aainstitutetext: Departament de Física Quàntica i Astrofísica and Institut de
Ciències del Cosmos, Universitat de Barcelona, Diagonal 647, E-08028
Barcelona, Spainbbinstitutetext: Institució Catalana de Recerca i Estudis
Avançats (ICREA), Pg. Lluis Companys 23, E-08010 Barcelona,
Spainccinstitutetext: C.N. Yang Institute for Theoretical Physics, Stony Brook
University, Stony Brook, NY 11794-3840, USAddinstitutetext: Instituto de
Física Teórica (IFT-CFTMAT), CSIC-UAM, Calle de Nicolás Cabrera 13–15, Campus
de Cantoblanco, E-28049 Madrid, Spaineeinstitutetext: Institute of Space
Sciences (ICE, CSIC), Campus UAB, Carrer de Can Magrans s/n, 08193 Cerdanyola
del Valls, Spainffinstitutetext: Institut d’Estudis Espacials de Catalunya
(IEEC), C/Gran Capità 2-4, 08034 Barcelona, Spain

# Status of Direct Determination of Solar Neutrino Fluxes after Borexino

M. C. Gonzalez-Garcia<EMAIL_ADDRESS>d Michele Maltoni
<EMAIL_ADDRESS>a João Paulo Pinheiro<EMAIL_ADDRESS>e,f Aldo M. Serenelli<EMAIL_ADDRESS>
###### Abstract

We determine the solar neutrino fluxes from the global analysis of the most
up-to-date terrestrial and solar neutrino data including the final results of
the three phases of Borexino. The analysis are performed in the framework of
three-neutrino mixing with and without accounting for the solar luminosity
constraint. We discuss the independence of the results on the input from the
Gallium experiments. The determined fluxes are then compared with the
predictions provided by the latest Standard Solar Models. We quantify the
dependence of the model comparison with the assumptions about the
normalization of the solar neutrino fluxes produced in the CNO-cycle as well
as on the particular set of fluxes employed for the model testing.

###### Keywords:

solar neutrinos

††preprint: IFT-UAM/CSIC-23-XXX, YITP-SB-23-39

##  1 Introduction

At present we know without a doubt that $(a)$ the Sun produces its energy as a
whole by fusing protons into $\mbox{}^{4}\text{He}$, $(b)$ electron neutrinos
are by-products of these processes, and $(c)$ an established fraction of such
neutrinos have either muon or tau flavour by the time they interact with the
detectors on Earth. These three facts, nowadays well established, are the
result of more than six decades of research in theoretical and experimental
astrophysics and particle physics.

More specifically the fusion of protons into $\mbox{}^{4}\text{He}$ is known
to proceed via two different mechanisms: the pp-chain and the CNO-cycle
Bethe:1939bt ; Bahcall:1989ks . In each of these two mechanisms electron
neutrinos are produced in a well-known subset of reactions. In particular, in
the pp-chain five fusion reactions among elements lighter than $A=8$ produce
neutrinos which are labeled by the parent reaction as pp,
$\mbox{}^{7}\text{Be}$, pep, $\mbox{}^{8}\text{B}$, and hep neutrinos. In the
CNO-cycle the abundance of C and N acts as a catalyst, and the
$\mbox{}^{13}\text{N}$ and $\mbox{}^{15}\text{O}$ beta decays provide the
primary source of neutrinos, while $\mbox{}^{17}\text{F}$ beta decay produces
a subdominant flux. For each of these eight processes the spectral energy
shapes of the produced neutrinos is known, but the calculation of the rate of
neutrinos produced in each reaction requires dedicated modeling of the Sun.

Over the years several Standard Solar Models (SSMs), able to describe the
properties of the Sun and its evolution after entering the main sequence, have
been constructed with increasing level of refinement Bahcall:1987jc ;
TurckChieze:1988tj ; Bahcall:1992hn ; Bahcall:1995bt ; Bahcall:2000nu ;
Bahcall:2004pz ; PenaGaray:2008qe ; Serenelli:2011py ; Vinyoles:2016djt . Such
models are numerical calculations calibrated to match present-day surface
properties of the Sun, and developed under the assumption that the Sun was
initially chemically homogeneous and that mass loss is negligible at all
moments during its evolution up to the present solar age $\tau_{\odot}=4.57$
Gyr. The calibration is done in order to satisfy the constraints imposed by
the current solar luminosity $L_{\odot}$, radius $R_{\odot}$, and surface
metal to hydrogen abundance ratio $(Z/X)_{\odot}$. Refinements introduced over
the years include more precise observational and experimental information
about the input parameters (such as nuclear reaction rates and the surface
abundances of different elements), more accurate calculations of constituent
quantities (such as radiative opacity and equation of state), the inclusion of
new physical effects (such as element diffusion), and the development of
faster computers and more precise stellar evolution codes.

The detection of solar neutrinos, with their extremely small interaction cross
sections, enable us to see into the solar interior and verify directly our
understanding of the Sun Bahcall:1964gx  – provided, of course, that one
counts with an established model of the physics effects relevant to their
production, interaction, and propagation. The Standard Model of particle
physics was thought to be such established framework, but it badly failed at
the first attempt of this task giving rise to the so-called “solar neutrino
problem” Bahcall:1968hc ; Bahcall:1976zz . Fortunately we lay here, almost
fifty years after that first realization of the problem, with a different but
well established framework for the relevant effects in solar neutrino
propagation. A framework in which the three flavour neutrinos ($\nu_{e}$,
$\nu_{\mu}$, $\nu_{\tau}$) of the Standard Model are mixed quantum
superpositions of three massive states $\nu_{i}$ ($i=1,2,3$) with different
masses. This allows for the flavour of the neutrino to oscillate from
production to detection Pontecorvo:1967fh ; Gribov:1968kq , and for non-
trivial effects (the so called LMA-MSW Wolfenstein:1977ue ; Mikheev:1986gs
flavour transitions) to take place when crossing dense regions of matter.
Furthermore, due to the wealth of experiments exploring neutrino oscillations,
the value of the neutrino properties governing the propagation of solar
neutrinos, mass differences and mixing angles, are now precisely and
independently known.

Armed with this robust particle physics framework for neutrino production,
propagation, and detection, it is possible to turn to the observation of solar
neutrino experiments to test and refine the SSM. Unfortunately soon after the
particle physics side of the exercise was clarified, the construction of the
SSM run into a new puzzle: the so called “solar composition problem”. In
brief, SSMs built in the 1990’s using the abundances of heavy elements on the
surface of the Sun from Ref. Grevesse1998  (GS98) had notable successes in
predicting other observations, in particular helioseismology measurements such
as the radial distributions of sound speed and density Bahcall:1992hn ;
Bahcall:1995bt ; Bahcall:2000nu ; Bahcall:2004pz . But in the 2000’s new
determinations of these abundances became available and pointed towards
substantially lower values, as summarized in Ref. Asplund2009  (AGSS09). The
SSMs built incorporating such lower metallicities failed at explaining the
helioseismic observations Bahcall:2004yr . For almost two decades there was no
successful solution of this puzzle as changes in the modeling of the Sun did
not seem able to account for this discrepancy Castro:2007 ; Guzik:2010 ;
Serenelli:2011py . Consequently two different sets of SSMs were built, each
based on the corresponding set of solar abundances Serenelli:2009yc ;
Serenelli:2011py ; Vinyoles:2016djt .

With this in mind, in Refs. Gonzalez-Garcia:2009dpj ; Bergstrom:2016cbh  we
performed solar model independent analysis of the solar and terrestrial
neutrino data available at the time, in the framework of three-neutrino masses
and mixing, where the flavour parameters and all the solar neutrino fluxes
were simultaneously determined with a minimum set of theoretical priors. The
results were compared with the two variants of the SSM, but they were not
precise enough to provide a significant discrimination.

Since then there have been a number of developments. First of all, a
substantial amount of relevant data has been accumulated, in particular the
full spectral information of the Phase-II Borexino:2017rsf  and Phase-III
BOREXINO:2022abl  of Borexino and their results based on the correlated
integrated directionality (CID) method Borexino:2023puw . All of them have
resulted into the first positive observation of the neutrino fluxes produced
in the CNO-cycle which are particularly relevant for discrimination among the
SSMs.

On the model front, an update of the AGSS09 results was recently presented by
the same group (AAG21) Asplund2021 , though leading only to a slight revision
upwards of the solar metallicity. Most interestingly, almost simultaneously a
new set of results (MB22) Magg:2022rxb , based on similar methodologies and
techniques but with different atomic input data for the critical oxygen lines
among other differences, led to a substantial change in solar elemental
abundances with respect to AGSS09 (see the original reference for details).
The outcome is a set of solar abundances based on three-dimensional radiation
hydrodynamic solar atmosphere models and line formation treated under non-
local thermodynamic equilibrium that yields a total solar metallicity
comparable to those of the “high-metallicity” results by GS98.

Another issue which has come up in the interpretation of the solar neutrino
results is the appearance of the so-called “gallium anomaly”. In brief, it
accounts for the deficit of the rate of events observed in Gallium source
experiments with respect to the expectation. It was originally observed in the
calibration of the gallium solar-neutrino detectors GALLEX GALLEX:1997lja ;
Kaether:2010ag  and SAGE SAGE:1998fvr ; Abdurashitov:2005tb  with radioactive
$\mbox{}^{51}\text{Cr}$ and $\mbox{}^{37}\text{Ar}$ sources, and it has been
recently confirmed by the BEST collaboration with a dedicated source
experiment using a $\mbox{}^{51}\text{Cr}$ source with high statistical
significance Barinov:2021asz ; Barinov:2022wfh .

The solution of this puzzle is an open question in neutrino physics (see Ref.
Brdar:2023cms  and reference therein for a recent discussion of – mostly
unsuccessful – attempts at explanations in terms of standard and non-standard
physics scenarios). In particular, in the framework of 3$\nu$ oscillations the
attempts at explanation (or at least alleviation) of the anomaly invoke the
uncertainties of the capture cross section Berryman:2021yan ; Giunti:2022xat ;
Elliott:2023xkb . With this motivation, in this work we have studied the
(in)sensitivity of our results to the intrinsic uncertainty on the observed
neutrino rates in the Gallium experiments posed by possible modification of
the capture cross section in Gallium, or equivalently, of the detection
efficiency of the Gallium solar neutrino experiments.

All these developments motivate the new analysis which we present in this
paper with the following outline. In Sec. 2 we describe the assumptions and
methodology followed in our study of the neutrino data. As mentioned above,
this work builds upon our previous solar flux determination in Refs. Gonzalez-
Garcia:2009dpj ; Bergstrom:2016cbh . Thus in Sec. 2, for convenience, we
summarize the most prominent elements common to those analyses, but most
importantly, we detail the relevant points in which the present analysis
method deviates from them. The new determination of the solar fluxes is
presented in Sec. 3 where we also discuss and quantify the role of the Gallium
experiments and address their robustness with respect to the Gallium anomaly.
In Sec. 4 we have a closer look at the determination of the neutrino fluxes
from the CNO-cycle and its dependence on the assumptions on the relative
normalization of the fluxes produced in the three relevant reactions. In Sec.
5 we compare our determined fluxes with the predictions of the SSMs in the
form of a _parameter goodness of fit_ test, and quantify the output of the
test for the assumptions in the analysis. We summarize our findings in Sec. 6.
The article is supplemented with a detailed Appendix A in which we document
our analysis of the Borexino III spectral data (and also their recent analysis
employing the CID method).

##  2 Analysis framework

In the analysis of solar neutrino experiments we include the total rates from
the radiochemical experiments Chlorine Cleveland:1998nv , Gallex/GNO
Kaether:2010ag , and SAGE Abdurashitov:2009tn , the spectral and day-night
data from the four phases of Super-Kamiokande Hosaka:2005um ; Cravens:2008aa ;
Abe:2010hy ; SK:nu2020 , the results of the three phases of SNO in terms of
the parametrization given in their combined analysis Aharmim:2011vm , and the
full spectra from Borexino Phase-I Bellini:2011rx , Phase-II Borexino:2017rsf
, and Phase-III BOREXINO:2022abl , together with their latest results based on
the correlated integrated directionality (CID) method Borexino:2023puw .
Details of our Borexino Phase-III and CID data analysis, which is totally
novel in this article, are presented in Appendix A.

In the framework of three neutrino masses and mixing the expected values for
these solar neutrino observables depend on the parameters $\Delta m^{2}_{21}$,
$\theta_{12}$, and $\theta_{13}$ as well as on the normalizations of the eight
solar fluxes. Thus besides solar experiments, we also include in the analysis
the separate DS1, DS2, DS3 spectra from KamLAND Gando:2013nba  which in the
framework of three neutrino mixing also yield information on the parameters
$\Delta m^{2}_{21}$, $\theta_{12}$, and $\theta_{13}$.

In what follows we will use as normalization parameters for the solar fluxes
the reduced quantities:

$f_{i}=\frac{\Phi_{i}}{\Phi_{i}^{\text{ref}}}$ (1)

with $i={\text{pp}}$, $\mbox{}^{7}\text{Be}$, pep, $\mbox{}^{13}\text{N}$,
$\mbox{}^{15}\text{O}$, $\mbox{}^{17}\text{F}$, $\mbox{}^{8}\text{B}$, and
hep. In this work the numerical values of $\Phi_{i}^{\text{ref}}$ are set to
the predictions of the latest GS98 solar model, presented in Ref. Magg:2022rxb
; B23Fluxes . They are listed in Table 1. The methodology of the analysis
presented in this work builds upon our previous solar flux determination in
Refs. Gonzalez-Garcia:2009dpj ; Bergstrom:2016cbh , which we briefly summarize
here for convenience, but it also presents a number of differences besides the
additional data included as described next.

The theoretical predictions for the solar and KamLAND observables depend on
eleven parameters: the eight reduced solar fluxes $f_{{\text{pp}}}$, …,
$f_{{\text{hep}}}$, and the three relevant oscillation parameters $\Delta
m^{2}_{21}$, $\theta_{12}$, $\theta_{13}$. In our analysis we include as well
the complementary information on $\theta_{13}$ obtained after marginalizing
over $\Delta m^{2}_{3\ell}$, $\theta_{23}$ and $\delta_{\text{CP}}$ the
results of all the other oscillation experiments considered in NuFIT-5.2
nufit-5.2 . This results into a prior $\sin^{2}\theta_{13}=0.0223\pm 0.0006$,
i.e., $\theta_{13}=8.59^{\circ}\,(1\pm 0.014)$. Given the weak dependence of
the solar and KamLAND observables on $\theta_{13}$, including such prior
yields results which are indistinguishable from just fixing the value of
$\bar{\theta}_{13}=8.59^{\circ}$.

Throughout this work, we follow a frequentist approach in order to determine
the allowed confidence regions for these parameters (unlike in our former
works Gonzalez-Garcia:2009dpj ; Bergstrom:2016cbh  where we used instead a
Bayesian analysis to reconstruct their posterior probability distribution
function). To this end we make use of the experimental data from the various
solar and KamLAND samples ($D_{\text{solar}}$ and $D_{\text{KamLAND}}$,
respectively) as well as the corresponding theoretical predictions (which
depends on ten free parameters, as explained above) to build the $\chi^{2}$
statistical function

$\chi^{2}_{\text{global}}(\vec{\omega}_{\text{flux}},\,\vec{\omega}_{\text{osc}})\equiv\chi^{2}_{\text{solar}}(D_{\text{solar}}\,|\,\vec{\omega}_{\text{flux}},\,\vec{\omega}_{\text{osc}})+\chi^{2}_{\text{KamLAND}}(D_{\text{KamLAND}}\,|\,\vec{\omega}_{\text{osc}})\,,$
(2)

with
$\vec{\omega}_{\text{flux}}\equiv(f_{{\text{pp}}},\,\dots,\,f_{{\text{hep}}})$
and $\vec{\omega}_{\text{osc}}\equiv(\Delta
m^{2}_{21},\,\theta_{12},\,\bar{\theta}_{13})$. In order to scan this
multidimensional parameter space efficiently, we make use of the MultiNest
Feroz:2013hea ; Feroz:2008xx and Diver Martinez:2017lzg algorithms.

Flux |  $\Phi_{i}^{\text{ref}}$ [$\text{cm}^{-2}\,\text{s}^{-1}$] |  $\alpha_{i}$ [MeV] | $\beta_{i}$
---|---|---|---
pp |  $5.960\times 10^{10}$ |  $13.0987$ | $9.1862\times 10^{-1}$
$\mbox{}^{7}\text{Be}$ |  $4.854\times 10^{9\hphantom{0}}$ |  $12.6008$ | $7.1972\times 10^{-2}$
pep |  $1.425\times 10^{8\hphantom{0}}$ |  $11.9193$ | $1.9986\times 10^{-3}$
$\mbox{}^{13}\text{N}$ |  $2.795\times 10^{8\hphantom{0}}$ |  $\hphantom{0}3.4577$ | $1.1372\times 10^{-3}$
$\mbox{}^{15}\text{O}$ |  $2.067\times 10^{8\hphantom{0}}$ |  $21.5706$ | $5.2464\times 10^{-3}$
$\mbox{}^{17}\text{F}$ |  $5.350\times 10^{6\hphantom{0}}$ |  $\hphantom{0}2.3630$ | $1.4876\times 10^{-5}$
$\mbox{}^{8}\text{B}$ |  $5.025\times 10^{6\hphantom{0}}$ |  $\hphantom{0}6.6305$ | $3.9205\times 10^{-5}$
hep |  $7.950\times 10^{3\hphantom{0}}$ |  $\hphantom{0}3.7370$ | $3.4959\times 10^{-8}$
Table 1: The reference neutrino fluxes $\Phi_{i}^{\text{ref}}$ used for
normalization (from Ref. Magg:2022rxb ; B23Fluxes ), the energy $\alpha_{i}$
provided to the star by nuclear fusion reactions associated with the
$i^{\text{th}}$ neutrino flux (taken from Ref. Bahcall:2001pf ), and the
fractional contribution $\beta_{i}$ of the $i^{\text{th}}$ nuclear reaction to
the total solar luminosity.

The allowed range for the solar fluxes is further reduced when imposing the
so-called “luminosity constraint”, i.e., the requirement that the overall sum
of the thermal energy generated together with each solar neutrino flux
coincides with the solar luminosity Spiro:1990vi :

$\frac{L_{\odot}}{4\pi\,(\text{A.U.})^{2}}=\sum_{i=1}^{8}\alpha_{i}\Phi_{i}\,.$
(3)

Here the constant $\alpha_{i}$ is the energy released into the star by the
nuclear fusion reactions associated with the $i^{\text{th}}$ neutrino flux;
its numerical value is independent of details of the solar model to an
accuracy of one part in $10^{4}$ or better Bahcall:2001pf . A detailed
derivation of this equation and the numerical values of the coefficients
$\alpha_{i}$ is presented in Ref. Bahcall:2001pf , with a refinement following
in 2021JPhG...48a5201V . Here, we use the original formulation from
Bahcall:2001pf . The coefficients are listed in Table 1. In terms of the
reduced fluxes Eq. (3) can be written as:

$1=\sum_{i=1}^{8}\beta_{i}f_{i}\quad\text{with}\quad\beta_{i}\equiv\frac{\alpha_{i}\Phi_{i}^{\text{ref}}}{L_{\odot}\big{/}[4\pi\,(\text{A.U.})^{2}]}$
(4)

where $\beta_{i}$ is the fractional contribution to the total solar luminosity
of the nuclear reactions responsible for the production of the
$\Phi_{i}^{\text{ref}}$ neutrino flux. In Refs. Gonzalez-Garcia:2009dpj ;
Bergstrom:2016cbh  we adopted the best-estimate value for the solar luminosity
$L_{\odot}\big{/}[4\pi\,(\text{A.U.})^{2}]=8.5272\times
10^{11}\,\text{MeV}\,\text{cm}^{-2}\,\text{s}^{-1}$ given in Ref.
Bahcall:2001pf , which was obtained from all the available satellite data
Frohlich1998 . This value was revised in Ref. kopp2011 using an updated
catalog and calibration methodology (see Ref. Scafetta_2014 for a detailed
comparative discussion), yielding a slightly lower result which is now the
reference value listed by the PDG Workman:2022ynf and leads to
$L_{\odot}\big{/}[4\pi\,(\text{A.U.})^{2}]=8.4984\times
10^{11}\,\text{MeV}\,\text{cm}^{-2}\,\text{s}^{-1}$. In this work we adopt
this new value when evaluating the $\beta_{i}$ coefficients listed in Table 1.
Furthermore, in order to account for the systematics in the extraction of the
solar luminosity we now assign an uncertainty of $0.34\%$ to the constraint in
Eq. (4), which we conservatively derive from the range of variation of the
estimates of $L_{\odot}$. In what follows we will present results with and
without imposing the luminosity constraint. For the analysis including the
luminosity constraint we add a prior

$\chi^{2}_{\text{LC}}(\vec{\omega}_{\text{flux}})=\frac{\left(\displaystyle
1-\sum_{i=1}^{8}\beta_{i}f_{i}\right)^{2}}{(0.0034)^{2}}$ (5)

Besides the imposition of the luminosity constraint in some of the analysis,
the flux normalizations are allowed to vary freely within a set of physical
constraints. In particular:

  * •

The fluxes must be positive:

$\Phi_{i}\geq 0\quad\Rightarrow\quad f_{i}\geq 0\,.$ (6)

  * •

Consistency in the pp-chain implies that the number of nuclear reactions
terminating the pp-chain should not exceed the number of nuclear reactions
which initiate it Bahcall:1995rs ; Bahcall:2001pf :

$\Phi_{{\mbox{}^{7}\text{Be}}}+\Phi_{{\mbox{}^{8}\text{B}}}\leq\Phi_{{\text{pp}}}+\Phi_{{\text{pep}}}\\\
\Rightarrow\quad 8.12\times 10^{-2}f_{{\mbox{}^{7}\text{Be}}}+8.42\times
10^{-5}f_{{\mbox{}^{8}\text{B}}}\leq f_{{\text{pp}}}+2.38\times
10^{-3}f_{{\text{pep}}}\,.$ (7)

  * •

The ratio of the pep neutrino flux to the pp neutrino flux is fixed to high
accuracy because they have the same nuclear matrix element. We have
constrained this ratio to match the average of the values in the five B23 SSMs
(Sec. 5), with $1\sigma$ Gaussian uncertainty given by the difference between
the values in the five models

$\frac{f_{{\text{pep}}}}{f_{{\text{pp}}}}=1.004\pm 0.018\,.$ (8)

Technically we implement this constraint by adding a Gaussian prior

$\chi^{2}_{\text{pep/pp}}(f_{{\text{pp}}},\,f_{{\text{pep}}})\equiv\left(\frac{f_{{\text{pep}}}\big{/}f_{{\text{pp}}}-1.004}{0.018}\right)^{2}.$
(9)

  * •

For the CNO fluxes ($f\Phi_{{\mbox{}^{13}\text{N}}}$,
$\Phi_{{\mbox{}^{15}\text{O}}}$, and $\Phi_{{\mbox{}^{17}\text{F}}}$) a
minimum set of assumptions required by consistency are:

    * –

The ${\mbox{}^{14}\text{N}}(p,\gamma){\mbox{}^{15}\text{O}}$ reaction must be
the slowest process in the main branch of the CNO-cycle Bahcall:1995rs :

$\Phi_{{\mbox{}^{15}\text{O}}}\leq\Phi_{{\mbox{}^{13}\text{N}}}\quad\Rightarrow\quad
f_{{\mbox{}^{15}\text{O}}}\leq 1.35\,f_{{\mbox{}^{13}\text{N}}}$ (10)

    * –

the CNO-II branch must be subdominant:

$\Phi_{{\mbox{}^{17}\text{F}}}\leq\Phi_{{\mbox{}^{15}\text{O}}}\quad\Rightarrow\quad
f_{{\mbox{}^{17}\text{F}}}\leq 40\,f_{{\mbox{}^{15}\text{O}}}\,.$ (11)

The conditions quoted above are all dictated by solar physics. However, more
practical reasons require that the CNO fluxes are treated with a special care.
As discussed in detail in Appendix A.2, the analysis of the Borexino Phase-III
spectra in Ref. BOREXINO:2020aww (which we closely reproduce) has been
optimized by the collaboration to maximize the sensitivity to the overall CNO
production rate, and therefore it may not be directly applicable to a
situation where the three $\mbox{}^{13}\text{N}$, $\mbox{}^{15}\text{O}$ and
$\mbox{}^{17}\text{F}$ flux normalizations are left totally free, subject only
to the conditions in Eqs. (10) and (11). Hence, following the approach of the
Borexino collaboration in Ref. BOREXINO:2020aww , we first perform an analysis
where the three CNO components are all scaled simultaneously by a unique
normalization parameter while their ratios are kept fixed as predicted by the
SSMs. In order to avoid a bias towards one of the different versions of the
SSM we have constrained the two ratios to match the average of the five B23
SSMs values

$\frac{\Phi_{{\mbox{}^{15}\text{O}}}}{\Phi_{{\mbox{}^{13}\text{N}}}}=0.73\enspace\text{and}\enspace\frac{\Phi_{{\mbox{}^{17}\text{F}}}}{\Phi_{{\mbox{}^{13}\text{N}}}}=0.016\quad\Rightarrow\quad\frac{f_{{\mbox{}^{15}\text{O}}}}{f_{{\mbox{}^{13}\text{N}}}}=0.98\enspace\text{and}\enspace\frac{f_{{\mbox{}^{17}\text{F}}}}{f_{{\mbox{}^{13}\text{N}}}}=0.85\,.$
(12)

In these analysis, which we label <<CNO-Rfixed>>, the conditions in Eq. (12)
effectively reduces the number of free parameters from ten to eight, namely
the two oscillation parameters in $\vec{\omega}_{\text{osc}}$ and six flux
normalizations in

$\vec{\omega}_{\text{flux}}^{\text{CNO-
Rfixed}}\equiv(f_{{\text{pp}}},\;f_{{\mbox{}^{7}\text{Be}}},\;f_{{\text{pep}}},\;f_{{\mbox{}^{13}\text{N}}},\;f_{{\mbox{}^{15}\text{O}}}=0.98\,f_{{\mbox{}^{13}\text{N}}},\;f_{{\mbox{}^{17}\text{F}}}=0.85\,f_{{\mbox{}^{13}\text{N}}},\;f_{{\mbox{}^{8}\text{B}}},\;f_{{\text{hep}}})\,.$

In Sec. 4 we will discuss and quantify the effect of relaxing the condition of
fixed CNO ratios.

##  3 New determination of solar neutrino fluxes

Figure 1: Constraints from our global analysis on the solar neutrino fluxes
for the analysis with luminosity constraint and fixed ratios of the CNO fluxes
(see Eq. (13)). Each panel shows a two-dimensional projection of the allowed
multidimensional parameter space after minimization with respect to the
undisplayed parameters. The regions correspond to 90% and 99% CL (2 d.o.f.).
The curves in the rightmost panels show the marginalized one-dimensional
$\Delta\chi^{2}_{\text{wLC,CNO-Rfixed}}$ for each of the parameters.

We present first the results of our analysis with the luminosity constraint
and the ratios of the CNO fluxes fixed by the relations in Eq. (12), so that
altogether for this case we construct the $\chi^{2}$ function

$\chi^{2}_{\text{wLC,CNO-
Rfixed}}\equiv\chi^{2}_{\text{global}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}}^{\text{CNO-
Rfixed}})+\chi^{2}_{\text{pep/pp}}(f_{{\text{pp}}},\,f_{{\text{pep}}})+\chi^{2}_{\text{LC}}(\vec{\omega}_{\text{flux}}^{\text{CNO-
Rfixed}})\,.$ (13)

The results of this analysis are displayed in Fig. 1, where we show the two-
and one-dimensional projections of $\Delta\chi^{2}_{\text{wLC,CNO-Rfixed}}$.
From these results one reads that the ranges at $1\sigma$ (and at the $99\%$
CL in square brackets) for the two oscillation parameters are:

$\displaystyle\Delta m^{2}_{21}$
$\displaystyle=7.43_{-0.30}^{+0.30}\,[{}_{-0.49}^{+0.44}]\times
10^{-5}\text{eV}^{2}\,,$ (14) $\displaystyle\sin^{2}\theta_{12}$
$\displaystyle=0.300_{-0.017}^{+0.020}\,[{}_{-0.027}^{+0.031}]\,,$

which are very similar to the results of NuFIT-5.2 nufit-5.2  with the
expected slight enlargement of the allowed ranges. In other words, within the
$3\nu$ scenario the data is precise enough to simultaneously constraint the
oscillation parameters and the normalizations of the solar flux components
without resulting into a substantial degradation of the former. As for the
solar fluxes, the corresponding ranges read:

$\displaystyle f_{{\text{pp}}}$
$\displaystyle=0.9969_{-0.0039}^{+0.0041}\,[{}_{-0.0092}^{+0.0095}]\,,\qquad$
$\displaystyle\Phi_{{\text{pp}}}$
$\displaystyle=5.941_{-0.023}^{+0.024}\,[{}_{-0.055}^{+0.057}]\times
10^{10}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ (15) $\displaystyle
f_{{\mbox{}^{7}\text{Be}}}$
$\displaystyle=1.019_{-0.017}^{+0.020}\,[{}_{-0.041}^{+0.047}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{7}\text{Be}}}$
$\displaystyle=4.93_{-0.08}^{+0.10}\,[{}_{-0.20}^{+0.23}]\times
10^{9}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle f_{{\text{pep}}}$
$\displaystyle=1.000_{-0.018}^{+0.016}\,[{}_{-0.042}^{+0.041}]\,,\qquad$
$\displaystyle\Phi_{{\text{pep}}}$
$\displaystyle=1.421_{-0.026}^{+0.023}\,[{}_{-0.060}^{+0.058}]\times
10^{8}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{13}\text{N}}}$
$\displaystyle=1.25_{-0.14}^{+0.17}\,[{}_{-0.40}^{+0.47}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{13}\text{N}}}$
$\displaystyle=3.48_{-0.40}^{+0.47}\,[{}_{-1.10}^{+1.30}]\times
10^{8}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{15}\text{O}}}$
$\displaystyle=1.22_{-0.14}^{+0.17}\,[{}_{-0.39}^{+0.46}]\qquad$
$\displaystyle\Phi_{{\mbox{}^{15}\text{O}}}$
$\displaystyle=2.53_{-0.29}^{+0.34}\,[{}_{-0.80}^{+0.94}]\times
10^{8}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{17}\text{F}}}$
$\displaystyle=1.03_{-0.20}^{+0.20}\,[{}_{-0.48}^{+0.47}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{17}\text{F}}}$
$\displaystyle=5.51_{-0.63}^{+0.75}\,[{}_{-1.75}^{+2.06}]\times
10^{7}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{8}\text{B}}}$
$\displaystyle=1.036_{-0.020}^{+0.020}\,[{}_{-0.048}^{+0.047}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{8}\text{B}}}$
$\displaystyle=5.20_{-0.10}^{+0.10}\,[{}_{-0.24}^{+0.24}]\times
10^{6}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle f_{{\text{hep}}}$
$\displaystyle=3.8_{-1.2}^{+1.1}\,[{}_{-2.7}^{+2.7}]\,,\qquad$
$\displaystyle\Phi_{{\text{hep}}}$
$\displaystyle=3.0_{-1.0}^{+0.9}\,[{}_{-2.1}^{+2.2}]\times
10^{4}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,.$

Notice that in Fig. 1 we separately plot the ranges for the three CNO flux
normalization parameters, however they are fully correlated since their ratios
are fixed, which explains the thin straight-line shape of the regions as seen
in the three corresponding panels. Compared to the results from our previous
analysis we now find that all the fluxes are clearly determined to be non-
zero, while in Refs. Gonzalez-Garcia:2009dpj ; Bergstrom:2016cbh  only an
upper bound for the CNO fluxes was found. This is a direct consequence of the
positive evidence of neutrinos produced in the CNO cycle provided by Borexino
Phase-III spectral data, which is here confirmed in a fully consistent global
analysis. We will discuss this point in more detail in Sec. 4. We also observe
that the inclusion of the full statistics of Borexino has improved the
determination of $f_{{\mbox{}^{7}\text{Be}}}$ by a factor $\mathcal{O}(3)$.

Figure 1 exhibits the expected correlation between the allowed ranges of the
pp and pep fluxes, which is a consequence of the relation (8). This
correlation is somewhat weaker than what observed in the corresponding
analysis in Ref. Bergstrom:2016cbh  because the spectral information from
Borexino Phase-II and Phase-III provides now some independent information on
$f_{{\text{pep}}}$. We also observe the presence of anticorrelation between
the allowed ranges of the two most intense fluxes, pp and
$\mbox{}^{7}\text{Be}$, as dictated by the luminosity constraint (see
comparison with Fig. 2). Finally we notice that the allowed ranges of
$f_{{\mbox{}^{7}\text{Be}}}$ and $f_{{\mbox{}^{8}\text{B}}}$ – the two most
precise directly determined flux normalizations irrespective of the luminosity
constraint (see Fig. 2) – are anticorrelated. This is a direct consequence of
the different dependence of the survival probability with
$\sin^{2}\theta_{12}$ in their respective energy ranges. $\mbox{}^{8}\text{B}$
neutrinos have energies of the order of several MeV for which the flavour
transition occurs in the MSW regime and the survival probability
$P_{ee}\propto\sin^{2}\theta_{12}$. Hence an increase in $\sin^{2}\theta_{12}$
must be compensated by a decrease of $f_{{\mbox{}^{8}\text{B}}}$ to get the
correct number of events, which leads to the anticorrelation between the
$\sin^{2}\theta_{12}$ and $f_{{\mbox{}^{8}\text{B}}}$ seen in the
corresponding panel in Fig. 1. On the contrary, most $\mbox{}^{7}\text{Be}$
neutrinos have 0.86 MeV (some have 0.38 MeV) and for that energy the flavour
transition occurs in the transition regime between MSW and vacuum average
oscillations for which $P_{ee}$ decreases with $\sin^{2}\theta_{12}$. Hence
the correlation between $\sin^{2}\theta_{12}$ and $f_{{\mbox{}^{7}\text{Be}}}$
seen in the corresponding panel. Altogether, this leads to the anticorrelation
observed between $f_{{\mbox{}^{7}\text{Be}}}$ and $f_{{\mbox{}^{8}\text{B}}}$.
This was already mildly present in the results in Ref. Bergstrom:2016cbh  but
it is now a more prominent feature because of the most precise determination
of $f_{{\mbox{}^{7}\text{Be}}}$.

All these results imply the following share of the energy production between
the pp-chain and the CNO-cycle

$\frac{L_{\text{pp-
chain}}}{L_{\odot}}=0.9919_{-0.0030}^{+0.0035}\,[{}_{-0.0077}^{+0.0082}]\quad\Longleftrightarrow\quad\frac{L_{\text{CNO}}}{L_{\odot}}=0.0079_{-0.0011}^{+0.0009}\,[{}_{-0.0026}^{+0.0028}]$
(16)

in perfect agreement with the SSMs which predict
$L_{\text{CNO}}\big{/}L_{\odot}\leq 1\%$ at the $3\sigma$ level. Once again we
notice that in the present analysis the evidence for $L_{\text{CNO}}\neq 0$
clearly stands well above 99% CL.

Figure 2: Same as Fig. 1 but without imposing the luminosity constraint (see
Eq. (17)).

We next show in Fig. 2 the results of the analysis performed without imposing
the luminosity constraint – but still with the ratios of the CNO fluxes fixed
by the relations in Eq. (12) – for which we employ

$\chi^{2}_{\text{woLC,CNO-
Rfixed}}\equiv\chi^{2}_{\text{global}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}}^{\text{CNO-
Rfixed}})+\chi^{2}_{\text{pep/pp}}(f_{{\text{pp}}},\,f_{{\text{pep}}})\,.$
(17)

The allowed ranges for the fluxes in this case are:

$\displaystyle f_{{\text{pp}}}$
$\displaystyle=1.038_{-0.066}^{+0.076}\,[{}_{-0.16}^{+0.18}]\,,\qquad$
$\displaystyle\Phi_{{\text{pp}}}$
$\displaystyle=6.19_{-0.39}^{+0.45}\,[{}_{-1.0}^{+1.1}]\times
10^{10}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ (18) $\displaystyle
f_{{\mbox{}^{7}\text{Be}}}$
$\displaystyle=1.022_{-0.018}^{+0.022}\,[{}_{-0.042}^{+0.051}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{7}\text{Be}}}$
$\displaystyle=4.95_{-0.089}^{+0.11}\,[{}_{-0.22}^{+0.25}]\times
10^{9}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle f_{{\text{pep}}}$
$\displaystyle=1.039_{-0.065}^{+0.082}\,[{}_{-0.16}^{+0.19}]\,,\qquad$
$\displaystyle\Phi_{{\text{pep}}}$
$\displaystyle=1.48_{-0.09}^{+0.11}\,[{}_{-0.22}^{+0.26}]\times
10^{8}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{13}\text{N}}}$
$\displaystyle=1.16_{-0.19}^{+0.19}\,[{}_{-0.45}^{+0.50}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{13}\text{N}}}$
$\displaystyle=3.32_{-0.54}^{+0.53}\,[{}_{-1.24}^{+1.40}]\times
10^{8}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{15}\text{O}}}$
$\displaystyle=1.16_{-0.19}^{+0.19}\,[{}_{-0.44}^{+0.49}]\qquad$
$\displaystyle\Phi_{{\mbox{}^{15}\text{O}}}$
$\displaystyle=2.41_{-0.39}^{+0.38}\,[{}_{-0.90}^{+1.02}]\times
10^{8}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{17}\text{F}}}$
$\displaystyle=1.01_{-0.16}^{+0.16}\,[{}_{-0.38}^{+0.45}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{17}\text{F}}}$
$\displaystyle=5.25_{-0.85}^{+0.84}\,[{}_{-1.97}^{+2.21}]\times
10^{6}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle
f_{{\mbox{}^{8}\text{B}}}$
$\displaystyle=1.034_{-0.021}^{+0.020}\,[{}_{-0.051}^{+0.052}]\,,\qquad$
$\displaystyle\Phi_{{\mbox{}^{8}\text{B}}}$
$\displaystyle=5.192_{-0.11}^{+0.10}\,[{}_{-0.26}^{+0.26}]\times
10^{6}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,,$ $\displaystyle f_{{\text{hep}}}$
$\displaystyle=3.6_{-1.1}^{+1.2}\,[{}_{-2.6}^{+3.0}]\,,\qquad$
$\displaystyle\Phi_{{\text{hep}}}$
$\displaystyle=2.9_{-0.9}^{+1.0}\,[{}_{-2.1}^{+2.4}]\times
10^{4}~{}\text{cm}^{-2}~{}\text{s}^{-1}\,.$

As expected, the pp flux is the most affected by the release of the luminosity
constraint as it is this reaction which gives the largest contribution to the
solar energy production and therefore its associated neutrino flux is the one
more strongly bounded when imposing the luminosity constraint. The pep flux is
also affected due to its strong correlation with the pp flux, Eq. (8). The CNO
fluxes are mildly affected in an indirect way due to the modified contribution
of the pep fluxes to the Borexino spectra.

Thus we find that the energy production in the pp-chain and the CNO-cycle
without imposing the luminosity constraint are given by:

$\frac{L_{\text{pp-
chain}}}{L_{\odot}}=1.030_{-0.061}^{+0.070}\,[{}_{-0.15}^{+0.17}]\qquad\text{and}\qquad\frac{L_{\text{CNO}}}{L_{\odot}}=0.0075_{-0.0013}^{+0.0013}\,[{}_{-0.0029}^{+0.0030}]\,.$
(19)

Comparing Eqs. (16) and (19) we see that while the amount of energy produced
in the CNO cycle is about the same in both analysis, releasing the luminosity
constraint allows for larger production of energy in the pp-chain. So in this
case we find that the present value of the ratio of the neutrino-inferred
solar luminosity, $L_{\odot}\text{(neutrino-inferred)}$, to the photon
measured luminosity $L_{\odot}$ is:

$\frac{L_{\odot}\text{(neutrino-
inferred)}}{L_{\odot}}=1.038_{-0.060}^{+0.069}\,[{}_{-0.15}^{+0.17}]\,.$ (20)

The neutrino-inferred luminosity is in good agreement with the one measured in
photons, with a $1\sigma$ uncertainty of $\sim 6\%$. This represents only a
very small variation with respect to the previous best determination
Bergstrom:2016cbh . Such result is expected because the determination of the
pp flux, which, as mentioned above gives the largest contribution to the
neutrino-inferred solar luminosity, has not improved sensibly with the
inclusion of the full statistics of the phases II and III of Borexino.

Figure 3: Dependence of the determination of the pp and pep fluxes on the
assumptions about the Gallium experiments included in the analysis. The upper
(lower) panels show the results of different variants of the analysis <<CNO-
Rfixed>> with (without) luminosity constraint. See text for details.

We finish this section by discussing the role of the Gallium experiments in
these results with the aim of addressing the possible impact of the Gallium
anomaly Laveder:2007zz ; Acero:2007su ; Giunti:2010zu . As described in the
introduction, this anomaly consists in a deficit of the event rate observed in
Gallium source experiments with respect to the expectation, which represents
an obvious puzzle for the interpretation of the results of the solar neutrino
Gallium measurements. In this work we assume the well established standard
$3\nu$ oscillation scenario and in this context the attempts at explanation
(or at least alleviation) of the anomaly invoke the uncertainties of the
capture cross section Berryman:2021yan ; Giunti:2022xat ; Elliott:2023xkb .
Thus the open question posed by the Gallium anomaly is the possible impact of
such modification of the cross section in the results of our fit.

In order to quantify this we performed two additional variants of our
analysis. In the first one we introduce an additional parameter,
$f_{\text{GA}}$, which multiplies the predicted event rates from all solar
fluxes in the Gallium experiments. This parameter is left free to vary in the
fits and would mimic an energy independent modification of the capture cross
section (or equivalently of the detection efficiency). In the second variant
we simply drop Gallium experiments from our global fit.

The results of these explorations are shown in Fig. 3 where we plot the most
relevant marginalized one-dimensional projections of $\Delta\chi^{2}$ for
these two variants. The upper (lower) panels correspond to analysis performed
with (without) the luminosity constraint. The left panel shows the projection
over the normalization parameter $f_{\text{GA}}$ obtained in the variant of
the analysis which makes use of this parameter. As seen from the figure, the
results of the fit favour $f_{\text{GA}}$ close to one, or, in other words,
the global analysis of the solar experiments do _not_ support a modification
of the neutrino capture cross section in Gallium (or any other effect inducing
an energy-independent reduction of the detection efficiency in the Gallium
experiments). This is so because, within the $3\nu$ oscillation scenario, the
global fit implies a rate of pp and $\mbox{}^{7}\text{Be}$ neutrinos in the
Gallium experiment which is in good agreement with the luminosity constraint
as well as with the rates observed in Borexino.

On the central and right panel of the figure we show the corresponding
modification of marginalized one-dimensional projections of $\Delta\chi^{2}$
on the pp and pep flux normalizations which are those mostly affected in these
variants. For the sake of comparison, in the upper and lower panels we also
plot the results for the $f_{\text{GA}}=1$ analysis (also visible in the
corresponding windows in Figs. 1 and 2, respectively). The figure illustrates
that once the luminosity constraint is imposed, the determination of the solar
fluxes is totally unaffected by the assumptions about the capture rate in
Gallium. As seen in the lower panels, even without the luminosity constraint
the impact on the pp and pep determination is marginal, which emphasizes the
robustness of the flux determination in Eqs. (15) and (18). This is the case
thanks to the independent precise determination of the pp flux in the phases I
and II in Borexino. Furthermore, the small modification is the same
irrespective of whether the Gallium capture rate is left free or completely
removed from the analysis; this is due to the lack of spectral and day-night
capabilities in Gallium experiments, which prevents them from providing
further information beyond the overall normalization scale of the signal.

##  4 Examination of the determination of the CNO fluxes

As mentioned above, one of the most important developments in the experimental
determination of the solar neutrino fluxes in the last years have been the
evidence of neutrinos produced in the CNO cycle reported by Borexino
BOREXINO:2020aww ; BOREXINO:2022abl ; Borexino:2023puw . The detection was
made possible thanks to a novel method to constrain the rate of the
$\mbox{}^{210}\text{Bi}$ background. In Ref. BOREXINO:2020aww , using a
partial sample of their Phase-III data, the collaboration found a $5.1\sigma$
significance of the CNO flux observation, which increased to $7\sigma$ with
the full Phase-III statistics BOREXINO:2022abl , and to about $8\sigma$ when
combined with the CID method Borexino:2023puw . See appendix A for details.

Key ingredients in the analysis performed by the collaboration in Refs.
BOREXINO:2020aww ; BOREXINO:2022abl ; Borexino:2023puw  (and therefore in the
derivation of these results) are the assumptions about the relative
contribution of the three reactions producing neutrinos in the CNO cycle, as
well as those about other solar fluxes in the same energy range, in particular
the pep neutrinos. In a nutshell, as mentioned above, the collaboration
assumes a common shift of the normalization of the CNO fluxes with respect to
that of the SSM, and it is the evidence of a non-zero value of such
normalization which is quantified in Refs. BOREXINO:2020aww ; BOREXINO:2022abl
; Borexino:2023puw . In what respects the rate from the pep flux, the SSM
expectation was assumed because the Phase-III data by itself does not allow to
constraint simultaneously the CNO and pep flux normalizations.

In this respect, the global analysis presented in the previous section are
performed under the same paradigm of a common shift normalization of the CNO
fluxes, but being global, the pep flux normalization is also simultaneously
fitted. For the sake of comparison we reproduce in Fig. 4 the projection of
the marginalized $\Delta\chi^{2}_{\text{wLC,CNO-Rfixed}}$ (13) and
$\Delta\chi^{2}_{\text{woLC,CNO-Rfixed}}$ (13) on the normalization parameters
for the three CNO fluxes. For convenience we also show the projections as a
function of the total neutrino flux produced in the CNO cycle. As seen in the
figure the results of the analysis (either with or without luminosity
constraint) yield a value of $\Delta\chi^{2}$ well beyond $3\sigma$ for
$\Phi_{{\text{CNO}}}=0$. A dedicated run for this no-CNO scenario case gives
$\Delta\chi^{2}=54$ ($33$) for the analysis with (without) luminosity
constraints, and it is therefore excluded at $7.3\sigma$ ($5.7\sigma$) CL.

In order to study the dependence of the results on the assumption of a unique
common shift of the normalization of three CNO fluxes we explored the
possibility of making a global analysis in which the three normalization
parameters are varied independently. As mentioned above, a priori the three
normalizations only have to be subject to a minimum set of consistency
relations in Eqs. (10) and (11). However, as discussed in detail in Sec. A.2,
the background model in Refs. BOREXINO:2020aww ; BOREXINO:2022abl ;
Borexino:2023puw  only assumes an upper bound on the amount of
$\mbox{}^{210}\text{Bi}$ and cannot be reliably employed to such general
analysis because of the larger degeneracy between the $\mbox{}^{210}\text{Bi}$
background and the $\mbox{}^{13}\text{N}$ flux spectra.

With this limitation in mind, we proceed to perform two alternative analysis
(with and without imposing the luminosity constraints) in which the
normalization of the three CNO fluxes are left free to vary independently but
with ratios constrained within a range broad enough to generously account for
all variants of the B23 SSM, but not to extend into regions of the parameter
space where the assumptions on the background model may not be applicable.
Conservatively neglecting correlations between their theoretical
uncertainties, the neutrino fluxes of SSMs presented in Ref. Magg:2022rxb  and
available publicly through a public repository B23Fluxes  verify

$\frac{f_{{\mbox{}^{15}\text{O}}}}{f_{{\mbox{}^{13}\text{N}}}}=\begin{cases}1.00\,(1\pm
0.24)\\\ 0.95\,(1\pm 0.22)\\\ 0.96\,(1\pm 0.21)\\\ 1.01\,(1\pm 0.23)\\\
1.00\,(1\pm
0.23)\end{cases}\qquad\frac{f_{{\mbox{}^{17}\text{F}}}}{f_{{\mbox{}^{13}\text{N}}}}=\begin{cases}1.00\,(1\pm
0.25)&\text{B23-GS98}\\\ 0.84\,(1\pm 0.23)&\text{B23-AGSS09-met}\\\
0.80\,(1\pm 0.20)&\text{B23-AAG21}\\\ 0.79\,(1\pm 0.22)&\text{B23-MB22-met}\\\
0.79\,(1\pm 0.22)&\text{B23-MB22-phot}\end{cases}$ (21)

Thus in these analyses, here onward labeled <<CNO-Rbound>>, we introduce two
pulls $\xi_{1}$ and $\xi_{2}$ for these two ratios so that

$\vec{\omega}_{\text{flu}}^{\text{CNO-
Rbound}}\equiv(f_{{\text{pp}}},\;f_{{\mbox{}^{7}\text{Be}}},\;f_{{\text{pep}}},\;f_{{\mbox{}^{13}\text{N}}},\;f_{{\mbox{}^{15}\text{O}}}=0.98\,\xi_{1}\,f_{{\mbox{}^{13}\text{N}}},\;f_{{\mbox{}^{17}\text{F}}}=0.85\,\xi_{2}\,f_{{\mbox{}^{13}\text{N}}},\;f_{{\mbox{}^{8}\text{B}}},\;f_{{\text{hep}}})$

and add two Gaussian priors for these pulls, so that the corresponding
$\chi^{2}$ function without the luminosity constraint is:

$\chi^{2}_{\text{woLC,CNO-
Rbound}}\equiv\chi^{2}_{\text{global}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}}^{\text{CNO-
Rbound}})+\chi^{2}_{\text{pep/pp}}(f_{{\text{pp}}},\,f_{{\text{pep}}})\\\
+\frac{(\xi_{1}-1)^{2}}{\sigma_{\xi_{1}}^{2}}+\frac{(\xi_{2}-1)^{2}}{\sigma_{\xi_{2}}^{2}}$
(22)

with $\sigma_{\xi_{1}}=0.26$ and $\sigma_{\xi_{2}}=0.48$, chosen to cover the
ranges in Eq. (21). In addition $f_{{\mbox{}^{13}\text{N}}}$,
$f_{{\mbox{}^{15}\text{O}}}$, and $f_{{\mbox{}^{17}\text{F}}}$ are required to
verify the consistency relations in Eqs. (10) and (11). The $\chi^{2}$
function with the luminosity constraint is obtained by further including the
$\chi^{2}_{\text{LC}}$ prior of Eq. (5):

$\chi^{2}_{\text{wLC,CNO-Rbound}}\equiv\chi^{2}_{\text{woLC,CNO-
Rbound}}+\chi^{2}_{\text{LC}}(\vec{\omega}_{\text{flux}}^{\text{CNO-
Rbound}})\,.$ (23)

Figure 4: One dimensional projections of the global $\Delta\chi^{2}$ of the
for the three neutrino fluxes produced in the CNO-cycle for different
assumptions as labeled in the figure. See text for details.

We plot in Fig. 4 the projection of the marginalized
$\Delta\chi^{2}_{\text{wLC,CNO-Rbound}}$ (23) and
$\Delta\chi^{2}_{\text{woLC,CNO-Rbound}}$ (23) on the normalization parameters
for the three CNO fluxes as well as on the total neutrino flux produced in the
CNO cycle. As seen in the figure, allowing for the ratios of the CNO
normalizations to vary within the intervals (21) has little impact on the
allowed range of the $\mbox{}^{15}\text{O}$ flux and on the lower limit of the
$\mbox{}^{13}\text{N}$ and $\mbox{}^{17}\text{F}$ fluxes. As a consequence,
the CL at which the no-CNO scenario can be ruled out is unaffected. On the
contrary, we see in Fig. 4 that the upper bound on the $\mbox{}^{13}\text{N}$
and $\mbox{}^{17}\text{F}$ fluxes, and therefore of the total neutrino flux
produced in the CNO-cycle, is considerably relaxed.111The allowed ranges for
the fluxes produced in the pp-chain are not substantially modified with
respect to the ones obtained from the ¡¡CNO-Rfixed¿¿ fits, Eqs. (15) and (18).
This is a consequence of the strong degeneracy between the spectrum of events
from these fluxes and those from the $\mbox{}^{210}\text{Bi}$ background
mentioned above, see Fig. 10 and discussion in Sec. A.2. Conversely the fact
that the range of the $\mbox{}^{15}\text{O}$ flux is robust under the
relaxation of the constraints on the CNO flux ratios, means that the high
statistics spectral data of the Phase-III of Borexino holds the potential to
differentiate the event rates from $\mbox{}^{15}\text{O}$ $\nu$’s from those
from $\mbox{}^{13}\text{N}$ and $\mbox{}^{17}\text{F}$ $\nu$’s. The reliable
quantification of this possibility, however, requires the knowledge of the
minimum allowed value of the $\mbox{}^{210}\text{Bi}$ background which so far
has not been presented by the collaboration.

So, let us emphasize that our <<CNO-Rbound>> analysis have been performed with
the aim of testing the effect of relaxing the severe constrains on the CNO
fluxes in the studies of the Borexino collaboration. Our conclusion is that
the statistical significance of the evidence of detection of events produced
by neutrinos from the CNO-cycle is affected very little by the relaxation of
the constraint on their relative ratios. However, their allowed range is, and
this can have an impact when confronting the results of the fit with the
predictions of the SSM as we discuss next.

##  5 Comparison with Standard Solar Models

Next we compare the results of our determination of the solar fluxes with the
expectations from the five B23 solar models: SSMs computed with the abundances
compiled in table 5 of Magg:2022rxb  based on the photospheric and meteoritic
solar mixtures (MB22-phot and MB22-met models, respectively), and with the
Asplund2021  (AAG21), the meteoritic scale from Asplund2009  (AGSS09-met), and
Grevesse1998  (GS98) compositions. We use both MB22-phot and MB22-met for
completeness, although the abundances are very similar in both scales, as
clearly reflected by the results in this section. A similar agreement would be
found using both the meteoritic and photospheric scales from AAG21, and
therefore we use only one scale in this case.222The structures of these
models, as well as the total neutrino fluxes and internal distributions are
available at B23Fluxes .

Figure 5: Marginalized one-dimensional probability distributions for the best
determined solar fluxes in our analysis as compared to the predictions for the
five SSMs in Ref. Magg:2022rxb ; B23Fluxes .

SSMs predict that nuclear energy accounts for all the solar luminosity (barred
about a few parts in $10^{4}$ that are of gravothermal origin) so for all
practical matters the neutrino fluxes predicted by SSMs satisfy the luminosity
constraint. Therefore we compare the expectations of the various SSM models
with the results of our analysis performed with such constraint. In what
respects the assumptions on the CNO fluxes, in order to explore the dependence
of our conclusions on the specific choice of flux ratios we quantify the
results obtained in both the <<FIT=CNO-Rfixed>> analysis (with
$\chi^{2}_{\text{FIT}}$ in Eq. (13)) and the <<FIT=CNO-Rbounded>> one (with
$\chi^{2}_{\text{FIT}}$ in Eq. (23)). For illustration we plot in Fig. 5 the
marginalized one-dimensional probability distributions for the best determined
solar fluxes in such two cases as compared to the predictions for the five B23
SSMs. The probability distributions for our fits are obtained from the one-
dimensional marginalized $\Delta\chi^{2}_{\text{FIT}}(f_{i})$ of the
corresponding analysis as
$P_{\text{FIT}}(f_{i})\propto\exp[-\Delta\chi^{2}_{\text{FIT}}(f_{i})/2]$
normalized to unity. To construct the analogous distributions for each of the
SSMs we use the predictions $\langle f_{i}^{\text{SSM}}\rangle$ for the
fluxes, the relative uncertainties $\sigma_{i}^{\text{SSM}}$ and their
correlations $\rho_{ij}^{\text{SSM}}$ as obtained from Refs. B23Fluxes , and
also assume gaussianity so to build the corresponding
$\chi^{2}_{\text{SSM}}(\vec{\omega}_{\text{flux}})$

$\chi^{2}_{\text{SSM}}(\vec{\omega}_{\text{flux}})=\sum_{i,j}(f_{i}-f_{i}^{\text{SSM}})C^{-1}_{ij}(f_{i}-f_{i}^{\text{SSM}})\quad\text{with}\quad
C_{ab}=\sigma_{a}^{\text{SSM}}\sigma_{b}^{\text{SSM}}\rho_{ab}\,,$ (24)

from which it is trivial to obtain the marginalized one-dimensional
$\Delta\chi^{2}_{\text{SSM}}(f_{i})$ and construct the probability
$P_{\text{SSM}}(f_{i})\propto\exp[-\Delta\chi^{2}_{\text{SSM}}(f_{i})/2]$.

In the frequentist statistical approach, quantitative comparison of a model
prediction for a set of fluxes with the results from the data analysis can be
obtained using the _parameter goodness of fit_ (PG) criterion introduced in
Ref. Maltoni:2003cu , by comparing the minimum value of $\chi^{2}$ function
for the analysis of the data with that obtained for the same analysis adding
the prior imposed by the model.333In this respect it is important to notice
that, in order to avoid any bias towards one of the models in the data
analysis, in both ¡¡CNO-Rfixed¿¿ and ¡¡CNO-Rbound¿¿ cases the assumptions on
the ratios of the three CNO fluxes have been chosen to be “model-democratic”,
i.e., centered at the average of the predictions of the models (and, in the
case of ¡¡CNO-Rbound¿¿, with $1\sigma$ uncertainties covering the $1\sigma$
range allowed by all SSM models). Thus, following Ref. Maltoni:2003cu , we
construct the test statistics

$\Delta\chi^{2}_{\text{FIT,SSM,SET}}=\left.\big{[}\chi^{2}_{\text{FIT}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}}^{\text{FIT}})+\chi^{2}_{\text{SSM,SET}}(\vec{\omega}_{\text{flux}}^{\text{FIT}})\big{]}\right|_{\text{min}}\\\
-\left.\chi^{2}_{\text{FIT}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}}^{\text{FIT}})\right|_{\text{min}}-\left.\chi^{2}_{\text{SSM,SET}}(\vec{\omega}_{\text{flux}}^{\text{FIT}})\right|_{\text{min}}$
(25)

where $\chi^{2}_{\text{SSM,SET}}(\vec{\omega}_{\text{flux}})$ is obtained as
Eq. (24) with $i,j$ (and $a,b$) fluxes restricted to a specific subset as
specified by “SET”. The minimization of each of the terms in Eq. (25) is
performed independently in the corresponding parameter space.
$\Delta\chi^{2}_{\text{FIT,SSM,SET}}$ follows a $\chi^{2}$ distribution with
$n$ degrees of freedom, which, in the present case, coincides with the number
of free parameters in common between
$\chi^{2}_{\text{FIT}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}}^{\text{FIT}})$
and $\chi^{2}_{\text{SSM,SET}}(\vec{\omega}_{\text{flux}}^{\text{FIT}})$.
Notice that, by construction, the result of the test depends on the number of
fluxes to be compared, i.e., on the fluxes in “SET”, both because of the
actual comparison between the measured and predicted values for those specific
fluxes, and because of the change in $n$ with which the $p$-value of the model
is to be computed. This is illustrated in Table 2 where we list the values of
$\Delta\chi^{2}_{\text{FIT,SSM,SET}}$ for different choices of “SET” which we
have labeled as:

SET | constrained fluxes
---|---
FULL | ($f_{{\text{pp}}}$, $f_{{\mbox{}^{7}\text{Be}}}$, $f_{{\text{pep}}}$, $f_{{\mbox{}^{13}\text{N}}}$, $f_{{\mbox{}^{15}\text{O}}}$, $f_{{\mbox{}^{17}\text{F}}}$, $f_{{\mbox{}^{8}\text{B}}}$, $f_{{\text{hep}}}$)
Be+B+CNO | ($f_{{\mbox{}^{7}\text{Be}}}$, $f_{{\mbox{}^{13}\text{N}}}$, $f_{{\mbox{}^{15}\text{O}}}$, $f_{{\mbox{}^{17}\text{F}}}$, $f_{{\mbox{}^{8}\text{B}}}$)
CNO | ($f_{{\mbox{}^{13}\text{N}}}$, $f_{{\mbox{}^{15}\text{O}}}$, $f_{{\mbox{}^{17}\text{F}}}$)
(26)

FIT | B23-SSM | FULL | Be+B+CNO | CNO
---|---|---|---|---
CNO-Rfixed |  | n=6 | n=3 | n=1
| $\Delta\chi^{2}$ | $p_{\text{GF}}$ |  CL [$\sigma$] | $\Delta\chi^{2}$ | $p_{\text{GF}}$ |  CL [$\sigma$] | $\Delta\chi^{2}$ | $p_{\text{GF}}$ |  CL [$\sigma$]
AGSS09-met | 14.5 | 0.024 | 2.3 | 9.8 | 0.020 |  2.30 | 7.2 | 0.0073 | 2.7
GS98 |  08.1 |  0.240 | 1.2 | 3.0 |  0.390 | 0.86 | 2.4 |  0.1200 | 1.5
AAG21 | 12.5 | 0.052 | 1.9 | 7.8 |  0.050 |  2.00 | 6.2 |  0.0130 | 2.5
MB22-met/phot |  07.1 |  0.310 | 1.0 | 2.2 |  0.530 | 0.62 | 2.0 |  0.1600 | 1.4
CNO-Rbound |  | n=8 | n=5 | n=3
| $\Delta\chi^{2}$ | $p_{\text{GF}}$ |  CL [$\sigma$] | $\Delta\chi^{2}$ | $p_{\text{GF}}$ |  CL [$\sigma$] | $\Delta\chi^{2}$ | $p_{\text{GF}}$ |  CL [$\sigma$]
AGSS09-met | 14.4 | 0.072 |  1.80 | 9.5 | 0.091 |  1.70 | 7.5 | 0.057 |  1.90
GS98 |  06.8 |  0.560 | 0.58 | 1.9 |  0.860 | 0.17 | 1.6 |  0.660 | 0.44
AAG21 | 11.9 |  0.150 |  1.40 | 7.0 |  0.220 |  1.20 | 6.1 |  0.110 |  1.60
MB22-met/phot |  06.1 |  0.640 | 0.47 | 1.2 |  0.940 | 0.07 | 1.1 |  0.780 | 0.28
Table 2: Results of the PG test for the different models and data samples
considered. Within the given accuracy the results for MB22-met and MB22-phot
models are the same.

Upon analyzing the data in the Table 2, it becomes evident that the B23-MB22
models (both the meteoritic and photospheric variations) exhibit a
significantly higher level of compatibility with the observed data, even
slightly better the B23-GS98 model. On the contrary the B23-AGSS09met and
B23-AAG21 models exhibits a lower level of compatibility with observations,
with B23-AAG21 model slightly better aligned with the data. Maximum
discrimination is provided by comparing mainly the CNO fluxes for which the
prediction of both models is mostly different. On the other hand, including
all the fluxes from the pp-chain in the comparison tends to dilute the
discriminating power of the test. The table also illustrates how allowing for
the three CNO fluxes normalizations to vary in the fit tends to relax the CL
at which the models are compatible with the observations.

Let us remember that our previously determined fluxes in Ref Bergstrom:2016cbh
when confronted with the GS98 and AGSS09 models of the time Serenelli:2011py
showed _absolutely_ no preference for either model. This was driven by the
fact that the most precisely measured $\mbox{}^{8}\text{B}$ flux (and also of
$\mbox{}^{7}\text{Be}$) laid right in the middle of the prediction of both
models. The new B16-GS98 model in Ref. Vinyoles:2016djt  predicted a slightly
lower value for $\mbox{}^{8}\text{B}$ flux in slightly better agreement with
the extracted fluxes of Ref Bergstrom:2016cbh , but the conclusion was still
that there was no significant preference for either model. Compared to those
results, both the most precisely determined $\mbox{}^{7}\text{Be}$ flux and,
most importantly, the newly observed rate of CNO events in Borexino have
consistently moved towards the prediction of the models with higher
metallicity abundances.

Let us finish commenting on the relative weight of the experimental precision
versus the theoretical model uncertainties in the results in Table 2. To this
end one can envision an ideal experiment which measures $f_{i}$ to match
precisely the values predicted by one of the models with infinite accuracy.
Assuming the measurements to coincide with the predictions of B23-GS98, one
gets $\Delta\chi^{2}_{\text{SSM,SET}}=17.1$ and $16.7$ for SSM=B23-AGSS09-met
and SSM=B23-AAG21 with SET=FULL, which means that the maximum CL at which
these two SSM can be disfavoured is $2.2\sigma$ and $2.1\sigma$. Choosing
instead SET=CNO these numbers become $\Delta\chi^{2}_{\text{SSM,SET}}=15.0$
and $14.1$ for SSM=B23-AGSS09-met and SSM=B23-AAG21, respectively,
corresponding to a $3.1\sigma$ and $3.0\sigma$ maximum rejection. This
stresses the importance of reducing the uncertainties in the model predictions
to boost the discrimination between the models.

##  6 Summary

In this work we have updated our former determination of solar neutrino fluxes
from neutrino data as presented in Refs. Gonzalez-Garcia:2009dpj ;
Bergstrom:2016cbh , by incorporating into the analysis the latest results from
both solar and non-solar neutrino experiments. In particular this includes the
full data from the three phases of the Borexino experiments which have
provided us with the first direct evidence of neutrinos produced in the CNO-
cycle.

We have derived the best neutrino oscillation parameters and solar fluxes
constraints using a frequentist analysis with and without imposing nuclear
physics as the only source of energy generation (luminosity constraint).
Compared to the results from previous analysis we find that the determination
of the $\mbox{}^{7}\text{Be}$ flux has improved by a factor $\mathcal{O}(3)$,
but most importantly we now find that the three fluxes produced in the CNO-
cycle are clearly determined to be non-zero, with $1\sigma$ precision ranges
between 20% to $\sim 100\%$ depending on the assumptions in the analysis about
their relative normalization. Conversely, in Refs. Gonzalez-Garcia:2009dpj ;
Bergstrom:2016cbh  only an upper bound for the CNO fluxes was found. This also
implies that it is solidly established that at 99% CL the solar energy
produced in the CNO-cycle is between $0.46\%$ and $1.05\%$ of the total solar
luminosity.

The observation of the CNO neutrinos is also paramount to discriminate among
the different versions of the SSMs built with different inputs for the solar
abundances, since the CNO fluxes are the most sensitive to the solar
composition. In this work we confront for the first time the neutrino fluxes
determined on a purely experimental basis with the predictions of the latest
generation of SSM obtained in Ref. Magg:2022rxb ; B23Fluxes . Our results show
that the SSMs built incorporating lower metallicities are less compatible with
the solar neutrino observations.

###### Acknowledgements.

This project is funded by USA-NSF grant PHY-1915093 and by the European Union
through the Horizon 2020 research and innovation program (Marie Skłodowska-
Curie grant agreement 860881-HIDDeN) and the Horizon Europe programme (Marie
Skłodowska-Curie Staff Exchange grant agreement 101086085-ASYMMETRY). It also
receives support from grants PID2019-110058GB-C21, PID2019-105614GB-C21,
PID2019-108892RB-I00, PID2019-110058GB-C21, PID2020-113644GB-I00,
PID2022-142545NB-C21, “Unit of Excellence Maria de Maeztu 2020-2023” award to
the ICC-UB CEX2019-000918-M, “Unit of Excellence Maria de Maeztu 2021-2025”
award to ICE CEX2020-001058-M, grant IFT “Centro de Excelencia Severo Ochoa”
CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033, as well as from
grants 2021-SGR-249 and 2021-SGR-1526 (Generalitat de Catalunya), and support
from ChETEC-INFRA (EU project no. 101008324). We also acknowledge use of the
IFT computing facilities.

##  Appendix A Details of Borexino analysis

A detailed description of our analysis of the full spectrum of the Phase-I
Bellini:2011rx ; Bellini:2008mr and Phase-II Borexino:2017rsf  of Borexino can
be found in Ref. Gonzalez-Garcia:2009dpj  and Ref. Coloma:2022umy
respectively. Here we document the details of our analysis of the Borexino
Phase-III data collected from January 2017 to October 2021, corresponding to a
total exposure of $\text{1431.6 days}\times\text{71.3 tons}$, which we perform
following closely the details presented by the collaboration in Refs.
BOREXINO:2020aww ; BOREXINO:2022abl .

###  A.1 Analysis of Borexino Phase-III spectrum

In our fit we use the Borexino spectral data as a function of the detected
hits ($N_{h}$) on the detector photomultipliers (including multiple hits on
the same photomultiplier) as estimator of the recoil energy of the electron.
At it was the case in Phase-II, Borexino divide their Phase-III data in two
samples: one enriched (tagged) and one depleted (subtracted) in the
$\mbox{}^{11}\text{C}$ events. The tagged sample picks up about $36\%$ of the
solar neutrino events, while the subtracted sample accounts for the remaining
$64\%$. In what follows we denote by $s=$”tagged” or $s=$”subtracted” each of
the two samples. The data and best fit components for the spectrum of the
subtracted sample are shown in Fig. 2(a) of Ref. BOREXINO:2022abl . The data
points for this sample can also be found in the data release material in Ref.
borexdata . The corresponding information for the tagged sample was kindly
provided to us by the Borexino collaboration borextag .

The number of expected events $T^{0}_{s,i}$ in some bin $i$ of data sample $s$
is the sum of the neutrino-induced signal and the background contributions.
The main backgrounds come from radioactive isotopes in the scintillator
$\mbox{}^{11}\text{C}$, $\mbox{}^{210}\text{Bi}$, $\mbox{}^{10}\text{C}$,
$\mbox{}^{210}\text{Po}$ and $\mbox{}^{85}\text{Kr}$. The collaboration
identifies one additional background due to residual external backgrounds.
With this

$T^{0}_{s,i}=\sum_{f}S^{f}_{s,i}+\sum_{c}B^{c}_{s,i}$ (27)

where the index
$f\in\\{{\mbox{}^{7}\text{Be}},{\text{pep}},{\mbox{}^{13}\text{N}},{\mbox{}^{15}\text{0}},{\mbox{}^{17}\text{F}},{\mbox{}^{8}\text{B}}\\}$
runs over the solar fluxes which contribute in the Borexino-III energy range
(see Fig. 8), while the index
$c\in\\{{\mbox{}^{11}\text{C}},\allowbreak{\mbox{}^{210}\text{Po}},\allowbreak{\mbox{}^{210}\text{Bi}},\allowbreak{\mbox{}^{85}\text{Kr}},\allowbreak{\mbox{}^{10}\text{C}},\allowbreak{\text{ext}}\\}$
runs over the background components.

We compute the solar neutrino signal from flux $f$ in bin $i$, $S^{f}_{s,i}$,
as

$S^{f}_{s,i}=\int_{N_{h,\text{min}}^{i}}^{N_{h,\text{max}}^{i}}\int\frac{\mathrm{d}S^{f}_{s}}{\mathrm{d}T_{e}}(T_{e})\,\frac{\mathrm{d}\mathcal{R}}{\mathrm{d}N_{h}}(T_{e},N_{h})\,\mathrm{d}T_{e}\,\mathrm{d}N_{h}$
(28)

where $\mathrm{d}S_{s}^{f}/\mathrm{d}T_{e}$ is the differential distribution
of neutrino-induced events from flux $f$ to sample $s$ as a function of recoil
energy of the scattered electrons ($T_{e}$)

$\frac{\mathrm{d}S^{f}_{s}}{\mathrm{d}T_{e}}(T_{e})=\mathcal{F}_{s}\,\mathcal{N}_{\text{tgt}}\,\mathcal{T}_{\text{run}}\,\mathcal{E}_{\text{cut}}\sum_{\alpha}\int\frac{\mathrm{d}\phi^{f}_{\nu}}{\mathrm{d}E_{\nu}}\,P_{e\alpha}(E_{\nu})\,\frac{\mathrm{d}\sigma^{\text{det}}(\nu_{\alpha})}{\mathrm{d}T_{e}}\,\mathrm{d}E_{\nu}\,,$
(29)

Here $\mathcal{F}_{s}=0.3572$ ($0.6359$) is the fraction for
$s=\text{``tagged''}$ (“subtracted”) signal events, $\mathcal{N}_{\text{tgt}}$
is the number of $e^{-}$ targets (i.e., the total number of electrons inside
the fiducial volume of the detector, corresponding to 71.3 ton of
scintillator), $\mathcal{T}_{\text{run}}=1431.6$ days is the data taking time,
$\mathcal{E}_{\text{cut}}=98.5\%$ is the overall efficiency (assumed to be the
same as Phase-II), $P_{e\alpha}(E_{\nu})$ is the transition probability
between the flavours $e$ and $\alpha$, and
$\mathrm{d}\sigma^{\text{det}}(\nu_{\alpha})/\mathrm{d}T_{e}$ is the flavour
dependent $\nu_{\alpha}-e^{-}$ elastic scattering detection cross section. The
calculation of $P_{e\alpha}(E_{\nu})$ is based on a fully numerical approach
which takes into account the specific distribution of the neutrino production
point in the solar core for the various solar neutrino flux components as
predicted by the SSMs; some technical details on our treatment of neutrino
propagation in the solar matter can be found in Appendix A of Ref.
Coloma:2022umy and in Sec. 2.4 of Ref. Maltoni:2023cpv . In addition Eq. (28)
includes the energy resolution function
$\mathrm{d}\mathcal{R}/\mathrm{d}T_{e}$ for the detector which gives the
probability that an event with electron recoil energy $T_{e}$ yields an
observed number of hits $N_{h}$. We assume it follows a Gaussian distribution

$\frac{\mathrm{d}\mathcal{R}}{\mathrm{d}N_{h}}=\frac{1}{\sqrt{2\pi}\,\sigma_{h}(T_{e})}\exp\bigg{[}-\frac{1}{2}\bigg{(}\frac{N_{h}-\bar{N}_{h}(T_{e})}{\sigma_{h}(T_{e})}\bigg{)}^{2}\bigg{]}$
(30)

where $\bar{N}_{h}$ is the expected value of $N_{h}$ for a given true recoil
energy $T_{e}$. We determine $\bar{N}_{h}$ via the calibration procedure
described in Ref. Coloma:2022umy , while $\sigma_{h}$ is slightly different
from Borexino Phase-II analysis. Concretely, we derive a relation between
$\bar{N}_{h}$ and $\sigma_{h}(\bar{N}_{h})$ which is

$\sigma_{h}(\bar{N}_{h})=1.21974+1.60121\sqrt{\bar{N}_{h}}-0.14859\bar{N}_{h}.$
(31)

In what respects the backgrounds, we have read the contribution $B^{c}_{s,i}$
for each component $c$ in each bin $i$ and for each data set $s$ from Fig.
2(a) of Ref. BOREXINO:2022abl  as well as the plot provided to us by the
collaboration borextag . These figures show the best-fit normalization of the
different background components as obtained by the collaboration, and we take
them as our nominal background predictions.444One technical detail to notice
is that the data in the tables are more thinly binned (817 bins) than the
corresponding figures from which we read the backgrounds (163 bins). Given the
relatively continuous spectra of the backgrounds, we have recreated the
background content of the 817 bins through interpolation.

Figure 6: Constraints from our analysis of Borexino Phase-III spectra obtained
with $\chi^{2}_{\text{BXIII,test}}$ in Eq. (35). Each panel shows a two-
dimensional projection of the allowed multi-dimensional parameter space after
minimization with respect to the undisplayed parameters. The regions
correspond to 90% and 99% CL (2 d.o.f.). The curves in the right-most panels
show the marginalized one-dimensional $\Delta\chi^{2}_{\text{BXIII,test}}$ for
each of the parameters.

Our statistical analysis is based on the construction of a $\chi^{2}$ function
built with the described experimental data, neutrino signal expectations and
sources of backgrounds. Following Refs. BOREXINO:2020aww ; BOREXINO:2022abl
we leave the normalization of all the backgrounds as free parameters with the
exception of $\mbox{}^{210}\text{Bi}$. The treatment of this background is
paramount to the positive evidence of CNO neutrinos. As described in
BOREXINO:2020aww , the extraction of the CNO neutrino signal from the Borexino
data faces two significant challenges: the resemblance between spectra of
CNO-$\nu$ recoil electrons and the $\mbox{}^{210}\text{Bi}$ $\beta^{-}$
spectra, and their pronounced correlation with the pep-$\nu$ recoil energy
spectrum. In order to surpass the first challenge, the collaboration
restricted the rate of $\mbox{}^{210}\text{Bi}$ for which it sets and upper
limit BOREXINO:2022abl :

$R({\mbox{}^{210}\text{Bi}})\leq(10.8\pm
1.0)~{}\text{cpd}\big{/}\text{100\,t}\,,$ (32)

while no constraint is imposed on its minimum value which is free to be as low
as allowed by the fit (as long as it remains non-negative). We will go back to
this point in Sec. A.2. In our analysis we implement this upper limit by
constraining the corresponding normalization factor
$f_{{\mbox{}^{210}\text{Bi}}}$ as

$f_{{\mbox{}^{210}\text{Bi}}}\leq\bigg{(}1\pm\frac{1.0}{10.8}\bigg{)},$ (33)

With this we construct the $\chi^{2}_{\text{BXIII}}$ as

$\chi^{2}_{\text{BXIII}}=\sum_{s,i}2\bigg{[}T^{0}_{s,i}-O_{s,i}+O_{s,i}\log\bigg{(}\frac{O_{s,i}}{T^{0}_{s,i}}\bigg{)}\bigg{]}+\bigg{(}\frac{f_{{\mbox{}^{210}\text{Bi}}}-1}{\sigma_{{\mbox{}^{210}\text{Bi}}}}\bigg{)}^{2}\,\Theta(f_{{\mbox{}^{210}\text{Bi}}}-1)\,,$
(34)

where $O_{s,i}$ is the observed number of events in bin $i$ of sample $s$, and
$\sigma_{{\mbox{}^{210}\text{Bi}}}=1.0/10.8$, and $\Theta(x)$ is the Heaviside
step function.

Constructed this way, $\chi^{2}_{\text{BXIII}}$ depends on 16 parameters: the
3 oscillation parameters ($\Delta m^{2}_{21}$, $\theta_{12}$, $\theta_{13}$),
6 solar flux normalizations ($f_{{\mbox{}^{7}\text{Be}}}$, $f_{{\text{pep}}}$,
$f_{{\mbox{}^{13}\text{N}}}$, $f_{{\mbox{}^{15}\text{O}}}$,
$f_{{\mbox{}^{17}\text{F}}}$, $f_{{\mbox{}^{8}\text{B}}}$) and 7 background
normalizations ($f_{{\mbox{}^{210}\text{Po}}}$,
$f_{{\mbox{}^{210}\text{Bi}}}$, $f_{{\mbox{}^{85}\text{Kr}}}$,
$f_{{\mbox{}^{10}\text{C}}}$, $f_{\text{ext}}$ and two different factors
$f_{{\mbox{}^{11}\text{C}}}^{\text{tag}}$ and
$f_{{\mbox{}^{11}\text{C}}}^{\text{sub}}$ for the tagged and subtracted
samples).

As a first validation of our $\chi^{2}$ function we perform an analysis
focused at reproducing the results on the solar neutrino fluxes found by the
Borexino collaboration in Ref. BOREXINO:2022abl , and in particular the
positive evidence of CNO neutrinos. In this test fit we fix the three
oscillation parameters to their best fit value ($\sin^{2}\theta_{13}=0.023$,
$\sin^{2}\theta_{12}=0.307$, $\Delta m^{2}_{21}=7.5\times 10^{-5}$), and
following the procedure of the collaboration we assume a common normalization
factor for the three CNO fluxes with respect to the SSM
($f_{{\mbox{}^{13}\text{N}}}=f_{{\mbox{}^{15}\text{O}}}=f_{{\mbox{}^{17}\text{F}}}\equiv
f_{{\text{CNO}}})$. Furthermore, in order to break the pronounced correlation
with the pep-$\nu$ recoil energy spectrum mentioned above, the collaboration
introduced a prior for the pep neutrino signal flux following the SSM. Thus we
define

$\chi^{2}_{\text{BXIII,test}}=\chi^{2}_{\text{BXIII}}+\bigg{(}\frac{f_{\text{pep}}-1}{\sigma_{{\text{pep}}}}\bigg{)}^{2}$
(35)

with $\sigma_{{\text{pep}}}=0.04/2.74$ (for concreteness we choose the
B16-GS98 model for this prior). The $\mbox{}^{7}\text{Be}$ and
$\mbox{}^{8}\text{B}$ fluxes are left completely free.

Figure 7: Dependence of $\Delta\chi^{2}$ of our fit to the Borexino Phase-III
spectra on the common normalization of the CNO fluxes (red line). For
comparison we plot (black dot-dashed line) the corresponding results in figure
2(b) of BOREXINO:2022abl  for their “Fit w/ Systematics”, divided by the
central value of the predicted CNO-$\nu$ rate of the B16-GS98 mode.

The results of this 11-parameter fit are shown in Figs. 6 and 7. In Fig. 6 we
plot the allowed ranges and correlations for the parameters. Notice that in
this figure all parameters are normalized to the best fit values obtained by
the corresponding analysis of the Borexino collaboration, hence a value of “1”
means perfect agreement. We observe a strong correlation between the
normalization of the CNO fluxes $f_{{\text{CNO}}}$ and the
$\mbox{}^{210}\text{Bi}$ background. This is expected because, as mentioned
before, the spectrum of CNO neutrinos and that of the $\mbox{}^{210}\text{Bi}$
background are similar (as can also be seen in Fig. 8 which shows our best
fitted spectra for the two samples). Still, the two spectra are different
enough so that, under the assumption of the upper bound on the
$\mbox{}^{210}\text{Bi}$ background, the degeneracy gets broken enough to lead
to a positive evidence of CNO neutrinos in an amount compatible with the
prediction of the SSMs.

A quantitative comparison with the results of the collaboration is shown in
Fig. 7 where we plot the dependence of our marginalized $\Delta\chi^{2}$ on
the common CNO flux normalization, $f_{{\text{CNO}}}$, together with that
obtained by the collaboration as extracted from Figure 2(b) of Ref.
BOREXINO:2022abl  (labeled “Fit w/ Systematics” in that figure).555Figure 2(b)
of Ref. BOREXINO:2022abl  shows their $\Delta\chi^{2}$ as a function of the
CNO-$\nu$ event rate which we divide by the central value of the expected rate
in the B16-GS98 model to obtain the black dot-dashed curve in Fig. 7.
Altogether, these figures show that our constructed event rates and the best-
fit normalization of the CNO flux reproduce with very good accuracy those of
the fit performed by the collaboration.

###  A.2 Allowing free normalizations for the three CNO fluxes

Figure 8: Spectrum for the best-fit normalizations of the different components
obtained from our fit to the Borexino Phase-III data for TFC-tagged (left) and
TFC-subtracted (right) events. In this figure “CNO” labels the events produced
by sum of the three fluxes produced in the CNO-cycle,
$\Phi_{{\mbox{}^{13}\text{N}}}+\Phi_{{\mbox{}^{15}\text{O}}}+\Phi_{{\mbox{}^{13}\text{F}}}$.

In their analysis of the different phases, the Borexino collaboration always
considers a common shift in the normalization for the three fluxes of
neutrinos produced in the CNO cycle with respect to their values in the SSM.
On the contrary the normalization of the fluxes produced in the pp-chain are
fitted independently.

Figure 9: Same as Fig. 6 but but allowing independent variation of the
normalizations of the three CNO fluxes, only subject to the consistency
conditions in Eqs. (10) and (11).

In principle, once one departs from the constraints imposed by the SSM, the
normalization of the three CNO fluxes could be shifted independently, subject
only to the minimum set of consistency relations in Eqs. (10) and (11). In
fact in our previous works Gonzalez-Garcia:2009dpj ; Bergstrom:2016cbh  we
could perform such general analysis. At the time there was no evidence of CNO
neutrinos and therefore those analysis resulted into a more general set of
upper bounds on their allowed values compared to those obtained assuming a
common shift. With this as motivation, one can attempt to perform an analysis
of the present BXIII spectra under the same assumption of free normalization.
However, within the present modelling of the backgrounds in the Borexino
analysis, optimized to provide maximum sensitivity to a positive evidence of
CNO neutrinos, such generalized analysis runs into trouble as we illustrate in
Fig. 9. As expected, allowing three free CNO flux normalizations results in a
weaker constraints on each of the three parameters. This is particularly the
case for the smaller $\mbox{}^{17}\text{F}$ flux which is allowed to take
values as large as $\sim 40$ times the value predicted by the SSM without
however yielding substantial $\chi^{2}$ improvements over the standard
$f_{{\mbox{}^{17}\text{F}}}=1$ value. In the same way
$f_{{\mbox{}^{15}\text{O}}}$ is compatible with the prediction of the SSM,
$\Delta\chi^{2}(f_{{\mbox{}^{15}\text{O}}}=1)\simeq 0$, with an upper bound
$f_{{\mbox{}^{15}\text{O}}}\lesssim 2$.666It is interesting to notice that the
Borexino bound on $\Phi_{{\mbox{}^{17}\text{F}}}$ is about one-half that on
$\Phi_{{\mbox{}^{15}\text{O}}}$. This is no surprise since the energy spectra
of $\mbox{}^{17}\text{F}$ and $\mbox{}^{15}\text{O}$ neutrinos are extremely
similar hence neither Borexino nor any other experiment can separate them and
what is actually constrained is their _sum_. This is reflected in the clear
anticorrelation visible in Fig. 9, while the factor of two stems from the
consistency condition in Eq. (11). On the contrary the fit results into a
favoured range for $\mbox{}^{13}\text{N}$ which, it taken at face value, would
imply an incompatibility with the SSM at large CL:
$\Delta\chi^{2}(f_{{\mbox{}^{13}\text{N}}}=1)\gtrsim 6$. This large
$\mbox{}^{13}\text{N}$ flux comes at a price of a very low value of the
$\mbox{}^{210}\text{Bi}$ normalization, which as seen in the figure is more
strongly correlated with $\mbox{}^{13}\text{N}$ than with
$\mbox{}^{15}\text{O}$ and $\mbox{}^{17}\text{F}$.

Figure 10: Spectrum of events in the TFC-subtracted sample for the best-fit
normalizations of the different components obtained from two fits to the
Borexino Phase-III data. In the left panel the fit is performed assuming a
common normalization shift for the 3 CNO fluxes, while on the right panel the
three normalizations are allowed to very free subject only the conditions Eqs.
(10) and (11).

To illustrate further this point we show in Fig. 10 our best fitted spectra of
the “subtracted” sample for the analysis where one common normalization for
the three CNO fluxes is used (left, in what follows “CNO” fit) and the one
where all the three normalizations are varied independently (right, in what
follows “N” fit). Thus the spectra in the left panel of Fig. 10 are the same
as the right panel of Fig. 8, except that now, for convenience, we plot
separately the events from each of the CNO fluxes. This highlights clearly the
different shape of the spectra of $\mbox{}^{15}\text{O}$ and
$\mbox{}^{13}\text{N}$, with $\mbox{}^{15}\text{O}$ extending to larger
energies. It is also evident that $\mbox{}^{13}\text{N}$ is the one mostly
affected by degeneracies with the $\mbox{}^{210}\text{Bi}$ background.
Comparing the two panels we see by naked eye that both spectra describe well
the data: in fact, the event rates for $\mbox{}^{15}\text{O}$ are comparable
in both panels. But in the right panel the normalization of the
$\mbox{}^{13}\text{N}$ events is considerably enhanced while the
$\mbox{}^{210}\text{Bi}$ background is suppressed: this is the option favoured
by the fit.

Figure 11: Left: Spectra of subtracted event rates from best fit CNO fluxes
and $\mbox{}^{210}\text{Bi}$ background in the range of $300\leq N_{h}\leq
400$ for the “CNO” fit (fit with a common normalization factor for the three
CNO fluxes, full lines) and the “N” fit (fit with three independent
normalizations, dashed lines). We also show the best fit spectra for both fits
compared to the data as labeled in the figure. Right: Difference of the value
of $\chi^{2}$ in both fits as a function of the maximum $N_{h}$ bin included
in the fit.

Upon closer examination we find that in the range of $N_{h}$ spanning between
300 and 400 photon hits, an increase in the value of
$f_{{\mbox{}^{13}\text{N}}}$ better fits the data while driving
$f_{{\mbox{}^{210}\text{Bi}}}$ towards 0. In Fig. 11 we show a blow-up of the
spectra in this $N_{h}$ window. To quantify the difference in the quality of
the fit for those two solutions and the relevant range of $N_{h}$ we plot in
the right panel the cumulative difference of $\chi^{2}_{\text{BXIII,test}}$
for the best fits of the “CNO” and “N” fits as a function of the maximum
$N_{h}$ bin included in the fit.

Clearly, this anomalously large $\mbox{}^{13}\text{N}$ solution is possible
only because the sole information included in the fit for the
$\mbox{}^{210}\text{Bi}$ background is the upper bound provided by the
collaboration. Such upper bound is enough to ensure a lower bound on the
amount of CNO neutrinos, and indeed it results in a positive evidence of CNO
fluxes (in good agreement with at least some of the SSMs) when a common
normalization for the three CNO fluxes is enforced, as reported by the
collaboration in Refs. BOREXINO:2020aww ; BOREXINO:2022abl  (and properly
reproduced by us, as described in the previous section). Our results show that
this is the case because the spectrum of $\mbox{}^{210}\text{Bi}$ and
$\mbox{}^{15}\text{O}$ are sufficiently different. However, once the
normalization of the three CNO fluxes are not linked together, the degeneracy
between the spectral shape of $\mbox{}^{13}\text{N}$ and
$\mbox{}^{210}\text{Bi}$ – together with the lack of a proper estimate for a
_lower_ bound on $\mbox{}^{210}\text{Bi}$ which is not quantified in Refs.
BOREXINO:2020aww ; BOREXINO:2022abl  – pushes the best-fit of
$\mbox{}^{13}\text{N}$ towards unnaturally large values. In other words, the
background model proposed in Refs. BOREXINO:2020aww ; BOREXINO:2022abl  cannot
be reliably employed for fits with independent $\mbox{}^{13}\text{N}$ and
$\mbox{}^{15}\text{O}$ normalizations.

We finish by noticing that this also implies that the high quality data of
Borexino Phase-III, besides having been able to yield the first evidence of
the presence of the CNO neutrinos, also holds the potential to discriminate
between the contributions from $\mbox{}^{13}\text{N}$ and
$\mbox{}^{15}\text{O}$, a potential which may be interesting to explore by the
collaboration.

###  A.3 Analysis with Correlated Integrated Directionality Method

In a very recent work Borexino:2023puw  the Borexino collaboration has
presented a combined analysis of their three phases making use of the
Correlated and Integrated Directionality (CID) method, which aims to enhance
the precision of the determination of the flux of CNO neutrinos. In a nut-
shell, the CID method exploits the sub-dominant Cherenkov light in the liquid
scintillator produced by the electrons scattered in the neutrino interaction.
These Cherenkov photons retain information of the original direction of the
incident neutrino, hence they can be used to enhance the discrimination
between the solar neutrino signal and the radioactive backgrounds.

Figure 12: Left: CID posterior probabilities for the number for Phase-I and
Phase-II+III of CNO-$\nu$ events after constraining pep and
$\mbox{}^{8}\text{B}$ neutrino events to their SSM expectation. Right:
Dependence of $\Delta\chi^{2}$ on the CNO flux normalization from our fit to
the Borexino Phase-III spectra combined with the CID constraint (blue line)
compared to that obtained by the Borexino collaboration (dotted black line).
For comparison we also show the result without CID information (dashed black
line).

Effectively, the CID analysis results into a determination of the total number
of solar neutrinos detected within a restricted range of $N_{h}$ which
corresponds to $0.85~{}\text{MeV}<T_{e}<1.3~{}\text{MeV}$ for Phase-I and
$0.85~{}\text{MeV}<T_{e}<1.29~{}\text{MeV}$ for Phase-II+III. In this range
the dominant contribution comes from CNO, pep and some $\mbox{}^{8}\text{B}$.
The increased fiducial volume for this analysis brings the exposures to
$740.7~{}\text{days}\times 104.3~{}\text{ton}\times 55.77\%$ for Phase-I and
$2888.0~{}\text{days}\times 94.4~{}\text{ton}\times 63.97\%$ for Phase-II+III.
The resulting number of solar neutrinos detected is
$N^{\text{P-I}}_{\text{obs}}=643^{+235}_{-224}\text{(stat)}^{+37}_{-30}\text{(sys)}$
for Phase-I and
$N^{\text{P-II+III}}_{\text{obs}}=2719^{+518}_{-494}\text{(stat)}^{+85}_{-83}\text{(sys)}$
for Phase-II+III. After subtracting the expected SSM contribution from pep and
$\mbox{}^{8}\text{B}$ the Borexino collaboration obtains the posterior
probability distributions for the number of CNO neutrinos shown in Fig. 9 of
Ref. Borexino:2023puw  (which we reproduce in the left panel in Fig. 12).
Furthermore, since this new directional information is independent of the
spectral information, the collaboration proceeded to combine these two priors
on $N_{{\text{CNO}}}$ with the their likelihood for the Borexino Phase-III
spectral analysis. This resulted in a slightly stronger dependence of the
combined likelihood on the CNO-$\nu$ rate shown in their Fig. 12 (which we
reproduce in the right panel in Fig. 12).

In order to account for the CID information in our analysis we try to follow
as closely as possible the procedure of the collaboration. With the
information provided on the covered energy range and exposures for the CID
analysis, we integrate our computed spectra of solar neutrino events in each
phase to derive the corresponding total number of expected events in Phase-I
and Phase-II+III. We then subtract the SSM predictions for pep and
$\mbox{}^{8}\text{B}$ neutrinos from the observed number of events to derive
an estimate for CNO neutrinos in in Phase-I and Phase-II+III, and construct a
simple Gaussian $\chi^{2}(N_{{\text{CNO}}})$ for Phase-Y (Y=I or II+III)

$\chi^{2}_{\text{CID,P-Y}}(N_{{\text{CNO}}})=\bigg{(}\frac{N_{\text{obs}}^{\text{P-Y}}-N^{\text{P-Y}}_{\text{SSM,pep}}-N^{\text{P-Y}}_{\text{SSM,{$\mbox{}^{8}\text{B}$}}}-N_{{\text{CNO}}}}{\sigma_{\text{P-Y}}}\bigg{)}^{2}$
(36)

where in $\sigma_{P-Y}$ we add in quadrature the symmetrized statistical and
systematic uncertainties in the number of observed events.

We plot in the left panel in Fig. 12 our inferred probability distributions
$P_{\text{P-Y}}(N_{{\text{CNO}}})\propto\exp[-\chi^{2}_{\text{P-Y}}/2]$
compared to those from Borexino in Fig. 9 of Ref. Borexino:2023puw . As seen
in the figure our simple procedure reproduces rather well the results of the
collaboration for the Phase-II+III but only reasonably for Phase-I. This may
be due to differences in the reanalysis of the Phase-I data by the
collaboration in the CID analysis compared to their spectral analysis of 2011.
Our simulations of the Phase-I event rates are tuned to their 2011 and there
is not enough information in Ref. Borexino:2023puw  to deduce what may have
changed. Thus we decide to introduce in our analysis the CID prior for the
Phase-II+III data but not for Phase-I.

We then combine the CID from Phase-II+III and Phase-III spectral information
as

$\chi^{2}_{\text{CID+BXIII,test}}=\chi^{2}_{\text{BX-
III}}+\bigg{(}\frac{f_{{\text{pep}}}-1}{\sigma_{{\text{pep}}}}\bigg{)}^{2}+\chi^{2}_{\text{CID,P-II+III}}\,.$
(37)

A quantitative comparison with the results of the collaboration for this
combined $\text{CID}+\text{Phase-III}$ spectrum analysis is shown in the right
panel of Fig. 12 where we plot the dependence of our marginalized
$\Delta\chi^{2}$ on the CNO flux normalization after including the CID
information compared to that obtained by the collaboration in Fig. 12 of Ref.
Borexino:2023puw . As seen in the figure, we reproduce well the improved
sensitivity for the lower range of the CNO flux normalization but our
constraints are more conservative in the higher range, though they still
represent an improvement over the spectrum-only analysis.

Altogether, after all these tests and validations we define the $\chi^{2}$ for
the full Borexino analysis as

$\chi^{2}_{\text{BX}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}})=\chi^{2}_{\text{BXI}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}})+\chi^{2}_{\text{BXII}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}})\\\
+\chi^{2}_{\text{BXIII}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}})+\chi^{2}_{\text{CID,P-II+III}}(\vec{\omega}_{\text{osc}},\,\vec{\omega}_{\text{flux}})\,.$
(38)

with
$\chi^{2}_{\text{BXIII}}(\vec{\omega}_{\text{osc}}\,\vec{\omega}_{\text{flux}})$
and
$\chi^{2}_{\text{CID,P-II+III}}(\vec{\omega}_{\text{osc}},\vec{\omega}_{\text{flux}})$
in Eqs. (34) and (36), respectively. We finish by noticing that the inclusion
of the CID information is not enough to break the large degeneracy between the
$\mbox{}^{13}\text{N}$ and $\mbox{}^{210}\text{Bi}$ contributions to the
spectra discussed in the previous section.

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