left=0.52in, top=0.52in, right=0.52in, bottom=1in

# Final results of Borexino on CNO solar neutrinos

D. Basilico  G. Bellini  J. Benziger  R. Biondi111Present affiliation: Max-
Planck-Institut für Kernphysik, 69117 Heidelberg, Germany B. Caccianiga  F.
Calaprice  A. Caminata  A. Chepurnov  D. D’Angelo  A. Derbin  A. Di Giacinto
V. Di Marcello  X.F. Ding222Present affiliation: IHEP Institute of High Energy
Physics, 100049 Beijing, China A. Di Ludovico333Present affiliation: INFN
Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy L. Di Noto  I.
Drachnev  D. Franco  C. Galbiati  C. Ghiano  M. Giammarchi  A.
Goretti44footnotemark: 4 M. Gromov  D. Guffanti555Present affiliation:
Dipartimento di Fisica, Università degli Studi e INFN Milano-Bicocca, 20126
Milano, Italy Aldo Ianni  Andrea Ianni  A. Jany  V. Kobychev  G. Korga  S.
Kumaran666Present affiliation: Department of Physics and Astronomy, University
of California, Irvine, California, USA M. Laubenstein  E. Litvinovich  P.
Lombardi  I. Lomskaya  L. Ludhova  I. Machulin  J. Martyn  E. Meroni  L.
Miramonti  M. Misiaszek  V. Muratova  R. Nugmanov  L. Oberauer  V. Orekhov  F.
Ortica  M. Pallavicini  L. Pelicci  Ö. Penek777Present affiliation: GSI
Helmholtzzentrum für Schwerionenforschung GmbH, 64291 Darmstadt, Germany L.
Pietrofaccia88footnotemark: 8 N. Pilipenko  A. Pocar  G. Raikov  M.T. Ranalli
G. Ranucci  A. Razeto  A. Re  N. Rossi  S. Schönert  D. Semenov  G.
Settanta999Present affiliation: Istituto Superiore per la Protezione e la
Ricerca Ambientale, 00144 Roma, Italy M. Skorokhvatov  A. Singhal  O. Smirnov
A. Sotnikov  R. Tartaglia  G. Testera  E. Unzhakov  F.L. Villante  A. Vishneva
R.B. Vogelaar  F. von Feilitzsch  M. Wojcik  M. Wurm  S. Zavatarelli  K. Zuber
G. Zuzel  The Borexino Collaboration AstroParticule et Cosmologie, Université
Paris Diderot, CNRS/IN2P3, CEA/IRFU, Observatoire de Paris, Sorbonne Paris
Cité, 75205 Paris Cedex 13, France  Joint Institute for Nuclear Research,
141980 Dubna, Russia  Dipartimento di Fisica, Università degli Studi e INFN,
16146 Genova, Italy  M. Smoluchowski Institute of Physics, Jagiellonian
University, 30348 Krakow, Poland  Institute for Nuclear Research of NASU,
03028 Kyiv, Ukraine  National Research Centre Kurchatov Institute, 123182
Moscow, Russia  National Research Nuclear University MEPhI (Moscow Engineering
Physics Institute), 115409 Moscow, Russia  INFN Laboratori Nazionali del Gran
Sasso, 67010 Assergi (AQ), Italy  Dipartimento di Fisica, Università degli
Studi e INFN, 20133 Milano, Italy  Dipartimento di Chimica, Biologia e
Biotecnologie, Università degli Studi e INFN, 06123 Perugia, Italy  St.
Petersburg Nuclear Physics Institute NRC Kurchatov Institute, 188350 Gatchina,
Russia  Physics Department, Princeton University, Princeton, NJ 08544, USA
Chemical Engineering Department, Princeton University, Princeton, NJ 08544,
USA  Amherst Center for Fundamental Interactions and Physics Department,
University of Massachusetts, Amherst, MA 01003, USA  Physics Department,
Virginia Polytechnic Institute and State University, Blacksburg, VA 24061, USA
Physik-Department, Technische Universität München, 85748 Garching, Germany
Lomonosov Moscow State University Skobeltsyn Institute of Nuclear Physics,
119234 Moscow, Russia  Gran Sasso Science Institute, 67100 L’Aquila, Italy
Department of Physics, Technische Universität Dresden, 01062 Dresden, Germany
Institute of Physics and Excellence Cluster PRISMA+, Johannes Gutenberg-
Universität Mainz, 55099 Mainz, Germany  Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany  III. Physikalisches Institut
B, RWTH Aachen University, 52062 Aachen, Germany  Department of Physics, Royal
Holloway University of London, Egham, Surrey,TW20 0EX, UK  Institute of
Nuclear Research (Atomki), Debrecen, Hungary  Dipartimento di Scienze Fisiche
e Chimiche, Università dell’Aquila, 67100 L’Aquila, Italy

###### Abstract

In this paper, we report the first measurement of CNO solar neutrinos by
Borexino that uses the Correlated Integrated Directionality (CID) method,
exploiting the sub-dominant Cherenkov light in the liquid scintillator
detector. The directional information of the solar origin of the neutrinos is
preserved by the fast Cherenkov photons from the neutrino scattered electrons,
and is used to discriminate between signal and background. The directional
information is independent from the spectral information on which the previous
CNO solar neutrino measurements by Borexino were based. While the CNO spectral
analysis could only be applied on the Phase-III dataset, the directional
analysis can use the complete Borexino data taking period from 2007 to 2021.
The absence of CNO neutrinos has been rejected with $>$5$\sigma$ credible
level using the Bayesian statistics. The directional CNO measurement is
obtained without an external constraint on the 210Bi contamination of the
liquid scintillator, which was applied in the spectral analysis approach. The
final and the most precise CNO measurement of Borexino is then obtained by
combining the new CID-based CNO result with an improved spectral fit of the
Phase-III dataset. Including the statistical and the systematic errors, the
extracted CNO interaction rate is
$R\mathrm{(CNO)}=6.7^{+1.2}_{-0.8}\,\mathrm{cpd/100\,tonnes}$. Taking into
account the neutrino flavor conversion, the resulting CNO neutrino flux at
Earth is $\Phi_{\mathrm{CNO}}=6.7^{+1.2}_{-0.8}\times
10^{8}\,\mathrm{cm^{-2}s^{-1}}$, which is found to be in agreement with the
high metallicity Standard Solar Models. The results described in this work
reinforce the role of the event directional information in large-scale liquid
scintillator detectors and open up new avenues for the next-generation liquid
scintillator or hybrid neutrino experiments. A particular relevance is
expected for the latter detectors, which aim to combine the advantages from
both Cherenkov-based and scintillation-based detection techniques.

###### Contents

  1. 1 Introduction
  2. 2 The Borexino experiment
  3. 3 Correlated and Integrated Directionality for CNO
    1. 3.1 CID strategy for the CNO measurement with the full dataset
    2. 3.2 N${}^{\text{th}}$-hit analysis approach
    3. 3.3 CID fit procedure
      1. 3.3.1 Fit in the RoI${}_{\text{gvc}}$
      2. 3.3.2 Fit in the RoI${}_{\text{CNO}}$
    4. 3.4 Systematic uncertainties
    5. 3.5 Results of the CID analysis
      1. 3.5.1 Effective $\text{gv}_{\text{ch}}$ calibration on the 7Be edge
      2. 3.5.2 CNO measurement with CID
  4. 4 Combined CID and multivariate analysis
    1. 4.1 Results
  5. 5 Conclusions

##  1 Introduction

Solar neutrinos are produced in the core of the Sun by nuclear reactions in
which hydrogen is transformed into helium. The dominant sequence of reactions
is the so-called $pp$ chain [1, 2] which is responsible for most of the solar
luminosity, while approximately 1 % of the solar energy is produced by the so-
called Carbon-Nitrogen-Oxygen (CNO) cycle. Even though the CNO cycle plays
only a marginal role in the solar fusion mechanisms, it is expected to take
over the luminosity budget for main sequence stars more massive, older, and
hotter than the Sun [3]. Solar neutrinos have proven to be a powerful tool to
study the solar core [4, 5, 6, 7] and, at the same time, have been of
paramount importance in shedding light on the neutrino oscillation phenomenon
[8, 9, 10, 11, 12, 13].

One important open question concerning solar physics regards the metallicity
of the Sun, that is, the abundance of elements with $Z>2$. In fact, different
analyses of spectroscopic data yield significantly different metallicity
results, that can be grouped in two classes: the so-called High-Metallicity
(HZ) [14, 15] and Low-Metallicity (LZ) [16, 17, 18] models. The solar neutrino
fluxes, in particular that from the CNO cycle reactions, can address this
issue. Indeed, the SSM predictions of the CNO neutrino flux depend on the
solar metallicity directly, via the abundances of C and N in the solar core,
and indirectly, via its effect on the solar opacity and temperature profile.

Borexino delivered the first direct experimental proof of the existence of the
CNO cycle in the Sun with a significance of $\sim$7 $\sigma$, also providing a
slight preference towards High-Metallicity models [4, 19]. This result was
obtained with a multivariate analysis of the energy and radial distributions
of selected events. To disentangle the CNO signal from the background, the
multivariate fit requires an independent external constraint on the $pep$
neutrino rate and on the 210Bi rate; the latter is obtained by tagging
210Bi-210Po coincidences in a temperature stabilized, layered scintillator
fluid (see [4, 19] for more details). For this reason, the CNO measurement has
been performed only on approximately one third of the Borexino data, the so-
called Phase-III.

In this paper, we present new results on CNO neutrinos obtained exploiting the
”Correlated and Integrated Directionality” (CID) technique, which uses the
directional information encoded in the Cherenkov light emitted alongside the
scintillation, to separate the solar signal from non-solar backgrounds.
Borexino demonstrated the viability of this technique using 7Be solar
neutrinos [20, 21]. Here we apply the CID technique to the CNO analysis,
obtaining two important results: we show that we can extract the evidence of
solar CNO neutrinos on the entire Borexino dataset following an alternative
approach with respect to the standard multivariate analysis and, consequently,
without the help of the 210Bi constraint; we also show that by combining the
information coming from the directionality with the standard multivariate
analysis performed on Phase-III data we obtain an improved measurement of the
CNO neutrino interaction rate.

The paper is structured as follows. Section 2 describes the Borexino detector
and summarizes the event reconstruction techniques. The CID analysis for the
CNO neutrino measurement is illustrated in Sec. 3, outlining the methods,
reporting the results, and detailing the main sources of systematic
uncertainties. Finally, in Sec. 4 we show our best result on CNO neutrinos
obtained combining the CID and the standard multivariate analysis.

##  2 The Borexino experiment

Figure 1: Scheme of the Borexino detector.

Borexino was a liquid scintillator (LS) neutrino detector [22] that ran until
October 2021 with unprecedented radiopurity levels [23, 5], a necessary
feature of its solar neutrino measurements. The detector was located deep
underground at the Laboratori Nazionali del Gran Sasso (LNGS) in Italy, with
about $3800\text{\,}\mathrm{m}$ water equivalent rock shielding suppressing
the cosmic muon flux by a factor of $\sim$$10^{6}$.

The detector layout is schematically shown in Fig. 1. The Stainless Steel
Sphere (SSS) with a 6.85 m radius supported 2212 8-inch photomultiplier tubes
(PMTs) and contained 280 tonnes of pseudocumene (1,2,4-trimethylbenzene, PC)
doped with 1.5% of PPO (2,5-diphenyloxazole) wavelenght shifter, confined in a
nylon inner vessel of 4.25 m radius. The density of the scintillator was
(0.878 $\pm$ 0.004) g cm-3 with the electron density of (3.307 $\pm$ 0.015)
$\times$ $10^{31}$ $e^{-}/100$ tonnes. The PC-based buffer liquid in the
region between the SSS and IV shielded the LS from external $\gamma$ radiation
and neutrons. The nylon Outer Vessel, that separated the buffer in two sub-
volumes, prevented the inward diffusion of 222Rn. The SSS itself is submerged
in a domed, cylindrical tank filled with $\sim$1 kton of ultra-pure water,
equipped with 208 PMTs. The water tank provided shielding against external
backgrounds and also served as an active Cherenkov veto for residual cosmic
muons passing through the detector.

Borexino detected solar neutrinos via their elastic scattering on electrons of
the LS, a process sensitive, with different probability, to all neutrino
flavors. Electrons, and charged particles in general, deposit their energy in
the LS, excite its molecules and the resulting scintillation light is emitted
isotropically. Using $n$$\approx$1.55 as scintillator index of refraction at
400 nm wavelength, sub-dominant but directional Cherenkov light is emitted
when the electron kinetic energy exceeds 0.165 MeV. Cherenkov light is emitted
over picosecond timescale while the fastest scintillation light component from
the LS has an emission time constant at the nanosecond level. The fraction of
light emitted as Cherenkov photons in Borexino was less than 0.5% for 1 MeV
recoiling electrons.

The effective total light yield was $\sim$500 photoelectrons per MeV of
electron equivalent deposited energy, normalized to 2000 PMTs [23]. The energy
scale is intrinsically non-linear due to ionization quenching and the emission
of Cherenkov radiation. The Geant4 based Monte Carlo (MC) software [24]
simulates all relevant physics processes. It is tuned using the data obtained
during calibration campaigns with radioactive sources [25]. Distinct energy
estimators have been defined, based on different ways of counting the number
of detected photons [23, 20]. The position reconstruction of each event is
performed by using the time-of-flight corrected detection time of photons on
hit PMT [23]. Particle identification is also possible in Borexino [23], in
particular $\alpha/\beta$ discrimination [4], by exploiting different
scintillation light emission time profiles.

The Borexino data-taking period is divided into three phases: Phase-I (May
2007-May 2010), Phase-II (December 2011-May 2016), and Phase-III (July
2016-October 2021). Phase-II started after the detector calibration [25] and
an additional purification of the LS, that enabled a comprehensive measurement
of the $pp$ chain solar neutrinos [5]. Phase-III is characterized by a
thermally stable detector with greatly suppressed seasonal convective
currents. This condition has made it possible to extract an upper limit
constraint on the 210Bi contamination in the LS, and thus, to provide the
first direct observation of solar CNO neutrinos [4].

##  3 Correlated and Integrated Directionality for CNO

Cherenkov photons emitted by the electrons scattered in neutrino interactions
retain information about the original direction of the incident neutrino.
Typically, in water Cherenkov neutrino detectors, this information is accessed
through an event-by-event direction reconstruction, as demonstrated by the
measurements of 8B neutrinos, at energies larger than
$3.5\text{\,}\mathrm{MeV}$ [26, 7]. Instead, the Borexino experiment has
provided a proof-of-principle for the use of this Cherenkov hit information in
a LS detector and at neutrino energies below $1\text{\,}\mathrm{MeV}$ through
the so-called ”Correlated and Integrated Directionality” (CID) technique. A
detailed explanation of the method can be found in [21, 20].

The CID method discriminates the signal originating in the Sun - due to solar
neutrinos - from the background. Cherenkov light is sub-dominant in Borexino,
but it is emitted almost instantaneously with respect to the slower
scintillation light. Consequently, directional information is contained in the
first hits of an event (after correcting for the time-of-flight of each
photon). The CID analysis is based on the $\cos\alpha$ observable: for a given
PMT hit in an event, $\alpha$ is the aperture angle between the Sun and the
hit PMT at the reconstructed position of the event (see also Fig. 3 in [20]).
For background events, the $\cos\alpha$ distribution is nearly uniform
regardless which hit is considered. For solar neutrino events, the
$\cos\alpha$ distribution is flat for scintillation photons which are emitted
isotropically, but has a characteristic non flat distribution peaked at
$\cos\alpha\sim 0.7$ for Cherenkov hits correlated with the position of the
Sun. Since we cannot distinguish Cherenkov and scintillation photons, in our
previous work [21] we have used only the $1^{\text{st}}$ and $2^{\text{nd}}$
hits of each event, which have the largest probability of being Cherenkov
hits. In the new analysis presented in this paper we fully exploit the
directional information contained in the first several hits. This choice is
supported by Monte Carlo simulations and sensitivity studies as discussed in
Sec. 3.2.

The solar neutrino signal is obtained by fitting the $\cos\alpha$
distributions of the selected first several hits, as a sum of signal and
backgrounds contributions. The expected $\cos\alpha$ distributions for signal
and background are obtained from Monte Carlo simulations. As in Ref. [20], for
each selected data event we simulate 200 MC events of solar neutrinos
(represented by 7Be or $pep$ according to RoI, see below) and the same amount
of the background events (represented by 210Bi). These events are simulated
with the same astronomical time as the data event and with the position
smeared around the reconstructed vertex. From the fit we then obtain the total
number of solar neutrinos $N_{\nu}$ detected in the RoI. The fit also includes
two nuisance parameters. The effective Cherenkov group velocity correction
$\text{gv}_{\text{ch}}$ nuisance parameter accounts for small differences in
the relative hit time distribution between scintillation and Cherenkov hits in
data relative to the MC. The second parameter is the event position mis-
reconstruction in the initial electron direction $\Delta r_{\text{dir}}$, an
indirect effect of the Cherenkov hits, where the reconstructed position is
slightly biased towards early hit PMTs of the corresponding event. Here
$\Delta r_{\text{dir}}$ is a free parameter of the fit, while
$\text{gv}_{\text{ch}}$ is obtained independently and is constrained in the
fit.

Figure 2: Illustration of the two RoIs used in the analysis on the energy
spectrum of the Phase-II+III data in a fiducial volume of 2.95 m radius. Monte
Carlo PDFs of different solar neutrino components are scaled to high-
metallicity SSM prediction [2]. The grey band shows the 7Be-$\nu$ edge region
used for the estimation of gv${}_{\text{ch}}$ correction
(RoI${}_{\text{gvc}}$), while the CNO region used to measure the CNO-$\nu$
rate is shown in yellow band (RoI${}_{\text{CNO}}$).

Compared to the previous proof-of-principle analysis [21, 20], the current CID
analysis has been improved in a variety of ways. The full detector live time
can be used now thanks to a novel $\text{gv}_{\text{ch}}$ correction
calibration. In the previous publication, $\text{gv}_{\text{ch}}$ has been
obtained using 40K $\gamma$ calibration data (see [20]). In this work instead
we calibrate $\text{gv}_{\text{ch}}$ by exploiting the 7Be solar neutrino
events which allows us to extend the analysis to the full Borexino dataset, as
explained in Sec. 3.1. Additionally, indirect Cherenkov information from the
systematic influence on the vertex reconstruction and consequently on the
$\cos\alpha$ distribution was exploited by the inclusion in the analysis of
later hits with negligible contribution of Cherenkov photons, see Sec. 3.2.
Technical details of the CID fitting procedure can be found in Sec. 3.3, while
different systematic effects are discussed in Sec. 3.4. The final CID results
regarding the CNO measurement are reported in Sec. 3.5.

### 3.1 CID strategy for the CNO measurement with the full dataset

In the previous Borexino works [21, 20], the calibration of the Cherenkov
light group velocity $\text{gv}_{\text{ch}}$ has been performed using $\gamma$
sources deployed during the Borexino calibration campaign in 2009 [20]. The
solar neutrino analysis was performed on the Phase-I dataset, that has been
taken close in time to the source calibration of the detector [25]. The
$\text{gv}_{\text{ch}}$ found in this way was used to obtain the first
measurement of 7Be solar neutrinos with the CID method [21]. For the CID
measurement of CNO in this paper, the entire Borexino dataset is used (from
2007 until 2021). Since the sub-nanosecond stability of the detector time
response cannot be guaranteed for long periods, and no more calibrations have
been performed after 2009, we developed a method to calibrate
$\text{gv}_{\text{ch}}$ on the 7Be shoulder data. This is done by using the
same RoI as in [21, 20] (here called RoI${}_{\text{gvc}}$ electron equivalent
energy range of $$0.5\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$\lesssim
T_{e}\lesssim$0.8\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$) and performing the
CID analysis where the 7Be is constrained to the Standard Model predictions
[2]. The $\text{gv}_{\text{ch}}$ correction extracted in this way is then used
in the CID analysis of the RoI${}_{\text{CNO}}$, in which the CNO contribution
is maximized, and which is fully independent from RoI${}_{\text{gvc}}$. This
step has been found to be justified according to MC studies, as the wavelength
distribution of the detected Cherenkov photons produced by electrons from
RoI${}_{\text{gvc}}$ and RoI${}_{\text{CNO}}$ is the same. With this new
strategy, the Cherenkov light $\text{gv}_{\text{ch}}$ can be calibrated on the
same data-taking period as the one used for the CNO analysis. Two analyses
have been performed in parallel for Phase-I (May 2007 to May 2010, 740.7 days)
and Phase-II+III (December 2011 to October 2021, 2888.0 days). The
$\text{gv}_{\text{ch}}$ correction obtained for Phase-I can be compared to the
one previously obtained from the 40K $\gamma$ source [20]. Additionally, the
analyses of the two independent data-sets allows for the investigation of any
variation of the detector response over time. The RoI${}_{\text{gvc}}$ and the
RoI${}_{\text{CNO}}$ are shown for the Phase-II+III dataset in Fig. 2 for
illustrative purposes. The results on $\text{gv}_{\text{ch}}$ are provided in
Sec. 3.5.1.

In the final analysis, the Three-Fold-Coincidence algorithm [4] is applied to
the RoI${}_{\text{CNO}}$ to suppress the cosmogenic 11C background, preserving
the exposure with a signal survival fraction of $55.77\%\pm 0.02\%$ for
Phase-I and $63.97\%\pm 0.02\%$ for Phase-II+III. The radial ($R_{\text{FV}}$)
and $T_{e}$ energy cuts of RoI${}_{\text{CNO}}$ were optimized considering the
expected number of solar neutrinos over the statistical uncertainty of the
total number of events. The optimized cuts are $R_{\text{FV}}<3.05\,(2.95)$ m
and $0.85\,(0.85)\,\text{MeV}<T_{e}<1.3\,(1.29)\,\text{MeV}$) for the Phase-I
(Phase-II+III). In addition, all other cuts including the muon veto and data
quality cuts have been applied as in Refs. [4, 19]. The overall exposures for
the CID CNO analysis are $740.7\,\text{days}\times 104.3\,\text{tonnes}\times
55.77\%$ for Phase-I and $2888.0\,\text{days}\times 94.4\,\text{tonnes}\times
63.97\%$ for Phase-II+III. The total exposure of Phase-II+III (477.81 years
$\times$ tonnes) is about four times larger than that of Phase-I (118.04 years
$\times$ tonnes).

### 3.2 N${}^{\text{th}}$-hit analysis approach

As mentioned above, the CID analysis is performed on the first several early
hits of ToF corrected hit times from each event in the RoI. In this subsection
we describe the optimization of the number of hits from each event to be used
in the CID analysis. The procedure is based on the comparison of the MC-
produced $\cos\alpha$ distributions of signal and background.

First, PMT hits of each individual event are sorted according to their ToF-
corrected hit times and are labeled in this order as ”N${}^{\text{th}}$-hit”,
with N = 1, 2, … up to the total number of hits. Second, the $\cos\alpha$
distributions are constructed for each N${}^{\text{th}}$-hit for both the
signal and background MC. Third, for each N${}^{\text{th}}$-hit $\cos\alpha$
distribution a number of 10,000 toy MC samples are simulated with the number
of events observed as in the real data. Next, we perform a direct signal to
background comparison, based on a standard $\chi^{2}$-test. Figure 3 shows the
resulting $\Delta\chi^{2}$ for Phase-II+III in the RoI${}_{\text{CNO}}$
averaged over the 10,000 toy datasets as a function of N${}^{\text{th}}$-hit.
A larger average $\Delta\chi^{2}$ corresponds to a greater difference between
the MC signal and background and thus a larger expected sensitivity for the
CID fit, independent of the true signal to background ratio. While only the
earliest $\sim$4 N${}^{\text{th}}$-hits have a relevant, direct contribution
of Cherenkov hits to the $\cos\alpha$ distribution of the neutrino signal,
later N${}^{\text{th}}$-hits also contribute to the CID sensitivity due to the
indirect Cherenkov influence on $\Delta r_{\text{dir}}$. A more in-depth
explanation of the $\Delta r_{\text{dir}}$ effect is shown in Fig. 9 in [20].

A possible impact of the $\text{gv}_{\text{ch}}$ and $\Delta r_{\text{dir}}$
nuisance parameters on the N${}^{\text{th}}$-hit selection has been
investigated and is presented in Fig. 3. The first hits of the events provide
the largest $\Delta\chi^{2}$ values thanks to the direct Cherenkov light. A
decrease of $\text{gv}_{\text{ch}}$ is decreasing the group velocity of
Cherenkov photons and thus their contribution at early hits. The impact of
$\Delta r_{\text{dir}}$ can be seen for N${}^{\text{th}}\text{-hit}>4$, where
the contribution of direct Cherenkov hits becomes negligible relative to the
scintillation hits, but the signal and background MC $\cos\alpha$ histograms
are still different from each other ($\Delta\chi^{2}>0$).

In conclusion, the early hits selection for the CID analysis in both
RoI${}_{\text{gvc}}$ and RoI${}_{\text{CNO}}$ is then performed from the first
hit up to the $\text{N${}^{\text{th}}$-hit(max)}=15,17$ for Phase-I and Phase-
II+III, respectively. This is an optimization where all direct and indirect
Cherenkov information is used, while at the same time this selection keeps the
contribution from delayed scintillation photons, undergoing various optical
process during the propagation through the detector, relatively small.

The Cherenkov-to-scintillation photon ratio as a function of
N${}^{\text{th}}$-hit has also been checked explicitly, as is shown in Fig. 4
for RoI${}_{\text{CNO}}$. As expected, it can be seen that the early
N${}^{\text{th}}$-hits benefit from the largest Cherenkov-to-scintillation
ratio of $\sim 13\%$ for the first hit. The overall total Cherenkov-to-
scintillation ratio is small and found to be $0.475\%$ in the MC.

Figure 3: $\Delta\chi^{2}$ between the Phase-II+III RoI${}_{\text{CNO}}$
neutrino signal and background MC $\cos\alpha$ distributions for different
selections of nuisance parameters. $\Delta r_{\text{dir}}=2.7\,\text{cm}$
corresponds to the nominal value observed in the neutrino MC.

### 3.3 CID fit procedure

The fitting strategy follows the procedure developed in our previous CID
analysis [20]. The data $\cos\alpha$ distributions from the selected RoI,
constructed for each N${}^{\text{th}}$-hit from the first up to the
N${}^{\text{th}}$-hit(max), are fitted simultaneously with the MC produced,
expected $\cos\alpha$ distributions of the neutrino signal and background,
where the signal $\cos\alpha$ distribution depends on $\text{gv}_{\text{ch}}$
and $\Delta r_{\text{dir}}$. The nuisance parameter $\Delta r_{\text{dir}}$
cannot be calibrated in Borexino and is left free to vary without a dedicated
pull term. The number of $\cos\alpha$ histogram bins used in the analyses is
$i=60$ for all energy regions and phases, as values of $i<30$ reduce the
expected CID sensitivity.

####  3.3.1 Fit in the RoI${}_{\text{gvc}}$

The CID analysis in RoI${}_{\text{gvc}}$ used for the $\text{gv}_{\text{ch}}$
calibration is based on the $\chi^{2}$-test:

$\displaystyle\begin{split}&\chi^{2}_{\text{gv}_{\text{ch}}}(N_{\nu},\text{gv}_{\text{ch}},\Delta
r_{\text{dir}})=\\\ &=\sum_{n=1}^{\text{
N${}^{\text{th}}$-hit(max)}}\sum_{i=1}^{I}\left(\frac{\left(\mathcal{N}\cdot
M_{i}^{n}-D_{i}^{n}\right)^{2}}{\mathcal{N}\cdot
M_{i}^{n}+\mathcal{N}^{2}\cdot
M_{i}^{n}}\right)-2\ln\left(P(N_{\nu})\right),\end{split}$ (1)

where $D_{i}^{n}$ and $M_{i}^{n}$ are the numbers of $\cos\alpha$ histogram
entries at bin $i$ for a given N${}^{\text{th}}$-hit $n$, for data and MC,
respectively. The term $\mathcal{N}$ is the scaling factor between the MC and
the data event statistics and the term ”$\mathcal{N}^{2}\cdot M_{i}^{n}$” in
the denominator takes into account the finite statistics of MC. The explicit
dependence of the fit on $N_{\nu}$, $\text{gv}_{\text{ch}}$, and $\Delta
r_{\text{dir}}$ can be expressed by decomposing the MC contribution to the one
from the signal $S$ and the background $B$:

$\displaystyle M_{i}^{n}=\frac{N_{\nu}}{N_{\text{data}}}\cdot
M_{\text{S},i}^{n}(\Delta
r_{\text{dir}},gv_{\text{ch}})+(1-\frac{N_{\nu}}{N_{\text{data}}})\cdot
M_{{\text{B}},i}^{n}.$ (2)

The number of neutrino events $N_{\nu}$ and $\Delta r_{\text{dir}}$ are
treated as nuisance parameters to produce the
$\chi^{2}(\text{gv}_{\text{ch}})$ profile, where $N_{\nu}$ is constrained by
the SSM expectation. For this, the neutrino prior probability distribution
$P(N_{\nu})$ is given by the sum of the Gaussian probability distributions
with mean and sigma from the high-metallicity (HZ) SSM and low-metallicity
(LZ) SSM [2] predictions on the number of 7Be+$pep$-$\nu$ in 7Be-$\nu$
shoulder region, which is then convoluted with a uniform distribution of
CNO-$\nu$ between zero and the HZ-SSM CNO expectation + $5\sigma$. In this
way, by leaving CNO reasonably free to vary, we avoid a potential correlation
of the $\text{gv}_{\text{ch}}$ calibration and the subsequent measurement of
the CNO-$\nu$ rate using this $\text{gv}_{\text{ch}}$ constraint.

####  3.3.2 Fit in the RoI${}_{\text{CNO}}$

Figure 4: Cherenkov-to-scintillation PMT hit ratio as a function of the time-
of-flight sorted N${}^{\text{th}}$-hits for the $pep$ neutrino Monte Carlo of
Phase II+III in the RoI${}_{\text{CNO}}$ (0.85 MeV – 1.3 MeV).

The $\chi^{2}$-test for the measurement of number of solar neutrinos
($N_{\nu}$) in RoI${}_{\text{CNO}}$ is:

$\displaystyle\begin{split}&\chi^{2}_{\nu}(N_{\nu},\text{gv}_{\text{ch}},\Delta
r_{\text{dir}})=\\\ &=\sum_{n=1}^{\text{
N${}^{\text{th}}$-hit(max)}}\sum_{i=1}^{I}\left(\frac{\left(\mathcal{N}\cdot
M_{i}^{n}-D_{i}^{n}\right)^{2}}{\mathcal{N}\cdot
M_{i}^{n}+\mathcal{N}^{2}\cdot
M_{i}^{n}}\right)+\Delta\chi_{\text{gv}_{\text{ch}}}^{2}\left(\text{gv}_{\text{ch}}\right)\end{split}$
(3)

using $\text{gv}_{\text{ch}}$ and $\Delta r_{\text{dir}}$ as nuisance
parameters. The $\text{gv}_{\text{ch}}$ parameter is now constrained by the
previous calibration in RoI${}_{\text{gvc}}$ through the pull term
$\Delta\chi_{\text{gv}_{\text{ch}}}^{2}\left(\text{gv}_{\text{ch}}\right)$.

### 3.4 Systematic uncertainties

In this work we have performed a detailed evaluation of the systematic
uncertainties. The quantitative evaluation of the systematic uncertainties on
the CID analysis results are given in Sec. 3.5.

It has been found that the choice of N${}^{\text{th}}$-hit(max) and the
histogram binning do not introduce any systematic uncertainty. Backgrounds
different than 210Bi also contribute in the analysed energy intervals, but
have been found to be indistinguishable in the CID analysis and do not
contribute to the systematic uncertainty budget. Even if the external $\gamma$
events are not uniformly distributed in the FV, due to their attenuation in
the LS, the difference in the $\cos\alpha$ distribution between these events
and uniform background events is found to be safely negligible, given the
statistics of the data.

The following effects have an impact on the final results:

PMT selection

Some PMTs feature an intrinsic misbehaviour of their hit time distribution, if
compared to all other PMTs. They are identified using an enriched sample of
11C events. In addition, a small number of PMTs feature inconsistency between
the data and MC in the relative contribution to the first hits. The systematic
effect has been evaluated by varying the selection of usable PMT.

Relative PMT hit time correction

It has been found by analyzing an enriched sample of 11C events, that the data
PMTs have a small relative time offset between each other with an average
value of 0.3 ns. This time offset has been measured with an uncertainty of up
to $\pm$0.1 ns. It is reasonable to correct this relative offset in data, as
it does not exists in MC. This makes it also necessary to propagate the
uncertainty of the PMT time correction through the entire analysis chain,
which introduces a systematic uncertainty.

Influence of low number of signal events

As described above, signal and background MC are produced on an event-by-event
basis. This could introduce an additional systematic uncertainty through the
particular choice of the event positions and neutrino directions used for the
production of the signal MC. This systematic uncertainty is estimated by
producing a large number of the signal MC $\cos\alpha$ distributions
corresponding to a random selection of the expected number of signal events
from data in each phase. The data is then analyzed again with these reduced
signal MC $\cos\alpha$ histograms. This effect contributes to the systematic
uncertainty only in the RoI${}_{\text{CNO}}$ for Phase-I.

CNO-$\nu$ and $pep$-$\nu$ $\cos\alpha$ distributions

The CNO-$\nu$ and $pep$-$\nu$ events show a significantly different energy
distribution in the selected RoI. The expected Cherenkov to scintillation hits
ratio for $pep$-$\nu$ events (0.475%) is higher than for CNO events (0.469%)
due to their different energy distribution in RoI${}_{\text{CNO}}$. Moreover,
the angular distribution of recoiled electrons by CNO-$\nu$ and $pep$-$\nu$ is
also different due to its dependence on energy distributions of the neutrino
and recoiled electron. The final fit on the number of solar neutrinos is
performed with the $pep$-$\nu$ MC and the systematic uncertainty is estimated
by performing the CID analysis again with the CNO-$\nu$ MC. The absolute
difference between the two analyses is used conservatively as the systematic
uncertainty. This systematic is negligible for the $\text{gv}_{\text{ch}}$
calibration, as CNO+$pep$ neutrinos are sub-dominant to the 7Be neutrinos.

Constraint on non-CNO neutrinos

The CID analysis is only sensitive to the measurement of the total number of
solar neutrinos $N_{\nu}$ and cannot differentiate between different solar
neutrino species. Therefore, a measurement of the number of CNO neutrinos
depends on the subtraction of the non-CNO neutrinos from $N_{\nu}$ obtained in
the RoI${}_{\text{CNO}}$. The number of $pep$ neutrinos is constrained by the
SSM predictions [2], while 8B is constrained using the high precision flux
measurement of Super-Kamiokande [6].

Exposure

In this category we cover systematic uncertainties on the determination of the
fiducial mass and on the fraction of solar neutrinos in the RoI. These
uncertainties are estimated using toy-MC studies, based on the results of the
calibration campaign [25] on the performance of the position reconstruction
and on the uncertainty on the energy scale described in [4], respectively. The
uncertainty on the fiducial mass includes also the uncertainty on the
scintillator density. Additionally, in the $\text{gv}_{\text{ch}}$ calibration
using the constraint on the expected number of all solar neutrino events, we
consider also the uncertainty on $\alpha/\beta$ discrimination applied to
suppress 210Po $\alpha$ decays in the RoI${}_{\text{gvc}}$.

### 3.5 Results of the CID analysis

####  3.5.1 Effective $\text{gv}_{\text{ch}}$ calibration on the 7Be edge

Figure 5: The CID data (black) and the best fit results (red) for the
measurement of the $\text{gv}_{\text{ch}}$ parameter. While the analysis is
done separately for Phase-I and Phase-II+III, here the sum of Phase-I+II+III
is shown for illustration purposes. There are in total 78632 events in the
RoI${}_{\text{gvc}}$. The best fit for the constrained number of neutrino
events is $N_{\nu}=50063$, while the best fit values for the parameter of
interest are $\text{gv}_{\text{ch}}=0.140\,\text{ns\,m}^{-1}$ for Phase-I and
$\text{gv}_{\text{ch}}=0.089\,\text{ns\,m}^{-1}$ for Phase-II+III. For
comparison, the background MC (blue) scaled to the same total number of events
is shown. (a) The sum of the first to fourth N${}^{\text{th}}$-hits
$\cos\alpha$ histograms shows the Cherenkov peak. (b) The sum of the fifth to
the N${}^{\text{th}}$-hit(max) $\cos\alpha$ histograms shows the effect the
$\Delta r_{\text{dir}}$ parameter on the later hits.

The effective calibration of the Cherenkov light as a results of the CID
analysis on the 7Be edge, using Eq. 1, is $\text{gv}_{\text{ch}}=(0.140\pm
0.029)\,\text{ns\,m}^{-1}$ for Phase-I and $\text{gv}_{\text{ch}}=(0.089\pm
0.019)\,\text{ns\,m}^{-1}$ for Phase-II+III, including the systematic
uncertainties summarized in Table I. The compatibility between the data and
the MC model, illustrated in Fig. 5, is good with
$\chi^{2}/\text{ndf}=874.9/897$, $p\text{ value}=0.70$ for Phase-I and
$\chi^{2}/\text{ndf}=1036.2/1017$, $p\text{ value}=0.33$ for Phase-II+III. The
$\chi^{2}/\text{ndf}$ and $p$ values have also been investigated for the
individual N${}^{\text{th}}$-hits $\cos\alpha$ histograms, with different
binning choices to investigate the fit performance. For all cases, the best
fit MC model $\cos\alpha$ distribution is always in agreement with the data.
Figure 5 shows an illustration of the best fit results (red) relative to a
pure background hypothesis (blue). For early hits, direct Cherenkov light
causes the peak around $\cos\alpha\sim 0.7$ and the influence of $\Delta
r_{\text{dir}}$ induces the negative slope for $\cos\alpha<0$. For later
N${}^{\text{th}}$-hits the Cherenkov peak washes away, but the indirect impact
of the Cherenkov hits on the position reconstruction bias $\Delta
r_{\text{dir}}$ makes it still possible to distinguish between the neutrino
signal and the background. The non-flat background $\cos\alpha$ distribution
originates from the live PMT non-isotropic distribution relative to the
position distribution of the Sun around Borexino. These
$\text{gv}_{\text{ch}}$ values for Phase-I and Phase-II+III differ by less
than $1.5\sigma$ and both are in agreement with the previous calibration
performed at the end of Phase-I using a 40K $\gamma$ source:
$\text{gv}_{\text{ch}}=(0.108\pm 0.039)\,\text{ns\,m}^{-1}$ (Fig. 13 in [20]).

|  |
---|---|---
Source of $\text{gv}_{\text{ch}}$ uncertainty | Phase-I | Phase-II+III
PMT selection | 2.1% | 1.6%
PMT time corrections | 3.7% | 2.1%
MLP event selection | 1.0% | 1.0%
Fiducial mass | $\left({}_{-1.2}^{+0.2}\right)\%$ | $\left({}_{-1.2}^{+0.2}\right)\%$
Fraction of neutrinos in RoI | 1.3% | 0.9%
|  |
Table I: Systematic uncertainties of the $\text{gv}_{\text{ch}}$ measurement
in the RoI${}_{\text{gvc}}$, relative to the best fit value.

####  3.5.2 CNO measurement with CID

Figure 6: Illustration of the CID data (black) and the best fit results (red)
summed for the Phase-I + Phase-II+III with a total of 8964 events in the
RoI${}_{\text{CNO}}$. The best fit of the total number of neutrino events is
$N_{\nu}=3519$ without any systematic correction. For comparison, the
background MC (blue) scaled to the same total number of events is shown. (a)
The sum of the first to fourth N${}^{\text{th}}$-hits $\cos\alpha$ histograms
shows the Cherenkov peak. (b) The sum of the fifth to the
N${}^{\text{th}}$-hit(max) $\cos\alpha$ histograms shows the effect the
$\Delta r_{\text{dir}}$ parameter on these later hits.

This section describes the results of the measurement of CNO solar neutrinos
with CID using the full Borexino live time from May 2007 to October 2021. The
$\text{gv}_{\text{ch}}$ values presented in Sec. 3.5.1 are used as independent
pull terms in the Eq. 3 for the fit in RoI${}_{\text{CNO}}$ of their
respective phases. This takes into account the potential systematic
differences of the detector response between Phase-I and Phase-II+III. The
resulting number of solar neutrino events $N_{\nu}$ in the
RoI${}_{\text{CNO}}$ can be converted into the number of CNO neutrinos
detected in the same energy region after constraining the contributions from
$pep$ and 8B neutrinos, but without any a-priori knowledge of the backgrounds.
This number of CNO events can be further transformed into the measurement of
the CNO-$\nu$ interaction rate in Borexino and the CNO flux at Earth.

The best fit values for the number of solar neutrinos in RoI${}_{\text{CNO}}$
are $N_{\nu}=691_{-224}^{+235}\,\text{(stat)}$ for Phase-I and
$N_{\nu}=2828_{-494}^{+518}\,\text{(stat)}$ for Phase-II+III without inclusion
of any systematic uncertainties or corrections. The compatibility betwwen the
data and the MC model is good with $\chi^{2}/\text{ndf}=884.8/897$, $p\text{
value}=0.61$ for Phase-I and $\chi^{2}/\text{ndf}=1000.7/1017$, $p\text{
value}=0.64$ for Phase-II+III. The MC model is able to reproduce the data
$\cos\alpha$ distribution, which has also been investigated for the individual
N${}^{\text{th}}$-hits $\cos\alpha$ histograms.

Figure 6 illustrates the best fit results (red) relative to a pure background
hypothesis (blue), in which the CID $\cos\alpha$ histograms of data (black)
are shown for the sum of Phase-I + Phase-II+III, as well as for the sum of the
early first to fourth N${}^{\text{th}}$-hits (top) and the sum of the later
N${}^{\text{th}}$-hits from the fifth to $\mathrm{N^{th}-hit(max)}$ (bottom).
The actual fit is performed on Phase-I and Phase-II+III independently. The
same observations made for Fig. 5 hold also true for Fig. 6, where the early
hits show the Cherenkov peak and the later hits show the impact of $\Delta
r_{\text{dir}}$.

##### Fit response bias correction

Toy-MC analyses found that the fit of the number of solar neutrinos in
RoI${}_{\text{CNO}}$ shows a small systematic shift between the injected
number of neutrinos and the best fit number of neutrinos, due to the
correlation between the nuisance parameters ($\text{gv}_{\text{ch}}$, $\Delta
r_{\text{dir}}$) and the relatively low total number of neutrino events. This
fit response bias is induced by the two nuisance parameters as they only
impact the shape of the neutrino signal MC $\cos\alpha$ distribution but not
that of background. We note that this effect was found to be negligible in
RoI${}_{\text{gvc}}$ due to the relative large number of signal events and the
large signal to background ratio.

|  |
---|---|---
Source of uncertainty | Phase-I | Phase-II+III
For Nν
PMT selection | 1.3% | 0.6%
PMT time corrections | 4.2% | 2.4%
Low number of signal events | 2.2% | –
CNO-$\nu$ vs. $pep$-$\nu$ MC | 2.2% | 2.0%
For N${}_{\text{CNO}}$
$pep$+8B-$\nu$ constraint | 4.6% | 1.8%
For R${}_{\text{CNO}}$
Fiducial mass | $\left({}_{-1.2}^{+0.2}\right)\%$ | $\left({}_{-1.2}^{+0.2}\right)\%$
Fraction of CNO-$\nu$ in RoI | 1.4% | 1.4%
|  |
Table II: Systematic uncertainties on the number of solar neutrino events
$N_{\nu}$ in RoI${}_{\text{CNO}}$, relative to the best fit value. The
uncertainty from $pep$+8B-$\nu$ constraint is relevant only for
N${}_{\text{CNO}}$. The last two rows are relevant only for the CNO-$\nu$ rate
(R${}_{\text{CNO}}$) calculation.

The value of the fit response bias in RoI${}_{\text{CNO}}$ is estimated using
the Bayesian posterior distribution of $N_{\nu}$ [27], which is produced
through a toy-MC rejection sampling, described in summary below. The prior
distribution for the number of neutrino events is chosen to be uniform between
zero and the number of selected data events (2990 for Phase-I and 5974 for
Phase-II+III), the prior distribution of $\Delta r_{\text{dir}}$ is also
uniform, and the prior distribution of $\text{gv}_{\text{ch}}$ is given by the
measurement at the 7Be-$\nu$ edge RoI${}_{\text{gvc}}$
$\left(P\left(\text{gv}_{\text{ch}}\right)\propto\exp\left(-\frac{1}{2}\Delta\chi^{2}(\text{gv}_{\text{ch}})\right)\right)$.
The pseudo-data inputs
$\left(N_{\nu}^{\text{sim}},\text{gv}_{\text{ch}}^{\text{sim}},\Delta
r_{\text{dir}}^{\text{sim}}\right)$ are sampled from the MC signal and
background $\cos\alpha$ distributions following these model parameter prior
distributions. The analysis is then performed in the same way as for the real
data and results in best fit values of the pseudo-data
$\left(N_{\nu}^{\text{fit}},\text{gv}_{\text{ch}}^{\text{fit}},\Delta
r_{\text{dir}}^{\text{fit}}\right)$. The real data result now defines a
multivariate Gaussian distribution
$\text{P}_{\text{accept}}(N_{\nu},\text{gv}_{\text{ch}},\Delta
r_{\text{dir}})$ with a mean value given by its best fit values and with a
standard deviation given by the systematic uncertainty of the PMT time
corrections. The sampled true values of the triplet
$\left(N_{\nu}^{\text{sim}},\text{gv}_{\text{ch}}^{\text{sim}},\Delta
r_{\text{dir}}^{\text{sim}}\right)$ are then saved only with a probability of
$\text{P}_{\text{accept}}(N_{\nu}^{\text{fit}},\text{gv}_{\text{ch}}^{\text{fit}},\Delta
r_{\text{dir}}^{\text{fit}})$, given by the best fit result of the pseudo-
data, otherwise they are rejected. The resulting distributions of the true
values for $\left(N_{\nu},\text{gv}_{\text{ch}},\Delta r_{\text{dir}}\right)$
then correspond to their Bayesian posterior distributions.

Figure 7: Illustration of the fit response bias for Phase II+III in the
RoI${}_{\text{CNO}}$. The data fit result, i.e. the likelihood $P(\nu)$ given
by the $\Delta\chi^{2}$ profile of Eq. 3 and averaged over 1000 fits with
different PMT time offsets is shown in blue. The posterior distribution of 20k
pseudo-data analyses, selected through rejection sampling, is shown in black.
The red line corresponds to the posterior distribution that includes only the
systematics from the PMT time alignment correction.

The fit response bias is illustrated in Fig. 7 for Phase II+III. The
likelihood distribution
$P(N_{\nu})\propto\exp\left(-\frac{1}{2}\Delta\chi^{2}(N_{\nu}\right)$ given
by the $\chi^{2}$ fit of data with Eq. 3 and averaged over the 1000 fits with
different PMT time offsets is shown in blue. The black distribution is given
by the simulation of 20k pseudo-data analyses, selected through the rejection
sampling MC described above. The red distribution is produced by shifting
$P(N_{\nu})$ by a value of $\Delta N_{\nu}=-109\pm 4$ and this distribution is
well in agreement with the black rejection sampled distribution. It is
therefore used as the posterior distribution of the CID analyses. We note that
for the Phase-I the situation is similar and the shift is found to be $-50\pm
4$ events.

##### Inclusion of systematics

The final result of the CID analysis for the number of solar neutrinos is
given by the Bayesian posterior distribution of $N_{\nu}$, marginalized over
the nuisance parameters and convoluted with the systematic uncertainties. The
relevant systematic uncertainties are shown in Table II and assumed to be
normally distributed.

The posterior distributions $P\left(N_{\nu}\right)$ in Phase-I and Phase-
II+III including these systematics are shown in Fig. 8. The resulting number
of solar neutrinos detected in the RoI${}_{\text{CNO}}$ is
$N_{\nu}=643_{-224}^{+235}\,\text{(stat)}_{-30}^{+37}\,\text{(sys)}$ for
Phase-I and
$N_{\nu}=2719_{-494}^{+518}\,\text{(stat)}_{-83}^{+85}\,\text{(sys)}$ for
Phase-II+III, including all systematics and correcting for the fit response
bias. The quoted uncertainties are calculated from the posterior distributions
using an $68\%$ equal-tailed credible interval (CI). The one-sided zero
neutrino hypothesis can be excluded with $P(N_{\nu}=0)=2.8\times 10^{-5}$
($\sim 4.2\sigma$) for Phase-I and $P(N_{\nu}=0)=6.4\times 10^{-11}$ ($\sim
6.5\sigma$) for Phase-II+III.

##### CID results on CNO

The interpretation of the CID results requires the correct treatment of the
physical boundaries of the analysis, i.e. $0\leq N_{\nu}\leq 2990$ (5974) for
Phase-I (Phase-II+III), respectively. This is done in a Bayesian
interpretation, based on the posterior distribution $P\left(N_{\nu}\right)$
shown in Fig. 8.

Figure 8: The CID measured posterior probability distributions for the number
of solar neutrinos $N_{\nu}$ in the RoI${}_{\text{CNO}}$ for Phase-I (blue)
and Phase-II+III (red). All systematic effects are included. Figure 9: The CID
measured posterior probability for the number of CNO-$\nu$ events after
constraining $pep$ and 8B neutrinos for Phase-I (blue) and Phase-II+III (red).
All systematic effects are included.  Figure 10: The combined CID Phase-I +
Phase-II+III CNO-$\nu$ rate posterior distribution is shown in red. The blue,
violet and grey bands show the 68% CI, for the low metallicity SSM
B16-AGSS09met ($(3.52\pm 0.52)\,\text{cpd\,/\,100\,tonnes}$), the high
metallicity SSM B16-GS98 ($(4.92\pm 0.78)\,\text{cpd\,/\,100\,tonnes}$)
predictions [2, 28], and the combined CID result, respectively. All systematic
effects are included.

Next, the distribution of the number of CNO-$\nu$ events is estimated by
constraining the expected number of $pep$ and 8B neutrino events
($N_{pep+^{8}\text{B}}$) where the constraint on the number of $pep$ neutrinos
uses the SSM predictions [2] and 8B is constrained using the high precision
flux measurement of Super-Kamiokande [6] including model uncertainties, the
difference between HZ-SSM and LZ-SSM predictions, as well as the Borexino FV
and energy systematic uncertainties from Table II. This is done through the
convolution of the $N_{\nu}$ posterior distributions from Fig. 8 with the
predicted $P(N_{pep+^{8}\text{B}})$ probability distribution:
$P(N_{\text{CNO}})=P(N_{\nu})\ast P(-N_{pep+^{8}\text{B}})$. The resulting
$P(N_{\text{CNO}})$ posterior distributions are shown in Fig. 9. The CID
measurement for the number of CNO-$\nu$ events is then
$N_{\text{CNO}}=270_{-169}^{+218}\,\text{(stat)}_{-25}^{+33}\,\text{(sys)}$
for Phase-I and
$N_{\text{CNO}}=1146_{-486}^{+518}\,\text{(stat)}_{-89}^{+92}\,\text{(sys)}$
for Phase-II+III, where the uncertainty corresponds to the equal-tail $68\%$
CI within the physical boundaries, including all systematics.

It has been observed that Phase-I and Phase-II+III do not show prohibitively
different behavior for the full CID analysis-chain and the MC model is well
able to reproduce the data $\cos\alpha$ histograms for both phase selections
and for each selected energy region. It is then reasonable to combine the
conditionally independent results of Phase-I and Phase-II+III, through the
convolution of both posterior distributions
$P\left(N_{\text{CNO}}\right)^{\text{I+II+III}}=P\left(N_{\text{CNO}}\right)^{\text{I}}\ast
P\left(N_{\text{CNO}}\right)^{\text{II+III}}$. The probability that exactly
zero CNO-$\nu$ events contribute to the measured data CID $\cos\alpha$
distribution is $P(N_{\text{CNO}}=0)=1.35\times 10^{-3}$ for Phase-I,
$P(N_{\text{CNO}}=0)=5.87\times 10^{-5}$ for Phase-II+III, and
$P(N_{\text{CNO}}=0)=7.93\times 10^{-8}$ for the combined result. This
corresponds to a one-sided exclusion of the zero-CNO hypothesis at about
$5.3\sigma$ credible level for the combination of Phase-I and Phase-II+III.

The CNO-$\nu$ rate probability density function is calculated from the
measured posterior distribution of CNO-$\nu$ events, using the exposure of the
respective phases. The effective exposure is given by the product of the
fiducial mass, the detector live time, the TFC-exposure, the trigger
efficiency, and the fraction of CNO-$\nu$ events within the selected energy
region. The final CID result for the CNO-$\nu$ rate, using the full dataset of
Phase-I + Phase-II+III, is $R_{\text{CNO}}^{\text{CID}}=7.2\pm
2.5\,\left(\text{stat}\right)\pm
0.4\,\left(\text{sys}\right)\,_{-0.8}^{+1.1}\,\left(\text{nuisance}\right)\text{
cpd/100\,tonnes}=7.2_{-2.7}^{+2.8}\,\text{ cpd/100\,tonnes}$ The quoted
uncertainties now also show the systematic uncertainties from Table II
separately from the influence of the nuisance parameters
$\text{gv}_{\text{ch}}$ and $\Delta r_{\text{dir}}$. The quoted statistical
uncertainty corresponds to a hypothetical, perfect calibration of these CID
nuisance parameters. The results are summarized in Table III.

|  |
---|---|---
CID results | $P(N_{\text{CNO}}=0)$ |  $R_{\text{CNO}}$ $\left[\frac{\text{cpd}}{$100\text{\,}\mathrm{t}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{s}$}\right]$
Phase-I | $1.35\times 10^{-3}$ | $6.4_{-4.1}^{+5.2}$
Phase-II+III | $5.87\times 10^{-5}$ | $7.3_{-3.2}^{+3.4}$
Combined | $7.93\times 10^{-8}$ | $7.2_{-2.7}^{+2.8}$
|  |
Table III: CID CNO-$\nu$ results with systematic uncertainties.

These CID results are well in agreement with the HZ-SSM prediction of
$(4.92\pm 0.78)\text{ cpd/100\,tonnes}$ ($0.6\,\sigma$), while the LZ-SSM
prediction $(3.52\pm 0.52)\text{ cpd/100\,tonnes}$ ($1.1\,\sigma$) is 1.7
times less likely to be true, given the results of the Borexino CID analysis.

##  4 Combined CID and multivariate analysis

In this Section we combine the CID analysis with the standard multivariate fit
of Phase-III to improve the result on CNO neutrinos. This is done by including
the posterior distributions of solar neutrinos from the CID analysis, shown in
Fig. 8, in the multivariate analysis likelihood. By statistically subtracting
the sub-dominant 8B neutrinos contribution and converting $N_{\text{CNO+pep}}$
to the corresponding interaction rate, it is possible to use these posterior
distributions as external likelihood terms in the minimization routine.
Following this procedure, two multiplicative pull terms constraining the
number of CNO and $pep$ neutrino events are used: the first one is related to
the Phase-I ($\mathcal{L}^{\mathrm{P-I}}_{\mathrm{CID}}$), while the second
one refers to Phase-II+III datasets
($\mathcal{L}^{\mathrm{P-II+III}}_{\mathrm{CID}}$).

The overall combined likelihood used for this analysis becomes:

$\mathcal{L}_{\mathrm{MV+CID}}=\mathcal{L}_{\mathrm{MV}}\cdot\mathcal{L}_{\mathrm{pep}}\cdot\mathcal{L}_{\mathrm{{}^{210}Bi}}\cdot\mathcal{L}^{\mathrm{P-I}}_{\mathrm{CID}}\cdot\mathcal{L}^{\mathrm{P-II+III}}_{\mathrm{CID}}$
(4)

where the first three terms correspond to an improved version of the standard
multivariate analysis described in [19]. This approach couples the one-
dimensional Poisson likelihood for the TFC-Tagged dataset with a two-
dimensional one (energy and radius) for the TFC-subtracted, to enhance the
separation between signal and backgrounds. We have improved the binning
optimization and used an updated version of Monte Carlo.

The $pep$ neutrinos interaction rate is constrained with 1.4% precision to the
$2.74\pm 0.04$ cpd/100 tonnes value, by combining the Standard Solar Model
predictions [2], the most current flavor oscillation parameters set [29] and
the solar neutrino data [30, 31]. This constraint is applied with the Gaussian
pull term $\mathcal{L}_{\mathrm{pep}}$. An upper limit on the 210Bi rate of
($10.8\pm 1.0$ cpd/100 tonnes) is applied with the half Gaussian term
$\mathcal{L}_{\mathrm{{}^{210}Bi}}$. This upper limit is obtained from the
rate of the 210Bi daughter 210Po (see [4, 19] for more details).

### 4.1 Results

As in [19], the energy RoI for the multivariate analysis is
$$0.32\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$<T_{e}<$2.64\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$$
for electron recoil kinetic energy. The reconstructed energy spectrum scale is
quantified in the $N_{h}$ estimator, representing total number of detected
hits for a given event (see Sec. 2). The dataset is the same one analyzed in
[19], in which the exposure amounts to 1431.6 days $\times$ 71.3 tonnes.

Along with CNO solar neutrinos, the free parameters of the fit are divided
into three categories: internal (85Kr and 210Po) and external (208Tl, 214Bi,
and 40K) backgrounds, cosmogenic backgrounds (11C, 6He, and 10C), and solar
neutrinos (7Be). Since 8B solar neutrinos exhibit a flat and marginal
contribution, the corresponding interaction rate is fixed at high-metallicity
expectations from Solar Standard Model. As discussed in Eq. 4, the interaction
rates of $pep$ neutrinos and 210Bi background are constrained with likelihood
pull terms, and CID results reported in Sec. 3.5 are accounted for as
additional external constraints. The result of the fit for the energy and
radial projections is shown in Fig. 11.

Figure 11: Multivariate fit results for the TFC-subtracted dataset, projected
over the energy (top panel) and the radius (bottom panel) dimensions. For both
projections, the sum of the individual components from the fit (magenta) is
superimposed on the data (grey points). CNO neutrinos, 210Bi and pep neutrinos
contributions are displayed in solid red, dashed blue and dotted green lines,
respectively, while the other spectral components (7Be and 8B neutrinos, other
backgrounds) are shown in grey. The analysis has been performed using $N_{h}$
as energy estimator and the conversion to keV energy scale was performed only
for the plotting purposes. The radial fit components, that are the uniform and
the external backgrounds contributions, are shown in solid blue and dashed
grey lines respectively.

The multivariate fit returns an interaction rate of CNO neutrinos of
$6.7^{+1.2}_{-0.7}\,\mathrm{cpd/100\,tonnes}$ (statistical error only). The
agreement between the model and data is quantified with a $p$ value of 0.2.

To account for sources of systematic uncertainty, the same Monte Carlo method
described in [4, 19] has been adopted. In a nutshell, hundred thousands Monte
Carlo pseudo-experiments were generated, including relevant effects able to
introduce a systematic error, such as the energy response non linearity and
non uniformity, the time variation of the scintillator light yield and the
different theoretical models for the 210Bi spectral shape. The analysis is
performed on these pseudo datasets assuming the standard response to study how
this impacts the final result on CNO, yielding a total systematic uncertainty
of ${}^{+0.31}_{-0.24}\,\mathrm{cpd/100\,tonnes}$. Other sources of systematic
error are included in the estimation of the upper limit on 210Bi
contamination, as discussed more in detail in [4, 19].

The negative log-likelihood profile as a function of the CNO rate is reported
in Fig. 12. The solid and dashed black lines show the results with and without
systematic uncertainty, respectively. The result without CID constraint
reported in [19] is included (blue line), for comparison. The improvement is
clear especially for the upper value of the CNO rate. The CNO interaction rate
is extracted from the 68% quantile of the likelihood profile convoluted with
the resulting systematic uncertainty, as
$R^{\mathrm{MV+CID}}\mathrm{(CNO)}=6.7^{+1.2}_{-0.8}\,\mathrm{cpd/100\,tonnes}$.
The significance to the no-CNO hypothesis reaches about 8$\sigma$ C.L., while
the resulting CNO flux at Earth is $\Phi(\mathrm{CNO})=6.7^{+1.2}_{-0.8}\times
10^{8}\,\mathrm{cm^{-2}\,s^{-1}}$. Following the same procedure used in [19],
we use this result together with the 8B flux obtained from the global analysis
of all solar data to determine the abundance of C + N with respect to H in the
Sun with an improved precision, for which we find $\rm{N_{\rm
CN}}=5.81^{+1.22}_{-0.94}\times 10^{-4}$. This error includes both the
statistical uncertainty due to the CNO measurement, and the systematic errors
due to the additional contribution of the SSM inputs, to the 8B flux
measurement, and to the 13N/15O fluxes ratio. Similar to what was inferred
from our previous publication, this result is in agreement with the High
Metallicity measurements [14, 15], and features a 2$\sigma$ tension with Low
Metallicity ones [16, 17, 18]. Similarly, if we combine the new CNO result
with the other Borexino results on 7Be and 8B in a frequentist hypothesis test
based on a likelihood ratio test statistics we find that, assuming the HZ-SSM
to be true, our data disfavours LZ-SSM at $3.2\sigma$ level.

Figure 12: CNO-$\nu$ rate negative log-likelihood ($-2\Delta\ln\mathcal{L}$)
profile obtained from the 2-dimensional multivariate spectral fit combined
with the CID analysis constraint, with and without folding in the systematic
uncertainties (black dashed and solid lines respectively). The blue line shows
the same profile obtained in the previously published analysis without CID
constraint [19]. The blue, violet, and gray vertical bands show 68% confidence
intervals (CI) for the low-metallicity SSM B16-AGSS09met ($3.52\pm 0.52$
cpd/100 tonnes) and the high-metallicity SSM B16-GS98 ($4.92\pm 0.78$ cpd/100
tonnes) predictions [2, 28], and the new Borexino result including systematic
uncertainty, respectively.

##  5 Conclusions

In this work we have presented the results on CNO solar neutrinos obtained
using the ”Correlated and Integrated Directionality” (CID) technique.

We have shown that the CID technique can be used to extract the CNO signal
without any a priori assumptions on the backgrounds, in particular that of
210Bi. The Phase-I (May 2007 to May 2010, 740.7 days) and Phase-II+III
(December 2011 to October 2021, 2888.0 days) datasets have been analyzed
independently to investigate possible variations of the detector response over
time. By adopting the Bayesian statistics, we have combined the conditionally
independent results of Phase-I and Phase-II+III: the resulting CNO rate
obtained with CID only is $7.2_{-2.7}^{+2.8}\,\text{cpd/100 \, tonnes}$. The
no-CNO hypothesis including the $pep$ constraint only is rejected at
5.3$\sigma$ level. This result, albeit less precise than the one published by
Borexino using the standard multivariate analysis, is the first obtained
without the application of a 210Bi constraint.

We have also obtained an improved CNO solar neutrino result by combining the
standard multivariate analysis with the CID technique. The CID technique helps
in separating the solar signal from non solar backgrounds, improving the
significance and precision of the CNO measurement with respect to the result
previously published by Borexino. The resulting CNO interaction rate is
$6.7^{+1.2}_{-0.8}\,\mathrm{cpd/100\,tonnes}$ and the significance against the
absence of a CNO signal, considered as the null hypothesis, is about
8$\sigma$. The C+N abundance with respect to H is calculated from this result
following the procedure adopted in [19] and is found to be $\rm{N_{\rm
CN}}=5.81^{+1.22}_{-0.94}\times 10^{-4}$, compatible with the SSM-HZ
metallicity measurements.

In conclusion, we have shown that the directional information of the Cherenkov
radiation can be effectively combined with the spectral information coming
from scintillation, for solar neutrino studies. This combined detection
approach provides a measurement that is more powerful than the individual
methods on their own. The sensitivity of the CID method could be significantly
improved in future liquid scintillator-based detectors by optimizing the
Cherenkov-to-scintillation ratio and by performing dedicated calibrations
campaigns.

Acknowledgments: We acknowledge the generous hospitality and support of the
Laboratori Nazionali del Gran Sasso (Italy). The Borexino program is made
possible by funding from Istituto Nazionale di Fisica Nucleare (INFN) (Italy),
National Science Foundation (NSF) (USA), Deutsche Forschungsgemeinschaft
(DFG), Cluster of Excellence PRISMA+ (Project ID 39083149), and recruitment
initiative of Helmholtz-Gemeinschaft (HGF) (Germany), Russian Foundation for
Basic Research (RFBR) (Grants No. 19-02-00097A), Russian Science Foundation
(RSF) (Grant No. 21-12-00063) and Ministry of Science and Higher Education of
the Russian Federation (Project FSWU-2023-0073) (Russia), and Narodowe Centrum
Nauki (NCN) (Grant No. UMO 2017/26/M/ST2/00915) (Poland). We gratefully
acknowledge the computing services of Bologna INFN-CNAF data centre and U-Lite
Computing Center and Network Service at LNGS (Italy).

## References

  * Bahcall [1989] J. N. Bahcall,  _Neutrino Astrophysics_ (Cambridge University Press, 1989).
  * Vinyoles  _et al._ [2017] N. Vinyoles _et al._ , Astrophysical Journal 835, 202 (2017).
  * Salaris and Cassisi [2005] M. Salaris and S. Cassisi,  _Evolution of Stars and Stellar Populations_ (John Wiley & Sons, Ltd, 2005).
  * Agostini  _et al._ [2020a] M. Agostini _et al._ (Borexino Collaboration), Nature 587, 577–582 (2020a).
  * Agostini  _et al._ [2018a] M. Agostini _et al._ (Borexino Collaboration), Nature 562, 505 (2018a).
  * Abe  _et al._ [2016a] K. Abe _et al._ (Super-Kamiokande Collaboration), Physical Review D 94, 052010 (2016a).
  * Anderson  _et al._ [2019] M. Anderson _et al._ (SNO+ Collaboration), Physical Review D 99, 012012 (2019).
  * Cleveland  _et al._ [1998] B. T. Cleveland _et al._ (Homestake Collaboration), Astrophysical Journal 496, 505 (1998).
  * Abdurashitov  _et al._ [2009] J. N. Abdurashitov _et al._ (SAGE Collaboration), Physical Review C 80, 015807 (2009).
  * Kaether  _et al._ [2010] F. Kaether, W. Hampel, G. Heusser, J. Kiko, and T. Kirsten, Physics Letters B 685, 47 (2010).
  * Araki  _et al._ [2005] T. Araki _et al._ (KamLAND Collaboration), Physical Review Letters 94, 081801 (2005).
  * Ahmad  _et al._ [2002] Q. R. Ahmad _et al._ (SNO Collaboration), Physical Review Letters 89, 011301 (2002).
  * Bellini  _et al._ [2011] G. Bellini _et al._ (Borexino Collaboration), Physical Review Letters 107, 141302 (2011).
  * Grevesse and Sauval [1998] N. Grevesse and A. Sauval, Space Science Reviews 85, 161 (1998).
  * Magg  _et al._ [2022] E. Magg _et al._ , Astron. Astrophys. 661, A140 (2022).
  * Caffau  _et al._ [2010] E. Caffau, H.-G. Ludwig, M. Steffen, B. Freytag, and P. Bonifacio, Solar Physics 268, 255 (2010).
  * Asplund  _et al._ [2009] M. Asplund _et al._ , Annu. Rev. Astron. Astrophys. 47, 481 (2009).
  * Asplund  _et al._ [2021] M. Asplund _et al._ , Astron. Astrophys. 653, A141 (2021).
  * Appel  _et al._ [2022] S. Appel _et al._ (Borexino Collaboration), Physical Review Letters 129, 252701 (2022).
  * Agostini  _et al._ [2022a] M. Agostini _et al._ (Borexino Collaboration), Physical Review D 105, 052002 (2022a).
  * Agostini  _et al._ [2022b] M. Agostini _et al._ (Borexino Collaboration), Physical Review Letters 128, 091803 (2022b).
  * Alimonti  _et al._ [2009] G. Alimonti _et al._ (Borexino Collaboration), Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 600, 568 (2009).
  * Bellini  _et al._ [2014] G. Bellini _et al._ (Borexino Collaboration), Physical Review D 89, 112007 (2014).
  * Agostini  _et al._ [2018b] M. Agostini _et al._ (Borexino Collaboration), Astroparticle Physics 97, 136 (2018b).
  * Back  _et al._ [2012] H. Back _et al._ (Borexino Collaboration), Journal of Instrumentation 7, P10018 (2012).
  * Abe  _et al._ [2016b] K. Abe _et al._ (Super-Kamiokande Collaboration), Physical Review D 94, 052010 (2016b).
  * D’Agostini [2003] G. D’Agostini, Reports on Progress in Physics 66, 1383 (2003).
  * Agostini  _et al._ [2020b] M. Agostini _et al._ (Borexino Collaboration), European Physics Journal C 80, 1091 (2020b).
  * Capozzi  _et al._ [2021] F. Capozzi, E. Di Valentino, E. Lisi, A. Marrone, A. Melchiorri, and A. Palazzo, Physical Review D 104, 083031 (2021).
  * Vescovi  _et al._ [2020] D. Vescovi, C. Mascaretti, F. Vissani, L. Piersanti, and O. Straniero, J. Phys. G 48, 015201 (2020).
  * Bergstrom  _et al._ [2016] J. Bergstrom, M. C. Gonzalez-Garcia, M. Maltoni, C. Pena-Garay, A. M. Serenelli, and N. Song, Journal of High Energy Physics 03, 132 (2016).