Model Independent Approach of the JUNO ⁸B Solar Neutrino Program

Electron neutrino fluxes are produced from thermal nuclear fusion reactions in the solar core, either through the proton-proton ($pp$) chain or the Carbon-Nitrogen-Oxygen (CNO) cycle. According to their production reactions, the solar neutrino species can be categorized as $pp$, ⁷Be, $pep$, ⁸B, $hep$ neutrinos of the $pp$ chain, and ¹³N, ¹⁵O, and ¹⁷F neutrinos of the CNO cycle. Before reaching the detector, solar neutrinos undergo the flavor conversion inside the Sun and the Earth during their propagation. Solar neutrino measurements have a long history starting with the measurements done by the Homestake experiment (Davis et al., 1968). Many measurements, such as Homestake (Davis et al., 1968), Kamiokande (Hirata et al., 1989), GALLEX/GNO (Anselmann et al., 1993; Altmann et al., 2000), SAGE (Abazov et al., 1991), and Super-Kamiokande (SK) (Fukuda et al., 1998, 2001), had observed the solar neutrino deficit problem: that is the amount of observed neutrinos originating from the Sun was much less than that expected from the Standard Solar Model (SSM). Subsequently, the Sudbury Neutrino Observatory (SNO) provided the first model-independent evidence of solar neutrino flavor conversion using three distinct neutrino interaction channels in heavy water (Chen, 1985; Ahmad et al., 2001, 2002; Ahmed et al., 2004; Aharmim et al., 2008, 2013a, 2013b). These reactions are the $\nu_{e}$ sensitive charged-current (CC) interaction, all flavor sensitive neutral-current (NC) interaction on Deuterium, and the elastic scattering (ES) interaction on electrons from all neutrino flavors with different cross sections.

Solar neutrino observations rely on the SSM flux predictions, the neutrino oscillation parameters and solar density model that determine the flavor conversion (Wolfenstein, 1978; Mikheev & Smirnov, 1985; Zyla et al., 2020). Thus although SK (Abe et al., 2016; Renshaw et al., 2014) and Borexino (Bellini et al., 2014; Agostini et al., 2020) experiments have made precision measurements on the ⁸B neutrinos via the ES interaction, the evaluation of the total amount of neutrinos produced inside the Sun relies on the input of solar neutrino oscillations (Zyla et al., 2020). The present most precise ⁸B neutrino flux is determined by SNO with the 1$\sigma$ confidence level uncertainly of around 3.8% (Ahmad et al., 2002; Ahmed et al., 2004; Aharmim et al., 2008, 2013a, 2013b), and it is the only existing model independent flux measurement. Therefore, a second independent measurement of the total ⁸B neutrino flux with the NC channel (Arafune et al., 1989; Ianni et al., 2005) would be important to answer relevant questions in the field of solar physics. For example, there is the solar abundance problem, in which the SSM based on the solar composition with a higher value of metallicity is inconsistent with the helioseismological measurements (Vinyoles et al., 2017). Note that a recent solar model is able to resolve the discord between the helioseismological and photospheric measurements (Magg et al., 2022), but lively discussions on this topic are still on-going (Buldgen et al., 2023; Yang, 2022).

In contrast, the neutrino oscillation parameters $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ have reached the 1$\sigma$ confidence level uncertainty of around 5% and 15% respectively, from the current global solar neutrino data (Esteban et al., 2020). The mixing angle $\sin^{2}\theta_{12}$ is extracted from the comparison of the observed fluxes of $pp$, ⁷Be, and ⁸B solar neutrinos to their respective total fluxes from the SSM. And the mass squared difference $\Delta m^{2}_{21}$ is measured from both the vacuum-matter transition of the ⁸B neutrino oscillations and the size of the day-night asymmetry. A direct comparison of oscillation parameters from the solar neutrino and reactor antineutrino oscillations is an unique probe of new physics beyond the Standard Model of particle physics. It would be excellent to have a new measurement of solar neutrino oscillations with high precision in this respect. This has triggered a variety of interesting discussions on the prospects of future large neutrino detectors (Capozzi et al., 2019; Abusleme et al., 2021a; Abe et al., 2018; Beacom et al., 2017).

The Jianmen Underground Neutrino Observatory (JUNO) is a liquid scintillator (LS) detector of 20 kton, which is located in South China and will start data taking by 2024. As a multiple-purpose neutrino experiment, JUNO is unique for the solar neutrino detection because of its large target mass, excellent energy resolution, and expected low background levels. With the analysis threshold cut of around 2 MeV for the recoiled electron energies in the ES channel, JUNO can make a high-statistics measurement of the flux and spectral shape of ⁸B solar neutrinos and will be able to extract the neutrino oscillation parameters $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ (Abusleme et al., 2021a). In addition to the high statistics measurement in the ES channel, the presence of a large mass of the ¹³C nuclei ($\sim$0.2 kt) makes it feasible to detect ⁸B solar neutrinos via CC and NC interactions on ¹³C. By combining the CC, NC and ES channels, we are able to perform a model independent measurement of the ⁸B solar neutrino flux and oscillation parameters $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$, which will add a unique contribution to the global solar neutrino program.

The paper is organized as follows. We illustrate the typical signatures of the CC and NC interactions of ⁸B solar neutrinos, and evaluate the corresponding backgrounds in the JUNO detector in Sec. 2. In Sec. 3, the physics potential of detecting the ⁸B solar neutrinos with different combinations of the CC, NC, and ES channels are presented, and the sensitivity to the ⁸B solar neutrino flux, $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ is reported. The concluding remarks of this study are presented in Sec. 4.

The JUNO experiment is building the world largest LS detector with the total target mass of 20 kt, in which the mass fraction of Carbon is 88%. Given that the natural abundance of ¹³C is 1.1%, the total mass of ¹³C reaches 193.6 ton, which is similar to the total Deuterium mass of 200 ton for the SNO detector. Considering the preferable cross sections of ¹³C at the solar neutrino energies (Fukugita et al., 1988; Suzuki et al., 2012, 2019), the CC and NC solar neutrino rates on ¹³C will be rather sizable in the JUNO detector.

In Table 1, we present the typical CC, NC and ES detection channels for ⁸B solar neutrinos in the LS medium. For each interaction channel, the reaction threshold is provided, together with the typical experimental signatures, and the expected event numbers for 10 years of data taking before event selection cuts. The spin and parity of the daughter nuclei at the ground (gnd) or excited state, denoted by the corresponding excited energies, are also provided. The unoscillated ⁸B solar neutrino $\nu_{e}$ flux (5.25$\times$10⁶ /cm²/s) is taken from the final result of SNO for this estimation (Aharmim et al., 2013b), and the spectrum is taken from  Bahcall et al. (1996); Bahcall (1997). The cross sections for these exclusive channels are taken from the calculation in  Fukugita et al. (1988); Suzuki et al. (2012, 2019), in which the uncertainties at the level of a few percent are considered to be achievable. Note that the standard Mikheev-Smirnov-Wolfenstein (MSW) effect of solar neutrino oscillations (Wolfenstein, 1978; Mikheev & Smirnov, 1985) and the neutrino oscillation parameters from  Zyla et al. (2020) are used in the signal calculations of the CC, NC, and ES channels.

There are no interactions on the ¹²C nuclei for most solar neutrinos because of the high energy threshold. Thus for the CC channel, we are left with the following two exclusive interactions:

where the final ${}^{13}{\rm N}$ is in the ground state and excited ${}^{13}{\rm N}\,({3}/{2}^{-};3.502\,{\rm MeV})$ state respectively. For the first reaction channel, the ground state of ${}^{13}{\rm N}$ undergoes a delayed $\beta^{+}$ decay ($Q$ = 2.2 MeV) with a lifetime of 863 s. The distinct signature for this channel is a coincidence of the prompt electron and delayed positron with stringent time, distance, and energy requirements. The expected number of events for ⁸B solar neutrinos in this coincidence channel is 3929 for 10 years of data taking. On the other hand, although the channel with an excited ${}^{13}{\rm N}\,({3}/{2}^{-};3.502\,{\rm MeV})$ has a comparable cross section as the ground-state channel (Suzuki et al., 2012), the corresponding signature after quenching is a single event since the deexcitation of ${}^{13}{\rm N}\,({3}/{2}^{-};3.502\,{\rm MeV})$ is dominated by a proton knockout, and thus cannot be distinguished from the recoiled electron of the ES channel and the single $\gamma$ of the NC channel on an event-by-event basis. Therefore, in the coincidence event category we focus on the CC channel with the ground state ${}^{13}{\rm N}$ and consider the channel with the excited ${}^{13}{\rm N}\,({3}/{2}^{-};3.502\,{\rm MeV})$ as a component of the total singles spectrum as illustrated in Fig. 2.

Among the five listed NC channels, the only one with a coincidence signature is the interaction of $\nu_{x}+^{13}{\rm C}\rightarrow\nu_{x}+n+^{12}{\rm C}$, with a prompt $\gamma$ energy of 4.44 MeV from ¹²C de-excitation and the delayed neutron capture. However, given that the background from the inverse beta decay interactions of reactor antineutrinos are overwhelming, where the signal to background ratio is at the level of 10^-4, and thus the event rate of this channel is unobservable. In this work we focus on the NC channels with the signature of single $\gamma$ deexcitation, among which the NC interaction with the ¹³C de-excited energy of 3.685 MeV:

is the dominant interaction channel and will be used to determine the ⁸B solar neutrino flux via the NC interaction.

Finally, we also consider the ES interaction channel on the electron,

where the signature is a single recoiled electron (Abusleme et al., 2021a). Using all the three channels of CC, NC, ES interactions, we are able to make a model independent measurement of the ⁸B solar neutrino flux, $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ with JUNO, which is useful to disentangle the solar dynamics and the neutrino oscillation effects. This measurement is expected to be the only model independent study after the SNO experiment (Ahmad et al., 2001, 2002; Ahmed et al., 2004; Aharmim et al., 2008).

To summarize, in this work we are going to employ the following three interaction channels for a model independent approach of the JUNO ⁸B solar neutrino program: i) the CC detection channel is sensitive to the $\nu_{e}$ component of solar neutrinos, ii) the NC channel is sensitive to all active neutrino flavors ($\nu_{e}$, $\nu_{\mu}$, $\nu_{\tau}$) with identical cross sections, iii) the ES channel is also sensitive to all active flavors, but with a preferred cross section for the $\nu_{e}$ flux [i.e., $\sigma(\nu_{\mu/\tau})\simeq 0.17\,\sigma(\nu_{e})$].

In this section, we study the physical potential for the model independent measurement of ⁸B solar neutrinos using CC, NC, and ES channels. Based on the typical event signatures, the full solar neutrino data can be separated into the correlated and single event data sets. As discussed in the previous section, all the three interaction channels from ⁸B solar neutrinos would contribute to the single event data set, while the correlated data set includes events from both the CC channel and the accidental coincidence of the ES channel.

In this analysis, we consider the following systematic uncertainties. First, the uncertainty of detection efficiency is estimated to be 2% (Abusleme et al., 2021a), which is fully correlated for the the signal and background components of each data set, but uncorrelated between the coincidence and single event data samples. Second, the current uncertainty of the ${}^{13}{\rm C}$ cross sections from the model calculation is at the level of several percents (Fukugita et al., 1988; Suzuki et al., 2012, 2019), but the precision could be reduced to 1% or better with large-scale modern shell-model calculations (Barrett et al., 2013). Therefore the uncertainties for the ${}^{13}{\rm C}$ CC and NC interaction are taken as 1% for the current study. A 0.5% cross section uncertainty is used for the ES channel (Tomalak & Hill, 2020). Third, the shape uncertainty of ⁸B solar neutrinos is taken from  Bahcall et al. (1996); Bahcall (1997), and the uncertainties for the radioactive and muon-induced backgrounds are the same as those in  Abusleme et al. (2021a), namely, 1% for ²³⁸U, ²³²Th and ¹²B decays, 3% for ⁸Li and ⁶He decays, and 10% for ¹⁰C and ¹¹Be decays. A 2% uncertainty is used for the single event from the reactor antineutrino ES interaction. In this work we treat the ⁸B solar neutrino flux as a free parameter since we are performing a model independent measurement. Only in the scenario of combining with the SNO flux measurement, an uncertainty of 3.8% is used as an informative prior.

The standard Poisson-type $\chi^{2}$ method using the Asimov data set (Zyla et al., 2020) is employed to estimate the sensitivity to measure the ⁸B solar neutrino flux and the oscillation parameters sin${}^{2}\theta_{12}$ and $\Delta m^{2}_{21}$, where different pull parameters are included in the $\chi^{2}$ function to account for the systematic uncertainties described in this section. More technical details on the construction of the $\chi^{2}$ function are provided in the Appendix. In order to identify the contribution of each interaction channel, we divide the whole data sets into the correlated events, the single events within [3.5, 4.1] MeV, and the single events outside [3.5, 4.1] MeV, which correspond to the ${\rm CC}$, ${\rm NC}$, and ${\rm ES}$ measurements respectively.

We illustrate in Figs 4-6 the two dimensional allowed ranges and the marginalized one dimensional curves on the sensitivity of the ⁸B neutrino flux, $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$, of which Fig. 4 is for the comparison of the ES and ES+NC measurements, Fig. 5 for the comparison the ES+NC and ES+NC+CC measurements, and Fig. 6 for the comparison of the JUNO and JUNO + SNO flux measurements. In addition, a summary of relative uncertainties on the ⁸B neutrino flux, $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ from the model independent approach is provided in Fig. 7. Several important observations and comments are presented as follows.

• •

  The NC measurement is accomplished based on the single events within [3.5, 4.1] MeV, where the background events are from the singles of ES and CC interactions of ⁸B solar neutrinos, together with the natural radioactivity and muon-induced unstable isotopes. The standard MSW effect of solar neutrino oscillations is used in the calculation of ES and CC interactions and the oscillation parameters sin${}^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ are marginalized. The ⁸B solar neutrino flux can be obtained with an accuracy of 10.6% with the NC measurement, which is comparable to the level of 8.6% from the NC measurement of the SNO Phase-III data (Aharmim et al., 2013a).

  

• •

  The ES measurement is based on the single events outside the energy range of [3.5, 4.1] MeV, in which the dominant background is from the natural radioactivity and muon-induced unstable isotopes, which are summarized in Fig. 2 and more details can be found in  Abusleme et al. (2021a). In the model independent approach of the ES measurement, the ⁸B neutrino flux and two oscillation parameters sin${}^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ are simultaneously constrained, where the relative uncertainties are derived as ${}^{+11\%}_{-8\%}$, ${}^{+17\%}_{-17\%}$, and ${}^{+45\%}_{-25\%}$, respectively. The uncertainties of sin${}^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ are larger than those obtained in  Abusleme et al. (2021a) by including the 3.8% SNO flux measurement because of the strong correlation between the flux and oscillation parameters in the model independent approach. When adding the JUNO NC measurement, the accuracy of the ⁸B neutrino flux can be improved to the level of ${}^{+6.0\%}_{-5.5\%}$, and the uncertainties of sin${}^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ are also improved to ${}^{+10\%}_{-10\%}$, and ${}^{+31\%}_{-21\%}$ respectively.

  

• •

  The CC measurement with the correlated events itself cannot simultaneously determine the ⁸B neutrino flux and oscillation parameters because of the high visible energy threshold. However, by combining the CC measurement with the single events of the NC+ES channels, it will help to break the correlation and possible degeneracy among different parameters, where the accuracy of the ⁸B neutrino flux can be further improved to 5%, while those of $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ are ${}^{+9\%}_{-8\%}$, and ${}^{+25\%}_{-17\%}$ respectively.

• •

  The expected 5% precision of the ⁸B neutrino flux obtained with all three detection channels is much better than that of 11.6% from the latest prediction of the SSM (Vinyoles et al., 2017). This will be the only model independent measurement after SNO (Aharmim et al., 2013b). In addition, the uncertainties of $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ from the ⁸B neutrino measurement at JUNO are at the levels of ${}^{+9\%}_{-8\%}$ and ${}^{+25\%}_{-17\%}$ respectively, which is comparable to the levels of ${}^{+5\%}_{-5\%}$, and ${}^{+20\%}_{-11\%}$ from the latest results of combined SK and SNO solar neutrino data (Nakajima, 2020). Considering that the reactor antineutrino measurement of JUNO will obtain sub-percent levels of $\sin^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ in the near future (Abusleme et al., 2022b), measurements of these parameters from future solar neutrino data would be important to test the CPT symmetry of fundamental physics and resolve the possible discrepancy between the neutrino and antineutrino oscillation channels.

  

• •

  Within the spirit of the model independent approach, one can also include the 3.8% ⁸B neutrino flux measurement of SNO as an informative prior, where even better precision levels of the flux and oscillation parameters can be achieved. In this scenario, the expected accuracy of the ⁸B solar neutrino flux would reach the level of 3%, and sin${}^{2}\theta_{12}$ and $\Delta m^{2}_{21}$ can be constrained with the precision of ${}^{+7.5\%}_{-6.5\%}$, and ${}^{+19\%}_{-15\%}$ respectively. These measurements are comparable to those from the current global solar neutrino data and would provide unique information to the future solar neutrino program.

• •

  It is noteworthy that the signal event statistics, detection efficiency and cross section uncertainties are the most crucial factors that affect the detection potential of the CC and NC detection channels. If the cross section uncertainties are 10%, instead of 1% assumed in this work, the uncertainty of the ⁸B neutrino flux will become ${}^{+6\%}_{-6\%}$.

• •

  In the CC detection channel, the observed energy of the prompt electron is directly related to the incoming neutrino energy, making it crucial to lower the prompt energy threshold to investigate the predicted increase in the solar neutrino survival probability at lower energies. For this analysis, we set a conservative prompt energy threshold at 5 MeV to optimize the trade-off between the signal detection efficiencies and background contamination. Regarding the accidental background, radioactivity is the primary source of the prompt signal below 3.5 MeV, where stringent background control measures are essential, as outlined in  Abusleme et al. (2021b). Conversely, solar neutrino ES events become the leading prompt signal above 3.5 MeV. For the prompt energy range from 3.5 to 5 MeV, the cosmogenic correlated background is significantly higher than that in the region above 5 MeV, as depicted in Fig. 1, while the signal efficiency is considerably lower between 3.5 and 5 MeV due to the multiplicity cut. Additional technical details in this regard will be reported elsewhere in the future.

In this work we have studied the physics potential of detecting ⁸B solar neutrinos at JUNO, in a model independent manner by using the CC, NC and ES detection channels. Because of its largest-ever mass of ¹³C and the expected low background level, excellent signal-to-background ratios can be achieved. Thus ⁸B solar neutrinos will be observable in all three interaction channels.

We have performed detailed evaluations of the background budgets and signal efficiencies of the CC, NC and ES channels at JUNO. With optimized selection strategies, we find that the expected ⁸B neutrino rates of the CC and NC channels are $\mathcal{O}(100)$ interactions per year after the event selection. It turns out that the signal event statistics, detection efficiency and cross section uncertainties are the most crucial factors that affect the detection potential of these two channels. We have carried out a combined analysis of both the coincidence and single events from all three detection channels, and shown that the ⁸B solar neutrino flux, $\sin^{2}\theta_{12}$, and $\Delta m^{2}_{21}$ can be measured to $\pm 5\%$, ${}^{+9\%}_{-8\%}$, and ${}^{+25\%}_{-17\%}$, respectively. When combined with the SNO flux measurement, the world-best precision of 3% can be achieved for the ⁸B neutrino flux.

In the history of solar neutrino experiments, the NC measurement is unique in decoupling the neutrino flux and oscillation parameters, and enabling the model independent approach of the solar neutrino program. SNO has been the only solar neutrino experiment in the past to achieve this goal, and JUNO would be the second one. In this work, we have demonstrated the feasibility of ⁸B solar neutrino measurements at JUNO, which, together with other large solar neutrino detectors (Capozzi et al., 2019; Abe et al., 2018; Beacom et al., 2017), will open a new era of solar neutrino observation and may uncover new directions for neutrino physics and solar physics.

We are grateful for the ongoing cooperation from the China General Nuclear Power Group. This work was supported by the Chinese Academy of Sciences, the National Key R&D Program of China, the CAS Center for Excellence in Particle Physics, Wuyi University, and the Tsung-Dao Lee Institute of Shanghai Jiao Tong University in China, the Institut National de Physique Nucléaire et de Physique de Particules (IN2P3) in France, the Istituto Nazionale di Fisica Nucleare (INFN) in Italy, the Italian-Chinese collaborative research program MAECI-NSFC, the Fond de la Recherche Scientifique (F.R.S-FNRS) and FWO under the “Excellence of Science – EOS” in Belgium, the Conselho Nacional de Desenvolvimento Científico e Tecnològico in Brazil, the Agencia Nacional de Investigacion y Desarrollo and ANID - Millennium Science Initiative Program - ICN2019_044 in Chile, the Charles University Research Centre and the Ministry of Education, Youth, and Sports in Czech Republic, the Deutsche Forschungsgemeinschaft (DFG), the Helmholtz Association, and the Cluster of Excellence PRISMA+ in Germany, the Joint Institute of Nuclear Research (JINR) and Lomonosov Moscow State University in Russia, the joint Russian Science Foundation (RSF) and National Natural Science Foundation of China (NSFC) research program, the MOST and MOE in Taiwan, the Chulalongkorn University and Suranaree University of Technology in Thailand, University of California at Irvine and the National Science Foundation in USA.

In this appendix, we present the technical details of the sensitivity study employed in this work. A Poisson-type least squares function, denoted as $\chi^{2}$, is defined as follows,

where $\chi^{2}_{\rm stat}({\rm CC})$, $\chi^{2}_{\rm stat}({\rm NC})$, and $\chi^{2}_{\rm stat}({\rm ES})$ are statistical parts of the CC, NC and ES channels in the $\chi^{2}$ function, respectively. These components are presented in the second and third rows of Eq. (5). The index $j_{\rm C}$ ranges from 1 to 90 for the CC measurement, representing the energy range from 5 MeV to 14 MeV with an equal bin width of 0.1 MeV. For the NC measurement, $j_{\rm S}$ ranges from 16 to 21, while for the ES measurement, $j_{\rm S}$ spans from 1 to 15 and from 22 to 140 covering the energy range from 2 MeV to 16 MeV with an equal bin width of 0.1 MeV. The predicted numbers of signal and background events, $N^{\rm C}_{\rm pre}(\theta^{i}_{z},E^{j_{\rm C}}_{\rm vis})$ and $N^{\rm S}_{\rm pre}(\theta^{i}_{z},E^{j_{\rm S}}_{\rm vis})$ are calculated for the $i$-th zenith angle bin and the $j_{\rm C}$-th or $j_{\rm S}$-th visible energy bin of the correlated and single event samples, respectively

where $S^{\rm CC}_{\rm pre}$, $S^{\rm NC}_{\rm pre}$, and $S^{\rm ES}_{\rm pre}$ represent the two-dimensional spectra of the ⁸B neutrino signals in the CC, NC, and ES channels, respectively, incorporating the fiducial volume and signal efficiencies. The projections of these spectra onto the visible energy axis are depicted in Fig. 1 for the CC channel and Fig. 2 for the NC and ES channels. Meanwhile, $B^{k_{\rm C}}_{\rm pre}$ and $B^{k_{\rm S}}_{\rm pre}$ correspond to the background components in the correlated and single event samples, respectively, with their visible energy spectra illustrated in the same figures. The calculations of the ⁸B neutrino signal spectra in the CC, NC, and ES channels are as follows:

The ⁸B neutrino signal spectra for the CC, NC, and ES channels are calculated by multiplying the ⁸B neutrino spectrum $S_{{}^{8}{\rm B}}$ with the neutrino oscillation probability $P_{e\alpha}$ (where $\alpha$ equals $e$ or $\mu+\tau$), and then convolving the resulting product with the differential interaction cross sections (namely, $\sigma_{\rm CC}$, $\sigma_{\rm NC}$, and $\sigma^{\nu_{\alpha}}_{\rm ES}$) as well as with the detector response matrix $M$. The neutrino oscillation probability $P_{e\alpha}$ includes both the standard MSW flavor conversion and terrestrial matter effects, and is a function of the neutrino energy $E_{\nu}$ and the zenith angle $\theta_{z}$, calculated within the three-neutrino oscillation framework. The detector response matrix $M$ accounts for the effects of energy resolution and energy non-linearity, as described in  Abusleme et al. (2021a). The observed spectra $N^{\rm C}_{\rm obs}(\theta^{i}_{z},E^{j_{\rm C}}_{\rm vis})$ and $N^{\rm S}_{\rm obs}(\theta^{i}_{z},E^{j_{\rm S}}_{\rm vis})$ are obtained from the corresponding predicted spectra by applying the true values of the ⁸B neutrino flux $\Phi_{{}^{8}{\rm B}}$, oscillation parameters $\sin^{2}\theta_{12}$, and $\Delta m^{2}_{21}$, and assuming negligible contributions from nuisance parameters. Note that, as discussed in Sec. 2, the ⁸B solar neutrino interactions may also contribute to the background components $B^{k_{\rm C}}_{\rm pre}$ (e.g., the green line in Fig. 1, the purple line in Fig. 2), In such instances, all correlations between the signal and background components are accounted for in the $\chi^{2}$ function.

The nuisance parameters $\varepsilon^{m}_{X}$ ($m$=CC, NC, ES), $\varepsilon^{k}_{\rm B}$, $\varepsilon^{n}_{\rm eff}$ ($n$=C, S) account for systematic uncertainties associated with the cross section, the backgrounds, and the detection efficiency, respectively, as discussed in the manuscript. The parameter $\delta^{\rm S}_{E_{\nu}}$ represents the 1$\sigma$ fractional variation of the ⁸B neutrino energy spectrum, as detailed in  Bahcall et al. (1996); Bahcall (1997), while $\varepsilon_{\rm s}$ denotes the magnitude of the ⁸B neutrino spectral uncertainty. A summary of the nuisance parameters and their corresponding uncertainties within the $\chi^{2}$ function is summarized in Table 3. For the sensitivity study that produced the results shown from Fig. 4 to Fig. 7, we selected data sets from one or a combination of the CC, NC, and ES measurements. We then activated the relevant nuisance parameters to account for systematic uncertainties in the corresponding $\chi^{2}$ function. During the calculation of the allowed regions for each analysis, the displayed parameters (one or two of the fitting parameters $\Phi_{{}^{8}{\rm B}}$, $\theta_{12}$, and $\Delta m^{2}_{21}$) were fitted, while all other physical and nuisance parameters were marginalized. The critical values of $\Delta\chi^{2}$ for various confidence levels are sourced from  Zyla et al. (2020).