^{2}\phi^{0}_{i}$, where $U$ represents the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. This yields the fluxes at terrestrial detectors [103, 104]

	$\displaystyle\phi_{\nu_{e}}$	$\displaystyle=$	$\displaystyle\phi^{0}_{\nu_{x}},$		(2.7)
	$\displaystyle\phi_{\bar{\nu}_{e}}$	$\displaystyle=$	$\displaystyle c_{12}^{2}\phi^{0}_{\bar{\nu}_{e}}+s_{12}^{2}\phi^{0}_{\nu_{x}},$		(2.8)
	$\displaystyle\phi_{\nu_{x}}$	$\displaystyle=$	$\displaystyle\frac{1}{4}\Big{[}\phi^{0}_{\nu_{e}}+s_{12}^{2}\phi^{0}_{\bar{\nu}_{e}}+(2+c_{12}^{2})\phi^{0}_{\nu_{x}}\Big{]}$		(2.9)

for the normal hierarchy (NH) and

	$\displaystyle\phi_{\nu_{e}}$	$\displaystyle=$	$\displaystyle s_{12}^{2}\phi^{0}_{\nu_{e}}+c_{12}^{2}\phi^{0}_{\nu_{x}},$		(2.10)
	$\displaystyle\phi_{\bar{\nu}_{e}}$	$\displaystyle=$	$\displaystyle\phi^{0}_{\nu_{x}},$		(2.11)
	$\displaystyle\phi_{\nu_{x}}$	$\displaystyle=$	$\displaystyle\frac{1}{4}\Big{[}c_{12}^{2}\phi^{0}_{\nu_{e}}+\phi^{0}_{\bar{\nu}_{e}}+(2+s_{12}^{2})\phi^{0}_{\nu_{x}}\Big{]}$		(2.12)

for the inverted hierarchy (IH), where $\phi^{0}_{\nu_{\alpha}}$ is the initial flux of (anti)neutrino flavor $\nu_{\alpha}$, $c_{12}^{2}=1-s_{12}^{2}$ and $s_{12}^{2}=0.310$ is the solar mixing angle. In this paper we assume the NH for the mass ordering and show the results for this case. The results for the IH are very similar.

Collective flavour oscillations also occur within a few hundred kilometers from the core due to $\nu-\nu$ coherent scattering [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. While these collective oscillations are currently the focus of much research, it is likely to be time dependent, meaning they will undergo some level of averaging out for the DSNB which detects the time-integrated neutrino spectrum. Also, during the late cooling phase of the protoneutron star, when some half of the neutrino emission occurs, the difference in spectra between flavors is much reduced, implying oscillation effects will not be pronounced. It has been predicted that collective neutrino oscillations would be subdominant compared to MSW and have an effect of less than $\sim 10\%$ [105]. However, the results after collective neutrino mixing, including its time and energy dependence, remain highly uncertain. Therefore, we do not include the uncertain collective neutrino oscillation effects in our analysis. Instead, we will estimate how well deviations from MSW can be probed with future datasets.

In principle massive neutrinos can decay to the lighter states. Within the SM this requires the lifetime to be larger than the age of the universe. However, neutrinos can decay faster if there are interactions beyond the standard model. We follow the decay mechanism described in Ref. [106]. Taking the NH to be the correct one, the third mass eigenstate can decay to $\nu_{1}$. Hence, this can significantly change the flavor content of the DSNB flux arriving to the earth. We assume that neutrinos are Majorana states and they can interact with a massless scalar $\phi$ within the following Lagrangian:

$\displaystyle\mathcal{L}\supset\frac{g}{2}{\nu}_{i}\nu_{j}\phi+{\rm{h.c.}}.$

(2.13)

We are interested in the case where $\nu_{3}$ with mass $m_{3}$ is produced as a left handed particle, where it can decay to $\nu_{1}$ as a left handed (helicity conserving) or right handed (helicity flipping) daughter particle. In the lab frame the decay width of these two processes are the same:

$\displaystyle\Gamma(E_{3})=\frac{g^{2}m_{3}^{2}}{32\pi E_{3}}=\frac{1}{E_{3}}\frac{m_{3}}{\tau_{3}},$

(2.14)

where $\tau_{3}$ is the lifetime. We can then calculate the flux of the mass eigenstates at the Earth as:

	$\displaystyle{\phi_{\nu_{3}}}(E)$	$\displaystyle=$	$\displaystyle c\int_{0}^{z_{max}}R_{CC}(z)\frac{dN_{\nu_{e}}}{dE^{\prime}}(1+z)\Big{|}\frac{dt}{dz}\Big{|}e^{-\Gamma(E)\zeta(z)}dz,$		(2.15)
	$\displaystyle{\phi_{\bar{\nu}_{3}}}(E)$	$\displaystyle=$	$\displaystyle c\int_{0}^{z_{max}}R_{CC}(z)\frac{dN_{\nu_{x}}}{dE^{\prime}}(1+z)\Big{|}\frac{dt}{dz}\Big{|}e^{-\Gamma(E)\zeta(z)}dz,$		(2.16)
	$\displaystyle{\phi_{\nu_{1}}}(E)$	$\displaystyle=$	$\displaystyle{\phi^{0}_{\nu_{x}}}(E)+\int_{0}^{z_{max}}dz\frac{1}{H(z)}\int_{{E^{\prime}}}^{\infty}dE_{3}\Gamma(E_{3})\Big{[}\frac{d\phi_{\bar{\nu}_{3}}}{dE_{3}}\psi_{h.c.}(E_{3},{E^{\prime}})+\frac{d\phi_{\nu_{3}}}{dE_{3}}\psi_{h.f.}(E_{3},{E^{\prime}})\Big{]},$
	$\displaystyle{\phi_{\bar{\nu}_{1}}}(E)$	$\displaystyle=$	$\displaystyle{\phi^{0}_{\bar{\nu}_{e}}}(E)+\int_{0}^{z_{max}}dz\frac{1}{H(z)}\int_{{E^{\prime}}}^{\infty}dE_{3}\Gamma(E_{3})\Big{[}\frac{d\phi_{\nu_{3}}}{dE_{3}}\psi_{h.c.}(E_{3},{E^{\prime}})+\frac{d\phi_{\bar{\nu}_{3}}}{dE_{3}}\psi_{h.f.}(E_{3},{E^{\prime}})\Big{]},$

where $\zeta(z)=\int_{0}^{z}dz^{\prime}H^{-1}(z^{\prime})(1+z^{\prime})^{-2}$, we have $E^{\prime}=E(1+z)$ and ${\phi^{0}_{\nu_{\alpha}}}$ expressions are given in Eq. (2.4). The other mass eigenstates are unaffected by the decay, e.g., ${\phi_{\nu_{2}}}(E)={\phi^{0}_{\nu_{x}}}(E)$. Finally, the energy distribution of the daughter particle for the helicity conserving and helicity flipping cases are given by

$\displaystyle\psi_{h.c.}(E_{3},E_{1})=\frac{2E_{1}}{E_{3}^{2}},\quad\psi_{h.f.}(E_{3},E_{1})=\frac{2}{E_{3}}\left(1-\frac{E_{1}}{E_{3}}\right).$

(2.17)

By inclusion of the neutrino decay the flux of $\nu_{\alpha}$ at the earth changes dramatically. In this case the spectrum of different flavors at the earth for the NH are given by:

	$\displaystyle\phi_{\nu_{e}}^{\rm{decay}}$	$\displaystyle=$	$\displaystyle c_{12}^{2}\phi_{\nu_{1}}+s_{12}^{2}\phi_{\nu_{2}},$
	$\displaystyle\phi_{\bar{\nu}_{e}^{\rm{decay}}}$	$\displaystyle=$	$\displaystyle c_{12}^{2}\phi_{\bar{\nu}_{1}}+s_{12}^{2}\phi_{\bar{\nu}_{2}},$		(2.18)
	$\displaystyle\phi_{\nu_{x}}^{\rm{decay}}$	$\displaystyle=$	$\displaystyle\frac{1}{4}\Big{[}s_{12}^{2}(\phi_{\nu_{1}}+\phi_{\bar{\nu}_{1}})+c_{12}^{2}(\phi_{\nu_{2}}+\phi_{\bar{\nu}_{2}})+\phi_{\nu_{3}}+\phi_{\bar{\nu}_{3}}\Big{]},$

where similar expressions can be obtained for the IH case.

One can be sensitive to neutrino decay of $\tau/m\sim 10^{5}$ s/eV from the neutronization burst of galactic SN [107]. The DSNB however, due to the further distances, can be sensitive to the neutrino lifetimes of $\tau/m\sim 10^{10}$ s/eV. It was shown in Ref. [106] that using the data of Hyper-K doped with Gd as well as THEIA one can reach a $3\sigma$ sensitivity of $\tau_{3}/m_{3}\lesssim 5\times 10^{9}$ after 20 years of data taking. Ref. [108] has also studied DSNB neutrino decay, but they have found one order of magnitude smaller sensitivity.

Using the LAr detector of DUNE, we will be able to detect the electron component of the DSNB flux via the CC interaction $\nu_{e}+Ar\rightarrow e^{-}+K^{+}$ [69]. The signature of this process is observing an electron accompanied by the decay products of the excited $K^{*}$. The DUNE far detector which has a fiducial mass of at least $40$ kt will be sensitive to DSNB neutrinos from around 5 MeV to a few tens of MeV. The main sources of background at this energy are the solar hep neutrinos which have an endpoint of $18.8$ MeV, and the atmospheric flux of electron neutrinos which rises in an energy around $40$ MeV. Since the exact details of the DSNB backgrounds at DUNE are still under investigation, we assume DUNE will have a background similar to ICARUS [94, 109]. We calculate the events in the range of $19-31$ MeV to get rid of these backgrounds. We take the CC cross section of neutrinos on Argon from Ref. [69] which we have shown in Fig. 2.

The number of expected DSNB events at the DUNE far detector is calculated by

$\displaystyle N_{i}^{\rm{DUNE}}={n}_{{\rm{DUNE}}}\times T\times\epsilon\int_{i}\frac{d\phi_{\nu_{e}}}{dE}\sigma(E)dE$

(3.1)

where ${n}_{{\rm{DUNE}}}$ is the total number of Argon targets at the detector mass and $T$ is the total lifetime of the experiment. The fiducial mass of the far detector is $40$ kt and we consider $20$ years of data taking. Therefore, we have

$\displaystyle{n}_{{\rm{DUNE}}}=6.02\times 10^{32}.$

(3.2)

Using this factor and assuming a detector efficiency of $\epsilon=86\%$ [69] we find a total of $15$ and $38$ $\nu_{e}$ events with the H18 and thermal FD fluxes, respectively. Figure 3 shows the expected number of events as a function of the neutrino energy for the two different neutrino emission models. For the thermal FD flux we have chosen $T_{\nu_{e}}=5$ MeV, $T_{\bar{\nu}_{e}}=6$ MeV and $T_{\nu_{x}}=7$ MeV, respectively. As can be seen, these temperatures result in much higher expected events than H18, even beyond the variability of the flux within its possible uncertainty. This is consistent with our choice of temperature which are on the large end of those predicted by simulations. For comparison, in the right panel we show the event rates for the decay scenario assuming different values for $\tau_{3}/m_{3}$, all for the H18 model. The shorter the lifetime, the faster the decay, and hence the higher the $\nu_{1}$ flux and higher the $\nu_{e}$ events on Earth; indeed, this is what we see in Fig. 3.

The Hyper-K experiment in Japan, which is the successor of the Super-K experiment, will be able to detect SN neutrinos down to $\sim 3$ MeV [67], although the threshold for the DSNB will be higher owing to backgrounds. It consists of two modules, each with a fiducial mass of $187$ kt. With a combined volume of 374 kt, Hyper-K will be sixteen times larger than Super-K, resulting in great sensitivity to the DSNB. Unlike the DUNE which will be sensitive to electron neutrinos, Hyper-K will detect $\bar{\nu}_{e}$ neutrinos through IBD scattering $\bar{\nu}_{e}+p\rightarrow e^{+}+n$. We get the IBD cross section from Ref. [110, 111] (see Fig. 2).

If Hyper-K is enriched with gadolinium (Gd) its sensitivity to DSNB events will be significantly improved due to the significantly reduced backgrounds. The CC atmospheric neutrinos and the lithium-9 spallation are subdominant backgrounds at energies around tens of MeV. On the other hand, the neutral current (NC) atmospheric neutrinos which are induced due to the $\gamma$-rays produced by the NC quasi elastic scattering, are non-negligible in this range. If we assume full tagging can be reached, we can ignore this background [94]. In practice, if the tagging efficiency will be comparable to Super-K, it will be in the 90% range. The invisible muons will be similarly suppressed by neutron tagging but given its much higher rate it will remain as the main source of background, which increases with energy. Therefore, we consider an upper bound of $E<24$ MeV to suppress the invisible muons. The modeling of backgrounds at Hyper-K enriched with Gd is taken from [94, 67]. There are other sources of background which will not be reduced by Gd, but a cut on the energy will avoid them. The dominant background at $E<10$ MeV is the $\bar{\nu}_{e}$ from reactor neutrinos, which is however suppressed at $E>10$ MeV. Therefore, we consider the range $10$ MeV $<E<24$ MeV.

The number of expected DSNB events through the IBD cross section at the Hyper-K detector at each bin of the positron energy is calculated by

$\displaystyle N_{i}^{\rm{HK-IBD}}(E^{+})={{n}}^{\rm{IBD}}_{{\rm{HK}}}\times T\times\epsilon\int_{i}\frac{d\phi_{\bar{\nu}_{e}}}{dE}\sigma(E^{+})dE^{+},$

(3.3)

where we have used $E=E^{+}+\Delta$ with $\Delta=1.3$ MeV in the flux to convert the neutrino energy to the positron energy. The total number of IBD targets are $n^{\rm{IBD}}_{\rm{HK}}=2.50\times 10^{34}$. Considering a tagging efficiency of 90% and event selection efficiency of 74%, which is slightly lower than $\sim 90$% efficiency of the latest Super-K DSNB analysis [112], we adopt an overall detector efficiency of $67\%$ [67]. The Hyper-K experiment will detect a total of $444$ (747) $\bar{\nu}_{e}$ events using the H18 (thermal FD) flux. Figure 4 shows the expected number of DSNB events as a function of the positron energy compared to relevant backgrounds for Hyper-K enriched with Gd. In the right panel, we also show how different values of $\tau_{3}/m_{3}$ can change the expected DSNB events under the decay scenario. Similar to $\nu_{e}$ events at DUNE, a faster neutrino decay can result in higher $\bar{\nu}_{e}$ events at Hyper-K.

Hyper-K can also be sensitive to the sum of all neutrino flavors due to the elastic scattering of neutrinos on electrons: $\nu+e\rightarrow\nu+e$, where the observable will be a forward going electron. Although the cross section will be almost three orders of magnitudes suppressed compared to IBD (see Fig. 2), in this way Hyper-K can be sensitive to $\nu_{e}$ and $\nu_{x}$ DSNB as well. The differential $\nu-e$ scattering cross section is given by [113],

$\displaystyle\frac{d\sigma}{dE_{R}}=\frac{2G_{F}^{2}m_{e}}{\pi}\left\{g_{1}^{2}+g_{2}^{2}\left(1-\frac{E_{R}}{E}\right)^{2}-g_{1}g_{2}\frac{m_{e}E_{R}}{E^{2}}\right\},$

(3.4)

where $G_{F}$ is the Fermi constant, $E$ is the energy of the incoming neutrino, and $m_{e}$ is the electron mass and $E_{R}$ is its recoil kinetic energy. The couplings $g_{1}$ and $g_{2}$ for different neutrino flavors depend on the weak angle $\sin^{2}\theta_{w}$ and are listed in Table 1 of [113].

The number of $\nu-e$ DSNB events at Hyper-K as a function of the electron recoil energy is given by

$\displaystyle N_{i}^{\rm{HK-\nu-e}}(E_{R})=\sum_{\alpha=\nu_{e},\bar{\nu}_{e},\nu_{x}}{{n}}^{\rm{\nu-e}}_{{\rm{HK}}}\times T\times\epsilon\int_{i}dE_{R}\int_{E_{R}}^{E_{R}^{\rm{max}}}dE\frac{d\phi_{\nu_{\alpha}}}{dE}\frac{d\sigma_{\nu_{\alpha}}}{dE_{R}},$

(3.5)

where the number of electron targets is ${{n}}^{\nu-e}_{{\rm{HK}}}=1.25\times 10^{35}$. Considering the recoil energy larger than 10 MeV we can get rid of the reactor neutrinos, and the solar angular cut can get rid of the solar neutrinos as the main sources of background. The invisible muons are however unavoidable, for which we show in Fig. 5 the results from Ref. [65]. Note however that this is the reduced rate considering coincidence tagging which applies to IBD events. For $\nu-e$ events the coincidence cut does not apply, and we have to remove the background reduction factor of five. Assuming a detection efficiency of $100\%$ above the threshold, we find that Hyper-K will be able to observe $17~{}(31)$ $\nu-e$ events using the H18 (thermal FD) flux, in the range of $10$ MeV$<E_{R}<20$ MeV, but this is small compared to the invisible muon events. Thus, Hyper-K will not be able to claim standalone detection, but would constrain extremely high fluxes of $\nu_{e}$ and $\nu_{x}$ when combined with its IBD channel (and when further combined with DUNE’s measurement of $\nu_{e}$, constrain high $\nu_{x}$ fluxes). We show in Fig. 5 the expected number of $\nu-e$ DSNB events at Hyper-K as a function of the electron recoil energy (left panel), as well as the comparison between the MSW with and without decays (right panel).

The JUNO detector in China has a fiducial volume of 17 kt, made of linear alkylbenzene liquid scintillator (C₆H₅C₁₂H₂₅) [68]. JUNO will be able to detect $\bar{\nu}_{e}$ events thought IBD interaction. We calculate the number of IBD events using Eq. (3.3) with ${n}^{\rm{IBD}}_{{\rm{JUNO}}}=1.21\times 10^{33}$ and a detection efficiency of $50\%$. The main sources of background here are the reactor electron anti-neutrinos, which we can get rid of with a cut of $E_{e^{+}}>10$ MeV, and the atmospheric $\bar{\nu}_{e}$ events as well as the NC atmospheric, which we take from Refs. [68, 94]. We show in Fig. 6 the expected number of IBD events at JUNO as a funciton of the positron energy as well as the backgrounds. In total we expect JUNO will be able to detect $19~{}(30)$ IBD events in 20 years of data taking, using the H18 (thermal FD) flux, in the range of $10$ MeV$<E_{e^{+}}<32$ MeV.

In addition to the IBD detection of $\bar{\nu}_{e}$, JUNO can also detect neutrino events with the neutrino-proton elastic scattering which is sensitive to the sum of all neutrino flavors. In this case the differential cross section is given by [114]

	$\displaystyle\frac{d\sigma}{dT_{\rm ke}}$	$\displaystyle=$	$\displaystyle\frac{G_{F}^{2}m_{p}}{\pi}\left\{c_{v}^{2}\left(1-\frac{m_{p}T_{\rm ke}}{2E^{2}}\right)^{2}+c_{a}^{2}\left(1+\frac{m_{p}T_{\rm ke}}{2E^{2}}\right)^{2}\right\}$		(3.6)
		$\displaystyle=$	$\displaystyle 4.83\times 10^{-42}\frac{{\rm{cm}}^{2}}{{\rm{MeV}}}\left(1+466\frac{T_{\rm ke}}{E^{2}}\right),$

where $m_{p}$ is the proton mass, $T_{\rm ke}$ is its kinetic energy, $c_{v}=0.04$ and $c_{a}=1.27/2$. The number of $\nu-p$ events at JUNO is given by

$\displaystyle N_{i}^{\rm{JUNO-\nu-p}}(T)=\sum_{\alpha=\nu_{e},\bar{\nu}_{e},\nu_{x}}{{n}}^{\rm{\nu-p}}_{{\rm{JUNO}}}\times T\times\epsilon\int_{i}dT_{\rm ke}\int_{\sqrt{m_{p}T_{\rm ke}/2}}^{T_{\rm ke}^{\rm{max}}}dE\frac{d\phi_{\nu_{\alpha}}}{dE}\frac{d\sigma}{dT_{\rm ke}},$

(3.7)

where ${{n}}^{\rm{\nu-p}}_{{\rm{JUNO}}}=1.21\times 10^{33}$ (the same as number of IBD targets) and we assume a detection efficiency of $100\%$. However, this channel is a big challenge; the main difficulty is that unlike IBD where a coincidence signature suppresses backgrounds, the proton scattering appears as a single flash of light with a much higher single-event backgrounds rate [68]. The relevant backgrounds arise from a variety of radioactive decays as well as cosmogenic backgrounds and solar neutrinos. The DSNB background in the $\nu-p$ channel of JUNO has not been studied in depth yet, and it remains to be determined how well factors such as material distillation, fiducial cuts, and pulse shape discrimination will mitigate these backgrounds [115]. As an optimistic scenario, we therefore follow the Galactic search where scintillator detectors are expected to significantly suppress background once a cut of $T_{\rm ke}>0.2$ MeV on the proton kinetic energy is imposed [114]. Hence, we assume this will be also the case at JUNO and calculate the events at the range $0.2-1.5$ MeV. We expect that JUNO will collect a total of $53~{}(98)$ $\nu-p$ events using the H18 (thermal FD) model. We show in Fig. 7 the distribution of events for total number of $\nu-p$ events. We will first show results without this JUNO channel, and later including this channel.