Sensitivity to neutrinos from the solar CNO cycle in Borexino

Neutrinos emitted in the carbon, nitrogen, oxygen (CNO) fusion cycle in the Sun are a sub-dominant, yet crucial component of solar neutrinos whose flux has not been measured yet. The Borexino experiment at the Laboratori Nazionali del Gran Sasso (Italy) has a unique opportunity to detect them directly thanks to the detector's radiopurity and the precise understanding of the detector backgrounds. We discuss the sensitivity of Borexino to CNO neutrinos, which is based on the strategies we adopted to constrain the rates of the two most relevant background sources, $pep$ neutrinos and $^{210}$Bi beta decays. Finally, the relevance of a detection of CNO neutrinos is extensively discussed in the context of the solar metallicity problem.


Introduction
The Sun releases energy mainly through a nuclear fusion process known as the proton-proton chain (pp chain). Another process, called the carbon-nitrogen-oxygen (CNO) cycle (see Fig. 1), is expected to contribute about one percent of the total energy and neutrino production [1,2]. The CNO cycle emits neutrinos with energies up to around 1.5 MeV and 1.7 MeV for its two main components, see Fig. 2. This process, thought to be the dominant energy production process for stars heavier than 1.3 solar masses [3] as well as solar-like stars in advanced evolutionary stages [4], has many implications for astrophysical problems. For example, the measurement of CNO neutrinos would allow the evaluation of the efficiency of the CNO cycle, helping with the determination of the age of globular clusters [5]. It would also provide a direct reading of the metal abundance in the solar core, which would in turn allow the study of the chemical evolution paradigm assumed by the Standard Solar Model (SSM) [6].
Currently the CNO neutrino rate is only measured to be less than 8.1 cpd/100 ton at 95% confidence level by Borexino [7]. This paper presents a detailed study of the sensitivity of the Borexino experiment to CNO neutrinos. The study relies on a precise and independent determination of the two main residual background components: 210 Bi contamination, which can be estimated 12 Fig. 2: Solar neutrino spectra predicted by the B16 (GS98)-HZ Standard Solar Model [6]. The numbers in parentheses represent the theoretical uncertainties on the flux predictions. For monochromatic lines, the vertical axis reports the flux in cm −2 s −1 . The so-called 13 N and 15 O neutrinos are produced by the β + -decays of Nitrogen-13 and Oxygen-15 in the CNO-I-branch of the CNO-cycle, respectively. The Fluorine-17 component is essentially negligible in the Sun.
using measurements of 210 Po decays (as suggested in [8]), and pep neutrinos, which are a part of the pp chain.
After a brief overview of the Borexino detector in Sec. 2, Sec. 3 discusses Borexino's sensitivity to CNO neutrinos. Sections 4-6 present the strategy to constrain the backgrounds, the influence of additional purification of the liquid scintillator, and the possible sensitivity without an estimation of the 210 Bi contamination, respectively. In Sec. 7 we comment on the relevance of the measurement of the flux of CNO neutrinos in the context of solar physics, with an emphasis on the "solar metallicity (or abundance) problem". This scientific puzzle originated when a re-determination of the sur-face metallicity of the Sun [9][10][11][12][13] indicated a lower value than previously assumed [14]. However, solar models incorporating these lower metal abundances [10] meet difficulties in reproducing the results from helioseismology, which support models with a higher metal content. Section 7.1 shows that, by following the approach proposed in [15,16], it is possible to infer the carbon and nitrogen contents of the solar core independently of the assumed opacity of solar plasma by combining a CNO neutrino flux measurement with the very precise measurement of the 8 B neutrino flux by the Super-Kamiokande collaboration [17] (about 2% precision). Finally, Sec. 7.2 discusses the possibility of using a CNO neutrino measurement to discriminate among SSMs with different hypotheses about the Sun's surface metallicity by combining a measurement of the CNO neutrino flux with existing 7 Be and 8 B data.

The Borexino Experiment
The Borexino detector [20] has been taking data since 2007 in the Laboratori Nazionali del Gran Sasso (Italy) at a depth of 3800 m.w.e. Borexino is an unsegmented calorimeter using about 300 ton of ultra-pure organic liquid scintillator. The scintillator consists of PC (pseudocumene, 1,2,4-trimethylbenzene) as a solvent with 1.5 g/l of fluor PPO (2,5diphenyloxazole) as a solute. This scintillator is contained in a 125 µm-thick nylon vessel with a nominal radius of 4.25 m. It is surrounded by non-scintillating buffer fluid contained in a 6.85 m radius stainless steel sphere, which supports 2212 8-inch ETL 9351 photomultipliers (PMTs). The stainless steel sphere is submerged in ultra-pure water serving as an active muon veto and a passive shield against external radiation.
Neutrinos are detected via the elastic scattering reaction on electrons in the liquid scintillator. The electrons deposit their energy in the scintillator, which results in scintillation light. To estimate this deposited energy, either the number of hits on the PMTs N h or the charge collected on the PMT anodes (or the number of photoelectrons N pe ) are used [18,19]. The interaction vertex is reconstructed via the arrival times of the scintillation photons on the PMTs. As a reference, the detected photoelectron yield is ∼500 photoelectrons/MeV (normalized to 2000 PMTs). The energy and position reconstruction uncertainties at 1 MeV are ∼ 50 keV and ∼ 10 cm, respectively.
The time distribution of the detected hits on the PMTs allows pulse-shape discrimination (PSD). With PSD, α particles, a product of 210 Po decay in the scintillator, can be discriminated from β particles on an event-by-event basis. It can also be used to constrain the ratio of the total counts of β + events, such as 11 C decays, to β − events in specific energy ranges [18,19].

Borexino sensitivity to CNO neutrinos
The determination of the upper limit on the interaction rate of CNO (R CNO ) reported in [7] is based on an analysis of the distributions of physical observables, it being impossible to distinguish individual neutrino events from background in Borexino.
In detail, as discussed in [18,19], we start from a background model consisting of a list of isotopes contaminating the scintillator (namely 14 C, 85 Kr, 210 Po, 210 Bi), isotopes of cosmogenic origin ( 11 C), and γ-rays emitted in the regions outside of the sensitive volume and referred to as external backgrounds (originating from decays from 208 Tl, 214 Bi, and 40 K).
We then perform a multivariate fit procedure of events selected in an optimized well-less fiducial volume in the central part of the detector that includes three observables: the energy spectra of the events in the region from 0.19 to 2.93 MeV, their reconstructed radial coordinate distribution, and that of a pulse-shape parameter. This fitting procedure makes it possible to disentangle the rates of events induced by neutrinos from those induced by backgrounds. At the same time, its successful convergence verifies the consistency of the background model.
The expected energy distribution of the signals and of the most relevant background components can be seen in Fig. 3, while the expected and measured interaction rates of solar neutrinos are listed in Table 1.
There is an intrinsic difficulty in the measurement of solar neutrinos due to the similarity of the energy spectra of 210 Bi electrons and those of electron recoils induced by pep and CNO neutrinos. This feature, which was discussed in [18,19], is clearly visible in Fig. 3. The partial degeneracy of these spectral shapes induces significant correlations among these three components. As a consequence, the measurements of R pp , R Be , and R pep presented in [7] were made possible only by constraining R CNO to the values expected from the HZ and LZ models. For the same reason, the limit on R CNO (also reported in [7]) was achieved by constraining the ratio of R pp and R pep to the SSM predictions.
However, the measurement of R CNO is now possible in Borexino in Borexino Phase-III, characterized by a stable thermal condition of the detector. We succeeded in thermally stabilizing the detector, and identified a strategy (discussed in Sec. 4) to measure or constrain the two main backgrounds, pep neutrinos and 210 Bi in the scintillator. The low internal background levels, great depth of LNGS, passive detector shielding,    The solar neutrino components are highlighted with bold red lines. The spectra of the relevant background contributions are scaled to existing measurements. The contribution of the decay of the 11 C isotope reported in the plot is reduced by the application of the Three Fold Coincidence (TFC) procedure described in [18,19]. Table 1: Expected integral interaction rates (without energy threshold), and corresponding Borexino results [7], in cpd/100 ton. The prediction assumes the Standard Solar Model [6] under the high (HZ) and low (LZ) metallicity hypotheses and the MSW-LMA paradigm with the oscillation parameters reported in [21]. The data was obtained with a total exposure of 1291.51 day×71.3 ton. The density of target electrons in the Borexino scintillator is (3.307 ± 0.003) × 10 31 e -/100 ton. The CNO interaction rate was constrained to the expected HZ or LZ values in the fit procedure thereby allowing the measurement of the pp, 7 Be, and pep interaction rates. As reported in the table, pep was the only fit component affected by the difference in the CNO values. The upper limit on the CNO neutrino interaction rate was obtained with the fit procedure by constraining the ratio of the pp and pep rates, with a Gaussian pull-term, to the values predicted by the SSM. and active removal (based on a three-fold-coincidence method [18]) of the cosmogenic 11 C background are also crucial.
Further, we observed an energy region from about 0.8 MeV to 1 MeV, hereinafter referred to as region of interest (ROI), that is dominated by R Bi , R pep , and R CNO (see Fig. 4). More precisely, in this ROI the background contributions are of the same order as the statistical fluctuations of the total CNO rate. Thus, it is also possible to extract the sensitivity of Borexino to CNO neutrinos through a simple procedure consisting only of determining the background rates and subsequently counting all the events in this energy region. Section 3.1 first discusses this counting analysis. Then, Sec. 3.2 presents a sensitivity study performed with the multivariate fit. Finally, Sec. 3.3-3.4 compare the resulting expected sensitivities and discovery significance.
In the following sections, the 210 Bi rate is assumed to be 17.5 cpd/100 ton as in [7], and an exposure of 1000 day×70 ton is used. The same constraints of pep neutrinos and 210 Bi are used, as they will be discussed in Sec. 4.1 and 4.2, respectively.  Fig. 3, events are selected with the fiducial volume cut and the contribution of 11 C is suppressed using the TFC procedure. The white region indicates the region of interest.

Borexino as a counting experiment
This approach consists of counting the total number of events in the ROI to obtain the interaction rate of CNO neutrinos as the difference between the total count rate and that of the other components.
Our key assumptions lie in the availability of a realistic background model and the ability to measure the rate of the residual background components independently (or to label them as negligible). Another assumption is the availability of a sufficiently accurate model of the detector response function, needed to determine the fraction of each spectral component in the ROI.
As shown in Fig. 3 and Fig. 4, there are six contributions to the event rate in the ROI: CNO neutrinos, 210 Bi electrons, pep neutrinos, residual 11 C positrons, 7 Be neutrinos, and external gamma backgrounds. The impact of the relevant parameters like the exposure, the choice of the ROI, and the precision of the estimation of pep neutrino and 210 Bi rates can be understood by writing R CNO (integrated over the ROI) and its statistical uncertainty σ count CNO as: where r tot and σ tot are the value and the uncertainty of the total event rate in the ROI, ε is the fraction of events for each component falling inside the ROI, R and σ are the rate and the uncertainty, respectively, of components in the full energy range, and r o is the total event rate of all other components in the ROI. The contributions to r o are either negligible and/or independently determined via their spectral features outside of the ROI. Variables with a tilde mark the values from external constraints and the subscripts indicate the spectral component. The symbol ⊕ is defined as The ratios of the four terms in Eq. (1), from left to right, are 9: 4: 2: 1. Whenσ Bi is 2 cpd/100 ton andσ pep is 1.4%·R pep , or 0.04 cpd/100 ton, the ratios of the four terms in Eq. (2), from left to right, are 12: 18: 1: 2. The largest contribution to σ count CNO in Eq. (2) is the 210 Bi term unless the uncertainty on the 210 Bi constraint is lower than 1.3 cpd/100 ton, in which case σ count CNO is limited by a statistical error. The resulting uncertainty on the CNO neutrino interaction rate as a functionσ Bi and σ pep can be seen in Fig. 5.

Multivariate fit analysis
We performed a study of the sensitivity of Borexino to CNO neutrinos by simulating tens of thousands of pseudo-experiments and fitting the pseudo-data with the full multivariate procedure as the one used for real data [7]. This approach requires maximizing a three dimensional likelihood function built using three observables (energy estimator, radial position, and pulseshape parameter). The simulation includes all background components and their rates were set to the values measured in [7]. We assumed, as throughout Sec. 3, that R pep and R Bi can be independently estimated. The constraints on R pep and R Bi were implemented by introducing two Gaussian pull terms in the likelihood function. The uncertainties of the two constraints are the most important parameters when determining the sensitivity.
Each simulated dataset was fitted assuming the same detector response function as that used in the simulation. The distribution of the best-fit results for the CNO interaction rate was used to assess the expected statistical uncertainty and discovery significance. The results were obtained using the multivariate fitting tools GooStats [22] and m-stats [23].
Similarly to the data analysis of [7], the study was conducted using both Monte Carlo and analytical methods that are complementary to each other. In the Monte Carlo approach the detector response was modeled using an ab initio simulation software [24] that takes into account realistic and microscopic descriptions of the energy deposition, scintillation light generation and propagation, electronics response, and energy and position reconstruction. In the analytical approach, the detector response was modeled using analytical functions [19]. The analytical procedure provides an independent way of evaluating the impact of the systematic uncertainty related to detector response parameters (e.g., the light yield). The two approaches gave consistent results for the expected uncertainty and discovery significance.

Uncertainty of the CNO rate
In this section we discussed the uncertainty of the CNO rate of both the counting and multivariate fit analyses. In the framework of the multivariate analysis, we built the distribution of the values of R CNO resulting from fits performed with fixed external constraints. The width of this distribution was used to estimate σ f it CNO . Figure 6 shows σ f it CNO as a function of the 210 Bi uncertainty for two different values ofσ pep as well as the uncertainty σ count CNO resulting from the counting analysis presented in Sec. 3.1. Figure 6 shows that the results of two analyses are fully compatible whenσ Bi 1.5 cpd/100 ton. One can conclude that the inclusion of all of the additional information used by the multivariate fit method only marginally helps to constrain the ratios R CNO /R Bi and R CNO /R pep . In the counting analysis, the rates of 210 Bi and pep are fixed by the external constraints. Thus, according to Eq. (1), a bias on the constraints will transfer linearly to the reconstructed CNO rate. The situation becomes slightly more complicated when a full spectral analysis, combined with constraints on the background, is performed. Indeed, in the case of a bias in the external constraints, the impact on the value of the extracted CNO rate is mitigated by the tension between the shape information and the biased values.
The systematic uncertainty of the CNO rate is dominated by the residual migration 210 Po (see Sec. 4.2), followed by the accuracy on the light yield.

Expected discovery significance to CNO neutrinos
A frequentist hypothesis test was performed to assess the expected discovery significance to CNO neutrinos.
In the search for the CNO signal, two hypotheses are considered: the null hypothesis, H 0 , where no CNO signal is present; and the alternative hypothesis, H 1 , that includes the presence of a CNO signal in addition to the background. We used the profile likelihood q 0 , defined as minus two times the logarithm of the ratio of the maximum value of the likelihood of H 0 and H 1 as the test statistic [25]. The significance of the signal is quantified by evaluating the compatibility of the observed data with the background-only hypothesis H 0 . The median discovery significance of a measurement is defined here as the median p-value, obtained when testing the background only hypothesis under the assumption of a SSM-CNO signal. This requires the PDFs of q 0 for both the background-only hypothesis f (q 0 |no CNO) and the signal hypotheses f (q 0 |HZ/LZ CNO).
The PDFs of the test statistics q 0 were obtained by analyzing simulated datasets with both the simplified counting analysis and full multivariate fit. The expected discovery significance was estimated assuming the CNO neutrino interaction rate predicted by the HZ and LZ SSMs. Figure 7 shows the results.
The expected discovery significance depends on the strength of the signal and on the precision of the external constraints: a higher rate of CNO neutrinos, as well as stronger constraints, results in a higher sensitivity. For the CNO rate predicted by the HZ (LZ) SSM and in the case of uncertainties of 0.04 cpd/100 ton and 2.5 cpd/100 ton on pep neutrinos and 210 Bi, respectively, the median discovery significance to CNO neutrinos is 3σ (2.1σ).
The difference between the median p-value obtained from the simple counting analysis and the one from the multivariate analysis gets smaller as the precision of the constraints increase. This effect is coherent with the results on the statistical uncertainty of CNO presented in Sec. 3.3, indicating that the impact of the spectral shape information is larger when the constraints are relatively weak (high uncertainty), while it becomes negligible when the background constraints get more stringent (low uncertainty).

Impact of an upper limit on 210 Bi rate
The method discussed in Sec. 4.2, used to obtain an independent measurement of the 210 Bi rate, can only provide an upper limit on the 210 Bi background under less stringent assumptions, that an additional contribution from migrating (diffusion and/or convection)  In this case, the constraint is implemented in the fitting procedure as a one-sided Gaussian penalty. In so doing, the upper limit on the 210 Bi rate is equivalent to a lower limit on the CNO rate. Therefore, if the method used to measure the 210 Bi rate is unbiased, the expected discovery significance for CNO is the same as would be obtained using a two-sided constraint on the 210 Bi rate. This was confirmed by performing a hypothesis test analogous to that described in Sec. 3.4. The results for varying widths (σ Bi ) on the 210 Bi penalty are shown in Fig. 8. The expected discovery significance obtained from the counting analysis and from applying a symmetric constraint are also displayed for comparison. As one can see, allowing an additional contribution from the migrating 210 Po barely decreases the expected discovery significance for CNO.
Therefore, an upper limit on the 210 Bi rate may be sufficient to claim detection of CNO neutrinos, but it would not allow a precise measurement of the CNO interaction rate required to solve the solar metallicity problem. Since there is no lower limit on the 210 Bi rate, the upper limit of the CNO neutrino rate will not be stringent. Figure 9 shows the distribution of the best fit estimates, obtained from the fit of simulated datasets when applying a constraint to the pep neutrino rate and leaving the 210 Bi rate free (Fig. 9a), constraining the 210 Bi rate with a symmetric Gaussian penalty (Fig. 9b), and applying the upper limit described in this section (Fig. 9c).
The expected confidence intervals in Fig. 9c are asymmetric: the upper limit of the confidence interval is similar to the one obtained when leaving the 210 Bi rate free, while its lower limit resembles the one resulting when the 210 Bi rate is constrained.

Strategy for establishing background assessment
This section discusses the strategy for setting constraints on the pep neutrinos and 210 Bi rates independently of both the counting analysis and the multivariate fit procedure.

pep constraint
The electron capture reaction p + e − + p → d + ν e , which generates pep neutrinos in the Sun, is linked to the β + decay process p + p → d + e + + ν e by wellknown nuclear physics, which is responsible for the pp neutrino production. Since the two processes depend on the same allowed nuclear matrix element, the ratio between their rates is determined by the available reaction phase spaces and by the electron density n e of the solar plasma only.
The ratio R pep /R pp was calculated in [26,27], and the effect of radiative corrections subsequently discussed in [28] (see, for example, [29] for a review). It can be determined with ∼ 1% precision for the conditions of the solar interior and is mildly dependent on the properties of the solar plasma, roughly proportional to T −1/2 c n e (where T c is the temperature of the core of the Sun). As a consequence, the ratio Φ pep /Φ pp between the pep and pp neutrino fluxes is a robust prediction of SSMs, and it can be used to improve the sensitivity to the CNO neutrino signal. With this approach, Borexino's direct observation of the pp neutrino flux [7] can be translated into a ∼ 10% determination of the pep-neutrino component, motivating the analysis performed in Sec. 3, which assumesσ pep = 0.28 cpd/100 ton.
The precision of the pp and pep neutrino flux determination can be further improved by performing a global analysis [30,31] on all neutrino experiment results applying the so-called solar luminosity constraint [32][33][34]. In this way, it was shown in [31] that the pep neutrino rate can be constrained with a precision of ∼ 1.4%, motivating the assumptionσ pep = 0.04 cpd/100 ton considered in Sec. 3. Because the contribution of CNO neutrinos to the solar luminosity is around 1%, neglecting the dependence of this pep constraint on the assumed R CNO has almost no effects on the significance to CNO neutrinos, as shown in Fig. 5.

210 Bi constraint 210
Bi is a β-emitting daughter of 210 Pb with a mean lifetime of 7.23 days and a Q value of 1160 keV. Since the lifetime of 210 Bi is small, it must be supported by its long-lasting parent nucleus 210 Pb (τ = 32.2 years) to maintain a constant decay rate. 210 Pb is a part of the 238 chain, and due to its low Q value it does not represent a background in this analysis. The presence of other isotopes above 210 Pb in the 238 U chain was ruled out by the absence of 226 Ra and 222 Rn, easily detectable through the fast 214 Bi-214 Po coincidence following their decays. The purification campaign in 2011 [35] significantly reduced in short order the 210 Pb contamination, leading to a residual rate in the range of a few tens of cpd/100 ton.  The strategy to independently determine the 210 Bi rate using the decay rate of its decay product, 210 Po, was suggested in [8]. The isotope 210 Po, with a mean lifetime of 199.6 days, ends the chain by decaying into stable 206 Pb thereby emitting a 5305 keV α particle (visible energy around 400 keV electron equivalent): When the above chain is in equilibrium, the 210 Bi rate is equal to that of 210 Po. The 210 Po rate can be precisely determined in Borexino because 210 Po emits α particles that can be identified event-by-event via pulse-shape discrimination. Using a pulse-shape analysis based on a multi-layer perceptron discriminator [19], 210 Po events in Borexino are identified with an efficiency very close to 1.
Unfortunately, the measured 210 Po rate is not only due to 210 Bi decays, but consists of three components: The 210 Bi rate is equal to the supported 210 Po rate and thus less than the total 210 Po rate in presence of the unsupported and migration terms. The rate of unsupported 210 Po, following the law of radioactive decay, gets asymptotically closer to zero over time. The migration 210 Po is the most intricate contribution to handle: In Borexino, diffusion of 210 Po into the fiducial volume is a completely negligible process. It depends on the convection currents induced by the temperature gradient in the detector; it is therefore time-and spacedependent and hard to model. These considerations are summarized through the following equation: where R tot Po is the observed 210 Po rate, R u Po is the initial rate of unsupported 210 Po decay, R m Po is the migration 210 Po decay rate, and R s Po is the supported 210 Po rate and is equal to the 210 Bi rate.
The convective motion of the liquid scintillator was suppressed in a major effort over several years since 2015 [36]. This included the installation of a passive thermal insulation system around the water tank and an active temperature control system to mitigate seasonal modulations. Our observations showed that the migration of 210 Po was significantly reduced in the Borexino Phase-III.
Once the rate of unsupported 210 Po is sufficiently low, it is possible to get a more precise estimate of the rate of supported 210 Po by identifying the region inside the fiducial volume, that is least affected by convective motions and thus with minimal migration 210 Po.
The residual migration 210 Po rate in this region can then be treated as a source of systematic uncertainty.
Neglecting the migration 210 Po rate, the statistical uncertainty of the supported term depends on the magnitude of the initial out-of-equilibrium contamination and on the exposure. As an example, with an unsupported rate of 50 cpd/100 ton and a supported rate of 17.5 cpd/100 ton, an precision of the order of 10% on the supported rate can be achieved in about 8 months of data-taking using the same fiducial volume of 71.3 ton as adopted in [7].
However, in order to extrapolate the value of 210 Bi obtained in the region with minimum 210 Po migration rate to the entire fiducial volume, two hypotheses must be verified: the 210 Bi spatial distribution must be uniform within the entire fiducial volume and its rate must follow the slow decay rate of 210 Pb, since there are no sources of either 210 Bi or 210 Pb in the liquid scintillator. The residual non-uniformity and instability of the 210 Bi rate can also be treated as systematic uncertainties of the 210 Bi constraint.

Impact of further scintillator purification
The purification campaign of the Borexino scintillator performed in the year 2011 [35], with 6 cycles of closedloop water extraction, significantly reduced the concentrations of several radioactive contaminants of the scintillator. The 210 Bi rate was reduced by a factor ∼ 2.3 [7,19]. In principle, another cycle of purification could decrease the amount of 210 Bi even further without introducing more 210 Po in the liquid scintillator. However, in order to keep the amount of migration 210 Po small, thus for the sensitivity to CNO neutrinos to benefit from further purification, the thermal stability of the detector must be maintained.
In order to quantify the effect of a reduced rate of 210 Bi, we rewrite Eq. 2 as where a 1 is the total rate of components other than 210 Bi in the ROI, a 2 is the total uncertainty of the last three terms in Eq. 2 and is dominated byσ Bi , and E is the exposure. From Eq. 4 we see that the importance of E and R Bi is greater when a 2 is smaller than the first term. Thus, in these conditions, reducing the rate of 210 Bi is equivalent to increasing the exposure. This can also be concluded from Fig. 10.

Sensitivity to CNO neutrinos without 210 Bi constraint
With a high exposure, the intrinsic small difference between the spectral shapes of CNO and 210 Bi makes it possible, in principle, to measure the interaction rate of CNO neutrinos without any constraint on R Bi . As shown in Fig. 6, even with the currently assumed exposure, the multivariate fit shows a better performance than the counting analysis when the 210 Bi constraint is not very strong. Quantitatively, a statistical significance of about 3σ can be reached for an exposure of 450 year × ton assuming CNO neutrinos from the HZ SSM. This statistical sensitivity improves as the inverse square root of exposure. However, the systematic uncertainties due to the detector response and energy scale modeling must be treated carefully.

CNO neutrinos in the Standard Solar Model
Despite the small contribution of the CNO cycle to the Sun's luminosity, a measurement of the CNO neutrino flux would be extremely valuable to expand our knowledge of the Sun.
Experimental measurements of solar neutrino fluxes are a fundamental test of the SSM, the benchmark of every stellar evolution model. Possible disagreements between predicted and measured neutrino fluxes may highlight issues in the model assumptions. Additionally, measurements of solar neutrino fluxes can be used to infer some properties of the Sun that are used as inputs for the SSM. This is particularly interesting in the context of the solar metallicity problem. As mentioned in the introduction, SSMs implementing the most recent determinations of the solar surface composition (e.g., the low metallicity (LZ) admixture described in [10]), fail to reproduce helioseismic observations, which agree better with models that assume a higher metal content, as prescribed by the older determinations from [14]. So far, all attempts to understand the origin of this discrepancy remain inconclusive. New experimental results on solar neutrinos could provide key pieces of information to solve this problem. This applies in particular to the flux of CNO neutrinos, whose value differs by almost 30% between the predictions of the SSM HZ and LZ models (see Table 1).
However, the dependence of predicted solar neutrino fluxes on the metallicity and other SSM parameters is non-trivial. There is indeed a strong degeneracy between the impact of metallicity and radiative opacity through their effect on the core temperature of the Sun T c (T c is determined by the opacity profile, which in turn depends on the chemical composition of the solar interior). Ultimately, T c acts as a proxy for the influence of those parameters (i.e., they impact neutrino fluxes via their effect on T c ). This degeneracy makes it difficult to disentangle metallicity from opacity with solar neutrino measurements, in particular after recent laboratory measurements [37] and theoretical advances [38][39][40][41][42] suggested that the uncertainty of the radiative opacity might be severely underestimated. Deviations of the radiative opacity from its nominal values can mimic the effect of a higher metal content in the Sun's core. As it was discussed in, for example, [43][44][45], the agreement of SSM calculations implementing the recent LZ composition and helioseismic data could be restored by considering suitable modifications to the radiative opacity. Similarly, neutrino fluxes expected by the HZ SSM can be obtained by assuming the LZ composition and a larger opacity of the solar plasma.
However, the case of the CNO neutrinos is somewhat different. In fact, the dependence of the CNO neutrino flux on metallicity is twofold: as for the other pp-chain neutrinos the CNO flux is indirectly linked to metallicity via T c , but it also depends directly on the amount of carbon and nitrogen in the core of the Sun. Considering the metallicity problem, other tests of the SSM that consider measurements of 8 B and 7 Be neutrino fluxes include a larger amount of information but are greatly affected by uncertainties on the radiative opacity. Section 7.1 shows that by exploiting this fact, a measurement of the CNO neutrino flux with Borexino can be used to infer the content of carbon and nitrogen in the Sun's core almost independently of the effect of radiative opacity, thus directly probing the solar metallicity. Sec. 7.2 discusses the impact of a future measurement of CNO neutrinos on the discrimination power of a hypothesis test between the HZ and LZ SSM.

CNO neutrinos as CN abundance messengers
As explained in [16], the temperature dependence for both the 8 B and CNO neutrino fluxes can be described by a power-law Φ i ∝ T γi c (i ∈ { 8 B; CNO}). Therefore, one can build a weighted ratio Φ CNO /(Φ8 B ) k , with k = γ CNO /γ8 B , that is very nearly (considering different uncertainties) independent of T c . This breaks the opacitycomposition degeneracy, making it possible to infer the abundance of carbon and nitrogen in the Sun in a way that is almost independent from variations in the SSM parameters that affect the temperature profile.
To elaborate, we followed the approach of [15,16], which adopts the practice of factorizing the dependence of the neutrino flux Φ i on the model's input parameters {β j }: where the x j terms are the model inputs normalised to their respective nominal SSM value (i.e., β j /β SSM j ). These parameters are typically grouped in three categories: (i ) the solar parameters, related to the Sun's astrophysical (age A , luminosity L ) and non-nuclear physical properties (diffusion D, radiative opacity κ), (ii ) the metallicity parameters (i.e., the abundances of C, N, O, F, Ne, Mg, Si, S, Ar, and Fe), and (iii ) the nuclear cross sections of the relevant processes, described by the astrophysical S-factors. The α(i, j) coefficients in the exponents are normalised logarithmic partial derivatives of the fluxes with respect to the input parameters β j which are typically provided along with SSM predictions. Using Eq. 5 to express solar neutrino fluxes, the aforementioned weighted ratio between the 15 O and 8 B fluxes reads: To reduce the dependence on T c , the weight k is chosen such that it minimizes the uncertainty of the quantity defined in Eq. 7 due to the so-called environmental parameters (i.e., those SSM inputs which affect the Sun's temperature profile the most [15,16]). These parameters include the solar and metallicity parameters discussed above, but without the C and N abundances.
When applying this strategy to Borexino we refer, for simplicity, to the counting analysis described in Sec. 3.1. In this narrow energy window, neglecting insignificant 17 F neutrinos, CNO neutrino events originate from contributions of 15 O and 13 N neutrinos. We thus define the flux of CNO neutrinos measured by Borexino Φ BX CNO as a combination of Φ15 O and Φ13 N . This can be written as: where ξ is the ratio between the event rate of 15 O neutrinos and all CNO neutrinos in the ROI, as expected from the SSM: Using the logarithmic derivatives of 15 O and 13 N neutrinos from the B16-SSM [6], we can describe the dependence of the measured CNO neutrino flux on the inputs of the SSM as Having computed those derivatives, we can express Eq. 7 using a CNO measurement by Borexino and the 8 B neutrino flux measured by Super-Kamiokande [17]. In the case of Borexino, the value of k, obtained from a minimization of the uncertainty from the environmental parameters, is found to be 0.735. To explicitly show the dependence on all SSM inputs: where the contributions of the environmental parameters 1 are grouped in the second and third rows on the right-hand side, while the contributions of nuclear cross sections are in the fourth row. Following [15,16], the exponents in Eq. 11 were obtained using the values of the logarithmic derivatives of the solar neutrino fluxes with respect to the SSM inputs used by the HZ SSM [6]. Assuming, for simplicity, that the C and N abundances are modified by the same factor (i.e., , where N C and N N indicate the number density of C and N with respect to hydrogen) and noticing in Eq. 11 that the sum of the exponents of x C and x N is 0.814 + 0.191 1, we can invert the formula, obtaining: where R BX CNO is the rate measured by Borexino and proportional to the flux Φ BX CNO . The quoted uncertainties were obtained by propagating the uncertainties of the environmental, nuclear, and diffusion parameters used in the SSM and by taking into account that the current uncertainty on the measurement of the boron neutrino flux is about three percent [31]. The dominant contributions to the uncertainty budget are from nuclear reactions, with the largest coming from S 114 (7.5%), S 34 (3.4%), and S 17 (3.6%).
The low level of the environmental contribution to the total error budget (which takes into account all SSM parameters except nuclear reactions and diffusion coefficients) demonstrates that a future CN flux measurement will be converted into an almost direct determination of the C+N content of the solar core, as discussed in [15,16].
The impact of the experimental and model uncertainties in the determination of the C+N abundance inferred from a CNO neutrino measurement, assuming the rate predicted by the HZ SSM, is represented in Fig. 11. While a 1.5 cpd/100 ton precision is translated, according to Eq. 12, to a precision that is about three to four times larger than the precision of the GS98 and AGSS09 catalogues, a future measurement of R CNO with a precision 0.5 cpd/100 ton (achievable by next generation experiments [46][47][48]) will be able to constrain the C+N fraction of the Sun with an uncertainty 15%, which is comparable to the precision of the current estimations obtained from measurements of the photosphere. These results have been obtained assuming the rate expected from the HZ SSM and using the result of [31] for the 8 B flux. The red and blue bands show the 68% confidence interval of the abundance of carbon and nitrogen reported in the GS98 [14] and in the AGS09 [10] catalogues respectively.

High versus Low Metallicity Standard Solar Model
If one considers the current treatment of the opacity and of its corresponding uncertainties in the SSM as reliable, then the measurements of neutrino fluxes due to the pp chain can also be used to test models of the composition of the Sun.
In the results of Borexino Phase-II [7], the measurement of the fluxes of 7 Be and 8 B neutrinos were used to this end because of their strong dependence on the Sun's core temperature, which is affected by the metal content as described above. Including CNO neutrinos in a similar test is particularly interesting since their flux depends on the carbon and nitrogen abundances in the Sun. This strong dependence on the SSM parameters leads to a 30% difference between the predictions of the CNO neutrino flux of the HZ and LZ SSMs (See Table 1).
In order to assess the relevance of a future CNO flux determination in this context, we considered the HZ and LZ SSMs as two alternative possibilities and we performed a frequentist hypothesis test analogue to the one described in [7]. Namely, we assumed a given model and determined the p-value against the alternative hypothesis when the 8 B, 7 Be, and CNO neutrino rates were measured with prescribed accuracies. For 8 B and 7 Be neutrinos, we adopted the rate uncertainties from [7]. This makes it possible to directly compare our results with the one from [7] that did not incorporate CNO neutrinos in the analysis. To simulate future CNO neutrino measurements, we used uncertainties of 1.5 and 0.5 cpd/100 ton, as discussed in the previous sections.
Our results are shown in Fig. 12, where we plot the distributions of the test statistic defined as the difference between the χ 2 computed for HZ and LZ predictions including both model and experimental uncertainties. The more the distributions are separated, the higher the probability of discriminating among the different hypothese is.
When taking into account values for the 8 B and 7 Be neutrino fluxes as predicted by the HZ SSM and measured with the same precision as in [7], the median pvalue for the LZ predictions is 0.057, corresponding to a 1.6σ significance in the exclusion of the wrong hypothesis. When including a measurement of CNO neutrinos with an uncertainty of 1.5 cpd/100 ton, the median discriminatory power for the LZ SSM hypothesis does not change significantly (p-value = 0.047, 1.7σ significance), as the experimental uncertainty is 2-3 times larger than the model precision. A larger increase in significance can be obtained if a 0.5 cpd/100 ton precision in the determination of the CNO neutrino flux is achieved. In this case, assuming the HZ SSM predictions, the median p-value for the LZ SSM is 0.016 (2.1σ). Similar significance levels were found when using the values of Φ8 B and Φ7 Be measured by Borexino in [7].
This test relies on stronger assumptions as the determination of C+N abundance presented in the previous section. It also considers a simplified "binary" hypothesis system where either the HZ and LZ SSM is assumed to be correct, with no possible alternative explanation. However, its discriminatory power is better than the one achievable from the inferred C+N abundance for the simple reason that while in the latter the impact of metallicity on the core temperature is cancelled out, in this study it is the dominant source of information, driving the sensitivity of the test through Φ8 B and Φ7 Be . In this context, a measurement of CNO neutrinos is expected to have a sensible impact on the test only if the experimental precision approaches model uncertainty (i.e., σ CNO 0.5 cpd/100 ton). The precision of the solar model predictions indeed poses a strong limit to the overall sensitivity of such a test. Currently, the largest sources of uncertainty-besides the plasma opacity and the carbon abundance-are the S 17 ( 8 B), S 34 ( 7 Be), and S 114 (CNO) astrophysical S-factors [6]. Therefore, a reduction of the nuclear cross section uncertainties is crucial to allow for a more significant test of the SSM based on solar neutrino measurements.  Fig. 12: PDFs of a test statistics suited to perform a hypothesis test between the HZ and LZ SSM. The solidline PDFs indicates the expected distributions obtained considering only 8 B and 7 Be neutrinos obtained with the same precision as reported by Borexino in [7], while the distributions with dashed and dotted lines are obtained including in the analysis a future measurement of CNO neutrinos with an precision of 1.5 and 0.5 cpd/100 ton, respectively.

Conclusions
Borexino has a a strong potential to detect CNO neutrinos due to its unique radiopurity, intrinsically low cosmogenic 11 C rate and successful TFC method for its further reduction, reduction of external gammas by passive detector shielding, as well as comprehensive detector and background modeling.
A detailed sensitivity study was performed which showed that if the backgrounds are constrained, the bulk of the sensitivity to the CNO neutrino signal can be established with a simple counting analysis. The multivariate analysis only mildly improves this sensitivity when the uncertainty on the background rates is relatively loose. The statistical uncertainty on the CNO rate depends on the uncertainty in the determination of the pep neutrinos and 210 Bi backgrounds. If R Bi and R pep can be constrained with a 2.5 cpd/100 ton and 0.04 cpd/100 ton precision, respectively, the median sensitivity to CNO neutrinos is 3σ (2.1σ) for the CNO flux expected by the HZ (LZ) SSM.
The sensitivity does not improve significantly if a new purification campaign were to further reduce the 210 Bi content, unless the absolute precision of the 210 Bi constraint is improved. This improvement is not guaranteed by the reduction of the 210 Bi content alone and is mainly related to the capability of stabilizing the detector temperature and suppressing convective motions. However, for a given uncertainty on the 210 Bi constraint and if the uncertainty on the rate of CNO neutrinos is dominated by statistical fluctuations, a reduction of 210 Bi would shorten the time needed to achieve a given precision on the CNO neutrino interaction rate.
In order to break the correlations, the pep and 210 Bi background rates must be independently constrained. The rate of pep neutrinos can be constrained by exploiting its relation with the pp neutrino rate as already done in the analysis presented in [7]. The uncertainty on the pep rate can be reduced to the level of around one percent by imposing an additional constraint based on solar luminosity measurements.
The 210 Bi isotope is supported by their long-lived parent 210 Pb (τ Pb = 32.2 years). The most promising method to constrain the 210 Bi background relies on a measurement of the daughter isotope 210 Po, which is expected to be in equilibrium with the 210 Pb when the liquid scintillator is thermally stable. 210 Po events can efficiently be tagged using pulse-shape discrimination. Since mid-2015, the Borexino collaboration has been focusing its effort on the thermally stabilizing the detector. At present, prospects of exploiting the 210 Bi constraint look good.
By combing the measurements of Φ8 B and Φ CNO , the carbon and nitrogen contents of the solar core can be determined independently of the opacity of solar plasma. Assuming R CNO is measured with an uncertainty 0.5 cpd/100 ton, the C+N fraction of the Sun can be determined within 15%, which is comparable to the precision of the current estimations obtained from measurements of the photosphere, and limited by the nuclear cross section precision.
The neutrino fluxes depend on the chemical composition of the Sun, and can be used to discriminate between SSM HZ and LZ models. Combining the measurements of Φ7 Be , Φ8 B , and Φ CNO , assuming the CNO rate to be determined at a precision of 0.5 cpd/100 ton, the median discrimination sensitivity is 2.1 σ and limited by the uncertainties of SSM.