The BOREXINO Collaboration – email<EMAIL_ADDRESS>
# Novel techniques for $\alpha/\beta$ pulse shape discrimination in Borexino

D. Basilico1 G. Bellini1 J. Benziger2 R. Biondi3,a B. Caccianiga1 F.
Calaprice4 A. Caminata5 A. Chepurnov6 D. D’Angelo1 A. Derbin7 A. Di Giacintov3
V. Di Marcello3 X.F. Ding4,b A. Di Ludovico4,c L. Di Noto5 I. Drachnev7 D.
Franco8 C. Galbiati4,9 C. Ghiano3 M. Giammarchi1 A. Goretti4 M. Gromov6 D.
Guffanti10,d Aldo Ianni3 Andrea Ianni4 A. Jany11 V. Kobychev12 G. Korga13,14
S. Kumaran15,16,3 M. Laubenstein3 E. Litvinovich17,18 P. Lombardi1 I.
Lomskaya7 L. Ludhova15,16 I. Machulin17,18 J. Martyn10 E. Meroni1 L.
Miramonti1 M. Misiaszek11 V. Muratova7 R. Nugmanov17 L. Oberauer19 V.
Orekhov10 F. Ortica20 M. Pallavicini5 L. Pelicci15,16 Ö. Penek15,f L.
Pietrofaccia4,c N. Pilipenko7 A. Pocar21 G. Raikov17 M.T. Ranalli3 G. Ranucci1
A. Razeto3 A. Re1 N. Rossi3 S. Schönert19 D. Semenov7 G. Settanta15,g M.
Skorokhvatov17,18 A. Singhal15,16 O. Smirnov22 A. Sotnikov22 R. Tartaglia3 G.
Testera5 E. Unzhakov7 A. Vishneva22 R.B. Vogelaar23 F. von Feilitzsch19 M.
Wojcik11 M. Wurm10 S. Zavatarelli5 K. Zuber24 G. Zuzel11 1Dipartimento di
Fisica, Università degli Studi e INFN, 20133 Milano, Italy  2Chemical
Engineering Department, Princeton University, Princeton, NJ 08544, USA  3INFN
Laboratori Nazionali del Gran Sasso, 67010 Assergi (AQ), Italy  4Physics
Department, Princeton University, Princeton, NJ 08544, USA  5Dipartimento di
Fisica, Università degli Studi e INFN, 16146 Genova, Italy  6Lomonosov Moscow
State University Skobeltsyn Institute of Nuclear Physics, 119234 Moscow,
Russia  7St. Petersburg Nuclear Physics Institute NRC Kurchatov Institute,
188350 Gatchina, Russia  8AstroParticule et Cosmologie, Université Paris
Diderot, CNRS/IN2P3, CEA/IRFU, Observatoire de Paris, Sorbonne Paris Cité,
75205 Paris Cedex 13, France  9Gran Sasso Science Institute, 67100 L’Aquila,
Italy  10Institute of Physics and Excellence Cluster PRISMA+, Johannes
Gutenberg-Universität Mainz, 55099 Mainz, Germany  11M. Smoluchowski Institute
of Physics, Jagiellonian University, 30348 Krakow, Poland  12Institute for
Nuclear Research of NASU, 03028 Kyiv, Ukraine  13Department of Physics, Royal
Holloway University of London, Egham, Surrey,TW20 0EX, UK  14Institute of
Nuclear Research (Atomki), Debrecen, Hungary  15Institut für Kernphysik,
Forschungszentrum Jülich, 52425 Jülich, Germany  16III. Physikalisches
Institut B, RWTH Aachen University, 52062 Aachen, Germany  17National Research
Centre Kurchatov Institute, 123182 Moscow, Russia  18National Research Nuclear
University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia
19Physik-Department, Technische Universität München, 85748 Garching, Germany
20Dipartimento di Chimica, Biologia e Biotecnologie, Università degli Studi e
INFN, 06123 Perugia, Italy  21Amherst Center for Fundamental Interactions and
Physics Department, University of Massachusetts, Amherst, MA 01003, USA
22Joint Institute for Nuclear Research, 141980 Dubna, Russia  23Physics
Department, Virginia Polytechnic Institute and State University, Blacksburg,
VA 24061, USA  24Department of Physics, Technische Universität Dresden, 01062
Dresden, Germany  aPresent address: Max-Planck-Institut für Kernphysik, 69117
Heidelberg, Germany  bPresent address: IHEP Institute of High Energy Physics,
100049 Beijing, China  cPresent address: INFN Laboratori Nazionali del Gran
Sasso, 67010 Assergi (AQ), Italy  dPresent address: Dipartimento di Fisica,
Università degli Studi e INFN Milano-Bicocca, 20126 Milano, Italy  ePresent
address: Department of Physics and Astronomy, University of California,
Irvine, California, USA  fPresent address: GSI Helmholtzzentrum für
Schwerionenforschung GmbH, 64291 Darmstadt, Germany  gPresent address:
Istituto Superiore per la Protezione e la Ricerca Ambientale, 00144 Roma,
Italy

###### Abstract

Borexino could efficiently distinguish between $\alpha$ and $\beta$ radiation
in its liquid scintillator by the characteristic time profile of their
scintillation pulse. This $\alpha/\beta$ discrimination, first demonstrated at
the tonne scale in the Counting Test Facility prototype, was used throughout
the lifetime of the experiment between 2007 and 2021. With this method,
$\alpha$ events are identified and subtracted from the $\beta$-like solar
neutrino events. This is particularly important in liquid scintillator as
$\alpha$ scintillation is quenched many-fold. In Borexino, the prominent 210Po
decay peak was a background in the energy range of electrons scattered from
7Be solar neutrinos. Optimal $\alpha/\beta$ discrimination was achieved with a
_multi-layer perceptron neural network_ , which its higher ability to leverage
the timing information of the scintillation photons detected by the
photomultiplier tubes. An event-by-event, high efficiency, stable, and uniform
pulse shape discrimination was essential in characterising the spatial
distribution of background in the detector. This benefited most Borexino
measurements, including solar neutrinos in the $pp$ chain and the first direct
observation of the CNO cycle in the Sun. This paper presents the key
milestones in $\alpha/\beta$ discrimination in Borexino as a term of
comparison for current and future large liquid scintillator detectors.

## Introduction

For as long as it operated, Borexino was the only detector capable of
measuring solar neutrino interactions (position and energy) on an event-by-
event basis with a threshold $\gtrsim 150$ keV, _i.e._ , down to the 14C
$\beta$-spectrum end-point. An important feature of Borexino was the
possibility to efficiently separate events initiated by recoiling electrons
($\beta$-like events) versus $\alpha$ particles. The former include solar
neutrino interactions as well as background from $\beta$ and $\gamma$ decays.
This is possible via pulse-shape discrimination (PSD) techniques that exploit
the different time profile of the scintillation emission for $\alpha$ and
$\beta$-like events (see, _e.g._ , [1]). The so-called $\alpha/\beta$
discrimination played an important role in solar neutrino measurements
throughout the Borexino data taking between 2007 and 2021. It is worth noting
that Borexino also achieved $\beta^{-}/\beta^{+}$ separation via PSD, as
reported in [2] and [3]; the latter topic is, however, outside the scope of
the present article.

PSD for $\alpha/\beta$ separation was first studied within the Borexino
program with the 4-tonne “Counting Test Facility” (CTF) prototype [5, 6]. The
original method is based on the _Gatti_ parameter [4] and enabled a
statistical subtraction of $\alpha$ background, especially from 210Po, from
the measured energy spectrum. Monochromatic, 5.3 MeV 210Po $alpha$s
($Q_{\alpha}$=5407 keV) appeared in the Borexino liquid scintillator as a peak
at $\sim$500 keV of electron-equivalent energy due to a greater than ten-fold
quenching of the scintillation for these highly ionizing tracks [6]. At the
start of the Borexino data taking, the 210Po rate was $\sim 8000$ counts per
day per 100 tonnes (hereafter, cpd/100 t). Quenching was also observed for
other $\alpha$ particles. These include those from the thoron (220Rn) and
radon (222Rn) decay chains which are handily identified using their time
coincidence, _e.g._ , 212Po (8954 keV), 214Po (7833 keV), and 218Po (6114
keV).

Because of its quenching, the 210Po peak falls within the 7Be solar neutrino
Compton-like energy spectrum, which presents a characteristic shoulder at 662
keV. Although in this case the 7Be shoulder appears at higher energy than the
210Po peak, making it possible for the multi-parameter spectral fit to clearly
identify these two separate components, the Borexino analysis was performed
both with and without bin-by-bin statistical $\alpha/\beta$ subtraction of the
210Po peak to ensure that there was no subtle bias due to the presence of the
$\alpha$ background. This was particularly true in Phase-I of the experiment,
when the 210Po activity was more than two orders of magnitude greater than the
7Be event rate ($\sim$50 cpd/100 t over the entire energy range). In each bin,
we assumed the Gatti parameter to be normally distributed with a mean value
linearly dependent on energy [7, 8, 9].

As data-taking progressed, the 210Po naturally decayed with a lifetime
$\tau=199.6$ d. The reduction of this background was, however, counterbalanced
by a progressive degradation of the energy resolution of the detector due to
the loss of photomultiplier tubes (PMTs). The count of working PMTs decreased
for $\sim$2000 units in mid 2007 to $\sim$1000 units at the end of 2021. This
effect and the need for a more uniform, stable, and higher efficiency
$\alpha/\beta$ discrimination for the study of CNO solar neutrinos suggested
exploring novel techniques based on neural networks already extensively
employed in particle physics. We pursued neural networks based on _multi-layer
perceptron_ (MLP). The subject of this paper is the description of the MLP
input parameters, structure, and training strategy given the Borexino
scintillator properties and layout. It also presents studies of the network’s
efficiency for $\alpha/\beta$ discrimination.

In Sec. I, the main characteristics of the Borexino detector and its main
physics results relevant to this article are briefly reviewed. In Sec. II, the
$\alpha/\beta$ PSD in Borexino using the Gatti parameter is presented. In Sec.
III, the implementation strategy of the MLP on the Borexino scintillator time
profile is described along with an evaluation of its performance and
efficiency. Finally, in Sec. IV, the impact of $\alpha/\beta$ MLP
discrimination on the CNO solar neutrino analysis and on other Borexino
results over its 14 years lifetime is discussed.

##  I The Borexino Experiment

Species | Rate [cpd/100 t ] | Flux [cm-2 s-1 ]
---|---|---
pp | $(134\pm 10)^{+6}_{-10}$ | $(6.1\pm 0.5)_{-0.5}^{+0.3}\times 10^{10}$
7Be | $(48.3\pm 1.1)^{+0.4}_{-0.7}$ | $(4.99\pm 0.11)_{-0.08}^{+0.06}\times 10^{9}$
pep (HZ) | $(2.7\pm 0.4)^{+0.1}_{-0.2}$ | $(1.3\pm 0.3)_{0.1}^{+0.1}\times 10^{8}$
8B($>3$ MeV) | $0.223_{-0.022}^{+0.021}$ | $5.68_{-0.44}^{+0.42}\times 10^{6}$
hep |  $<0.002$ (90% CL) |  $<1.8\times 10^{5}$ (90% CL)
CNO | $6.7_{-0.8}^{+1.2}$ | $6.7_{-0.8}^{+1.2}\times 10^{8}$
Table 1: Solar neutrino interaction rates in Borexino and extrapolated solar
neutrino fluxes for the different components of the $pp$ chain and CNO cycle.
Rates are reported in cpd/100 t, while fluxes are reported in cm-2s-1. N.B.:
HZ stands for _high metallicity_ assumption.

Borexino was located in the Hall C of Laboratori Nazionali Gran Sasso (LNGS)
of the Italian Institute of Nuclear Physics (INFN) [10]. The detector had been
taking data from mid-2007 to the end of 2021, and is currently under
decommissioning. The detector is made of concentric layers of increasing
radiopurity (see for details _e.g._ [11]): the innermost core, called Inner
Vessel (IV), consists of about 280 tons of liquid scintillator (_pseudocumene_
mixed with 1.5 g/l of PPO as scintillating solute) contained inside an ultra-
pure nylon vessel with a thickness of 125 $\upmu$m and a radius of 4.25 m. A
Stainless Steel Sphere (SSS), filled up with the remaining $1000$ m3 of buffer
liquid (_pseudocumene_ mixed with DMP quencher) is instrumented with more than
2000 PMTs for detecting the scintillation light inside the IV. Finally, the
SSS is immersed in an about 2000 m3 Water Tank (WT), acting as Cerenkov veto,
equipped with 200 PMTs. Using results from the study of internal residual
contaminations and from the 2010 calibration, it results that the detector is
capable of determining the event position with an accuracy of $\sim 10$ cm (at
1 MeV) and the event energy with a resolution following approximately the
relation $\sigma(E)/E\simeq 5\%/\sqrt{E/[MeV]}$.

The Borexino data set is divided, according to the internal conventional
subdivision of the experimental program, in three different phases: Phase-I,
from May 2007 to May 2010, ended with the calibration campaign, in which the
first measurement of the 7Be solar neutrino interaction rate [7, 8, 9] and the
first evidence of the $pep$ [2] were performed; Phase-II, from December 2011
to May 2016, started after an intense purification campaign with unprecedented
reduction of the scintillator radioactive contaminants, in which a 10% first
spectroscopic observation of the $pp$ neutrinos [12] was published, and later
updated in the solar neutrino comprehensive analysis of all $pp$ chain
neutrino fluxes [3, 13, 14]; finally, Phase-III, from July 2016 to October
2021, after the thermal stabilisation program, in which the first detection of
the CNO neutrinos [15] and its subsequent improvements [16, 17] were achieved.
The most important solar neutrino results in terms of interaction rate and
corresponding fluxes are summarised in Tab. 1. Thanks to its unprecedented
radio-purity, Borexino has also set a lot of limits on rare processes [19, 20,
21, 22, 23] and performed other neutrino physics studies, as _e.g._ geo-
neutrino detection (for review, see _e.g._ [24]). As it will be highlighted in
the Sec. IV, all these important results are strongly dependent on the
$\alpha/\beta$ PSD optimisation. In the following Section, the $\alpha/\beta$
discrimination problem is introduced, starting from the first method exploited
by the Collaboration, based on the Gatti parameter.

##  II $\alpha/\beta$ discrimination: the _Gatti_ parameter

Figure 1: The reference $P_{\alpha}(t)$ (red) and $P_{\beta}(t)$ (black) pulse
shapes obtained by tagging the 222Rn-correlated 214Bi-214Po coincidences. The
dip at 180 ns is due to the dead time on every individual electronic channel
applied after each detected hit. The small knee around 60 ns is due to the
reflected light on the SSS surface and on the PMTs photo-cathodes. Figure 2:
The distribution of $G_{\alpha}$ (red) and $G_{\beta}$ (black) (see Eq. 2) for
events obtained by tagging the radon correlated 214Bi–214Po coincidences.

The $\alpha$/$\beta$ discrimination in Borexino is possible thanks to the
sizeable difference between the time distributions of the scintillation light
(pulse shape) for $\alpha$ and $\beta$-like events (see Fig. 1). For each
event meeting the threshold condition of a few tens of photo-electrons (PE) in
100 ns ($\sim 60$ keV), the arrival times of PEs on each PMTs are recorded. As
described in detail in [1], for each detected PE, the arrival time and the
charge are measured by an analogue and digital electronics chain. When a
trigger occurs, the time and the charge of each PMT, that has detected at
least one photoelectron in a time gate of 7.2 $\upmu$s, is recorded. The time
is measured by a time-to-digital converter (TDC) with a resolution of about
0.5 ns, while the charge (after integration and pulse shaping) is measured by
an 8 bit analogue-to-digital converter (ADC). The time resolution is smaller
than the intrinsic PMT time jitter of about 1.1 ns.

For each event, time, charge, and position are reconstructed by the offline
software. The code identifies in the recorded time window of 16 $\upmu$s a
group of time correlated hits, called “cluster”. Each event is generally made
of a single cluster, but in the case of fast coincidences, like 214Bi-214Po
and 85Kr-85mRb close decays, or in case of accidental pile-ups (often
occurring with very frequent 14C events), more than one cluster could be
identified as superposition of two of more different events. The position of
events is determined via a photon time-of-flight maximum likelihood method
with probability density functions (PDF) based on experimental data and Monte
Carlo simulations, resulting in an uncertainty of 10 cm for each of the three
Cartesian spatial coordinates. The spatial resolution is expected to scale
naively as $1/\sqrt{NPE}$ where NPE is the number of detected photoelectrons
[25]. For the most important analyses in Borexino, the fundamental event
selection is based on the following criteria: internal only trigger (no muon
veto coincidence), event time 2 ms off a preceding muon event, single cluster
in the acquisition window and position reconstructed in $r\lesssim 3$ m. These
cuts guarantee that the selected event is a neutrino-like candidate, i.e. an
event occurred in the innermost part of the IV ($\lesssim$100 t) and far
enough from the external background coming from the SSS and from the IV
structures.

After the application of the selection criteria listed above, the typical
Borexino spectrum shows a prominent 210Po $\alpha$ peak at about 500 keV, that
falls inside the 7Be energy window, see _e.g._ [8]. At the beginning of
Phase-I the 210Po activity was of order $10^{4}$ cpd/100 t. At the beginning
of Phase-II, more than 4 years later, the activity went down by one order of
magnitude to $\sim 10^{3}$ cpd/100 t, a bit more than expected because a
little amount of 210Po was reintroduced by the water extraction campaign.
Finally in Phase-III, after more than 4 years and thanks to the thermal
insulation campaign, which reduced drastically the scintillator convective
motions (see Sec. IV for further details), the 210Po activity was
significantly lowered by another order of magnitude, namely $\sim 10^{2}$
cpd/100 t. This allowed one to reach the condition of the CNO measurement via
the so-called 210Bi-210Po link [15].

An estimation of the 210Po activity and its possible independent
quantification for the Borexino analysis can be done for example by defining a
simple parameter called tail-to-tot (t2t), which is defined as the fractional
portion of the time distribution of the hits above a given characteristic time
$t_{0}$ with respect to the beginning of the scintillation, namely

${\rm\
\texttt{t2t}}=\frac{\int\limits_{t_{0}}^{\infty}S(t)\mathrm{d}t}{\int\limits_{0}^{\infty}S(t)\mathrm{d}t},$
(1)

where $S(t)$ is the scintillation time distribution. The characteristic time
$t_{0}$ can be optimised by maximising the figure of merit defined as the
difference between the t2t populations for $\alpha$ and $\beta$ events. This
sort of parameter works very well for example for separating electron and
nuclear recoils in liquid argon scintillation chambers [26], where the
scintillation light is basically made of a combination of two typical
exponential decay times, differing 3 order of magnitude from each other
(typically 6 and 1600 ns). This is not the case of the Borexino scintillation,
where the time behaviour is more complicated and less specific for different
particle types [27]. As a consequence, t2t in Borexino gives a more mild
$\alpha/\beta$ separation rather than a real high efficiency event
classification.

Figure 3: Example of $\alpha/\beta$ separation in the Gatti-Energy(NPE) space
(first 300 days of Borexino Phase-II) . The big blob on the top represents the
$\alpha$ distribution (210Po), while the bottom horizontal belt represents the
$\beta$-like component. Figure 4: The raw NPE charge spectrum after the basic
selection criteria (black), after the fiducial volume cut (blue), and after
the statistical subtraction of the $\alpha$-emitting contaminants (red).

A more efficient identification of $\alpha$/$\beta$, instead, can be performed
using discriminating procedures like the Gatti optimal filter [4]. The latter
allows one to classify two types of events with different, but known, time
distributions of hits as a function of time. Their reference shapes
$P_{\alpha}(t)$ and $P_{\beta}(t)$ are created by averaging the time
distributions of a large sample of events selected independently, without any
use of pulse shape variables. A practical way to build the reference shapes is
to use the 214Bi-214Po fast coincidence, originating from the 222Rn events in
the scintillator. The 214Bi-214Po coincidences (a few hundreds of $\upmu$s) in
Borexino are tagged with a space-time correlation with about 90% efficiency,
basically limited by the trigger threshold for the preceding $\beta$ event.
The first of the two events in time provides a pure $\beta$-like sample with
events mostly located in the energy interval 1500-3000 keV as superposition of
different $\beta$ and $\gamma$ lines, while the second provides a pure
$\alpha$ sample with events peaked at about 800 keV and smeared only by the
detector resolution. The radon events in the IV (with about one week lifetime)
are strictly related to invasive operations on the scintillator (especially at
the beginning of Phase-I and during the WE campaign), while are basically
absent in quiet periods as Phase-II and especially Phase-III.

The functions $P_{\alpha}(t)$ and $P_{\beta}(t)$ represent the PDFs as a
function of time of detecting a PE for events of type $\alpha$ or $\beta$,
respectively. Let $e(t)$ be the normalised time distribution of the light for
each event. The Gatti parameter $G$ is defined as

$G=\int e(t)w(t)dt,$ (2)

where $w(t)$ are the weights given by

$w(t)=\frac{P_{\alpha}(t)-P_{\beta}(t)}{P_{\alpha}(t)+P_{\beta}(t)}\cdot$ (3)

The $G$ parameter follows a probability distribution with the mean value
$\langle G_{\alpha,\beta}\rangle$, which depends on the particle type, namely

$\langle G_{\alpha,\beta}\rangle=\int P_{\alpha,\beta}(t)w(t)dt.$ (4)

In the Borexino scintillation, the Gatti mean values are empirically found to
be linearly decreasing with energy. Finally, considering the Poissonian
statistical fluctuations of the entries in each time bin, the corresponding
variance, following the variance expansion identity, reads

$\sigma_{G_{\alpha,\beta}}^{2}=\int P^{2}_{\alpha,\beta}(t)w(t)dt-\langle
G_{\alpha,\beta}\rangle^{2}.$ (5)

In the real experimental case, the integration in Eqs. 4 and 5 are converted
into a sum over histograms binned at 1 ns from zero to about 1.5 $\upmu$s. In
the scintillator used by Borexino, $\alpha$ pulses are slower and have
therefore a longer tail with respect to $\beta$ pulses. This feature
represents basically the key for the $\alpha/\beta$ separation. Examples of
reference shapes $P_{\alpha}(t)$ and $P_{\beta}(t)$, from the 214Bi-214Po
tagging of the radon events from Phase-I, are shown in Fig. 1: the dip at 180
ns is due to the dead time on every individual electronic channel applied
after each detected hit. The small knee around 60 ns is due to the reflected
light on the SSS surface and on the PMTs photo-cathodes. The distributions of
the corresponding G parameters ($G_{\alpha}$ and $G_{\beta}$) for events with
respect to these reference PDFs are shown in Fig. 2. The two distributions,
resembling Gaussian shapes, are partially overlapped due to the sizeable $G$
variance. As a consequence, when the number of $\alpha$ events largely exceeds
that of the $\beta$’s, a high efficiency event-by-event $\alpha/\beta$
selection is anyway limited. In principle, a bin-by-bin statistical separation
of the two event populations is possible, whenever the $G_{\alpha,\beta}$
distribution are known either analytically or through a Monte Carlo
simulation. Since the mean values and the variances of $G_{\alpha,\beta}$ are
energy dependent, their distributions are fitted to two Gaussian models for
each bin in the energy spectrum of interest, and their value are forcibly
constraint around the linear dependence guess. The integral of the fitted
curves represents the relative contribution of each species in each energy
bin, and the $\alpha$ contribution is subtracted from the total bin content,
thus obtaining the $\beta$-like spectrum by statistical subtraction.

Figure 5: Example of $\alpha/\beta$ statistical subtraction with the
analytical curves for events in the energy range 200-205 NPE. The blue and red
lines show the individual Gaussian fits to the Gatti parameter distributions
for the $\beta$ and $\alpha$ components, respectively, while the green line is
the total fit.

We make the reasonable hypothesis that the underlying distributions are
Gaussian. The fit procedure also provides the error of the estimated particle
population to be replaced in the corresponding bin in which the subtraction is
performed. In bins where one species greatly outnumbers the other, for example
in the energy bins in which the 210Po is peaked, the mean values of the
Gaussian parameters are fixed to their predicted values, extrapolated from the
energy dependency trend, in order to avoid any possible bias in the
subtraction procedure. Figure 5 shows an example of the $G_{\alpha\beta}$
parameter in the energy range bin 200-205 NPE and its fit to the analytical
model. Furthermore any other possible double Gaussian fit bias, due to the
large difference in the two population statistics, is corrected according toy
Monte Carlo simulations with the same population ratio.

The statistical subtraction can be applied in the full 7Be energy window,
removing all $\alpha$ backgrounds coming mostly from 210Po, but also from
other 222Rn $\alpha$’s daughters such as 214Po and 218Po leaking the fast
coincidence cut. This secondary subdominant contamination is partially
affecting Phase-I, but is it completely negligible in Phase-II and Phase-III,
thanks to the better background condition achieved after the WE campaign and
210Po decay. The error associated to the statistical subtraction is propagated
as a systematic uncertainty on the final neutrino interaction rates [7, 8, 9].
It is worth mentioning that a possible bias due to the presence of the 210Po
peak is not negligible only in Phase-I when the 210Po activity is much larger
than 7Be neutrino interaction rates. In fact, in Phase-II and in Phase-III the
statistical subtraction of the $\alpha$ component is not applied and the 210Po
is simply quantified by the spectral fit, see _e.g._ [3, 15].

Figure 3 shows the Gatti distribution as a function of the event energy in NPE
for the first 300 days of Borexino Phase-II. The big blob on the top
represents the $\alpha$ distribution, consisting basically of 210Po events,
while the bottom horizontal belt represents the $\beta$-like component (solar
neutrinos and background). The Gatti parameter shows a neat separation of the
$\alpha/\beta$ population as a function of the energy.

Figure 4 shows the implementation of the $\alpha/\beta$ statistical
subtraction in Phase-I: the black curve represents the energy distribution of
all events before applying the basic selection criteria. The blue curve
represents the event energy distribution after the fiducial volume selection:
below 100 NPE the spectrum is dominated by 14C decay ($\beta^{-}$, $Q$=156
keV) [28] and the peak at 200 NPE is dominated by 210Po decays. The red curve
is the final spectrum after the statistical subtraction of the $\alpha$
component. The prominent feature around 300 NPE includes the Compton-like edge
due to 7Be solar neutrinos. Finally, the large bump peaked around $\sim$600
NPE is the spectrum of the cosmogenic 11C ($\beta^{+}$, $Q$=1.98 MeV, created
in situ by cosmic ray-induced showers).

Thanks to its relevant discrimination power, the $\alpha/\beta$ based on the
Gatti parameter has been applied in many Borexino analyses also as soft cut
for pre-selecting events and hard cut to locate the main $\alpha$
contaminants, as the 210Po within the fiducial volume, and for understanding
the nature of the main backgrounds [1], _e.g._ in the geo-neutrino analysis
[24]. In those cases, no statistical subtraction is applied, and the Gatti
parameter selects the $\alpha$ population with a given efficiency, depending
on the position of the cut itself.

The optimisation of the Gatti filter, already exploited in the Borexino CTF,
played a crucial role in many important Borexino studies. Nevertheless new
requirements and some drawbacks pushed the Collaboration to investigate other
novel techniques based on neural networks. As it will be described in Sec. IV,
the CNO feasibility study had been requiring, since the beginning of Phase-II,
a deep understanding of the spacial evolution of the 210Po contamination. This
analysis required, instead of a statistical subtraction, a high efficiency
event-by-event selection uniform in space, and easily modelable in energy and
time. The PMT loss, and the consequent resolution degradation, is affecting
the Gatti parameter distributions, but, more important, the Gatti has shown
since the beginning a spatial dependence, especially along the radial
direction. Figure 6 shows, indeed, the shift of the $G_{\alpha,\beta}$ as a
function of $r^{3}$ for 214Bi-214Po events (N.B.: plotting data as a function
of $r^{3}$ remove the spherical volume dependence over $r$). This dependence
is neither easily modelled nor completely reproducible in Monte Carlo
simulations.

Figure 6: Radial ($r^{3}$) dependence of the Gatti parameter $G$ (grey) for a
sample of 214Bi-214Po events. Red and blue points represent the mean values
with their uncertainty for $\alpha$ and $\beta$-like events, respectively.

Figure 7: Limitations of the t2t input variable in comparison with quantiles.
_Top:_ last t2t ($t>310$ [ns]) as a function of the energy in NPE for
$\alpha$’s (red) and $\beta$’s distribution. _Bottom:_ same distributions for
the last quantile (10 % tail of time PDF). Figure 8: Comparison between MLP
versions with test Sample-WEX. Form the top left: MLPv8, MLPv10, MLPv12 and
MLPv14. Red and blue PDFs represent $\alpha$’s and $\beta$’s events,
respectively.

Contrary to the Gatti filter, artificial neural networks, accepting plenty of
input parameters, and returning their corresponding ranking for the specific
case of the $\alpha/\beta$ selection, helped a lot in understanding the origin
of the radial dependence of the PSD. They offered a more uniform selector,
with a controllable dependence of the efficiency upon energy and time, as
discussed later in Sec. III.3. In the next Section, the strategy for the
implementation and tuning of a class of _multi-layer perceptron_ is reviewed.

##  III Improving the selection with MLP

###  III.1 Artificial neural networks

_Artificial neural networks_ (ANNs) are a major component of machine learning
and are designed to detect patterns in data [29]. This makes ANNs the optimal
solution for classifying (sorting data by predetermined categories), grouping
(finding similar characteristics among data and combining that data into
categories), and making forecasts based on the data.

An ANN is, more generally speaking, any simulated collection of interconnected
neurons, with each neuron producing a certain response at a given set of input
signals. The input data can be values for the characteristics of an external
data sample, such as images or documents, or it can be outputs from other
neurons.

By applying an external signal to some input neurons, the network is put into
a defined state that can be measured from the response of one or several
output neurons. One can therefore view the neural network as a mapping from a
space of input variables $x_{1},...,x_{n_{\rm var}}$ onto a _one-dimensional_
(_e.g._ in case of a signal-versus-background discrimination problem) or
_multi-dimensional_ space of output variables. The mapping is nonlinear if at
least one neuron has a nonlinear response to its input. It is important
noticing here that Gatti parameter is linear over the input parameters, as one
can easily see from Eq. (2), therefore, given the input data set, it cannot do
better than their intrinsic statistical power.

A multilayer perceptron (MLP) is a class of feed-forward artificial neural
network. An MLP consists of at least three layers of nodes: an input layer, a
hidden layer and an output layer. Except for the input nodes, each node is a
neuron that uses a nonlinear activation function. MLP utilises a supervised
learning technique called _back-propagation_ for training.

###  III.2 TMVA package

The Toolkit for Multivariate Analysis (TMVA) provides a ROOT-integrated
environment for the processing, parallel evaluation and application of
multivariate classification techniques [30]. TMVA is specifically designed for
the needs of high-energy physics (HEP) applications where the search for ever
smaller signals in ever larger data sets has become essential to extract a
maximum of the available information from the data. Multivariate
classification methods based on machine learning techniques have become an
essential ingredient in most of the HEP analyses. The package hosts a large
variety of multivariate classification algorithms, _e.g._ artificial neural
networks (three different MLPs implementations), support vector machines
(SVM), boosted decision trees (BDT), _etc._

Independent input data sets used for _training_ and _testing_ of multivariate
methods must be defined prior to the algorithm implementation. The most
important step is to identify such input parameters that are important in
order to obtain the highest possible efficiency of the pulse shape
discrimination of signals, and this can be done looking at the returned
ranking of the variable themselves at each controlled trial.

###  III.3 Selection of input variable and different versions

It is standard practice to normalise the input variables before integrating
them into the ANN. In the Borexino case of $\alpha/\beta$ discrimination, a
set of t2t variables were defined for ten different $t_{0}$, according to Eq.
(1). Due to the fact that the distributions for $\alpha$ and $\beta$ (Fig.1)
differ mainly in the tails, times after 10 ns were chosen, i.e. $t_{0}$ in the
set $\\{35,70,105,140,175,210,245,280,315,350\\}$ (ns). To this set, the root
mean square (RMS) and kurtosis of the photoelectron time distribution were
added. At this stage, having a set made of t2t’s, RMS and kurtosis in the
input vector, the MLP algorithm returns the discrimination efficiency similar
to Gatti, as expected. The statistical theory is absolutely constraining here,
so it was necessary to try to add information not present in the first trial.

Figure 9: Zoom of the ROC curves around the region of interest for Gatti and
four MLP versions according to the color legend inset. The analysis is
performed over the Sample-WEX test sample.

A breakthrough came when it was noticed that the time distribution of the
scintillation events is analysed after the event position correction,
determined by the time-of-flight of photons originated in a point-like
scintillation and propagated towards the PMTs. From subsequent trials, it was
observed that the mean-time variable of the hits calculated before the
position reconstruction (“non reconstructed cluster”) was adding some missing
information, possibly lost with the position correction. This recovered
information improved the $\alpha/\beta$ discrimination, even solving the
radial dependence observed in the Gatti parameter. This mean-time is basically
the mean of the temporal PDF of the scintillation events, in which the times
are associated with the photomultiplier reference system.

This finding also clarified why in CTF the $\alpha/\beta$ discrimination was
working in a more efficient way. In practice, since the CTF detector was a few
meters small, there was basically no bias due to off-centre event
reconstruction. This guess is also confirmed for events located in a region
very close to the centre of Borexino, where the Gatti parameter does not show
a substantial bias, and exhibits a very high efficiency.

Figure 10: Comparison between the radial efficiencies of the PSD cut for
MLPv8$<0.3$ (green) and Gatti$>0$ (red). In order to account for the cubic
increase of the statistics for the radial distribution, the efficiency is
reported as a function of $r^{3}$, instead of $r$. Figure 11: Example of ‘MLP-
subtracted” (red) and “MLP-complementary” (blue) histogram fitted with the
MLP-complementary method. The red and blue distributions represent $\alpha$
and $\beta$ complementary sets, whereas the violet and cyan curves represent
the fit models accounting for the exponential dependence of the MLP cut as in
Eq. 8. This analysis is performed over one-year period of Phase-II, using
MLPv8.

A further improvement is achieved in MLPs in which the ten t2t’s input
variables are replaced with the ten PDF quantiles. Quantiles gives same
statistical weight to the input variables, being indeed defined as one tenth
of PDF area. This definition avoids numerical quantisation problem of t2t’s,
coming from the integer definitions of $t_{0}$’s (see Fig. 7) and, more
important, remove the correlation present by definition on the t2t inputs,
basically due to the partial overlap of the integrals for different $t_{0}$.

###  III.4 MLP Test and Training from 222Rn events.

The Gatti parameter in Borexino was initially tuned on 7000 214Bi-214Po events
collected during the scintillator operations, before the official start of
data acquisition in mid-2007. Subsequently, during six cycles of the WE
campaign, occurred in between 2010 and 2011, another and bigger sample of
214Bi-214Po events was collected. This sample, on which the Gatti parameter
was upgraded and the MLP studies are based, contains in total 85.000 events,
whose 27000 events lies in the fiducial volume region $r\simeq 3$ m. The only
issue, that must be taken into account and controlled, is the fact that, in
both data-sets (Phase-I and WE), the radon events were observed mainly on the
detector top, in the region above the equator ($z>0$). This evidence,
supported also by the fluid dynamic simulations performed for the CNO analysis
[15], is a consequence of the effective separation between the two hemispheres
due to the fluid motion in relation with the spherical geometry. After the MLP
training, a slight top-bottom asymmetry was actually observed, but was found
not of practical relevance inside the analysis fiducial volume.

The final training sample (Sample-WE) contains in each MLP version 25000
events with $r<3$ m from the WE period. The comparison of performances, among
MLPs and Gatti, was done on a reduced sample made of about 15000 events for
training, and about 15000 events for test (Sample-WEX), both with larger radii
to study also the radial dependence. Another test samples (Sample-Ph23), for
double-checking the evolution of the efficiency in time, space and energy, was
chosen selecting an $\alpha/\beta$ sample, not from the 214Bi-214Po (basically
absent after WE), but from two energy intervals of the Borexino Spectrum in
the first 1000 days of Phase-II, in which the contribution of the 210Po
activity is still sizeable. The $\alpha$ sample is selected in a very narrow
region of the 210Po peak (209-210 NPE), with a very small contamination of the
underlying $\beta$-like component from solar neutrino and $\beta$ decays;
whereas the $\beta$ sample is selected in the 7Be shoulder region (320-400
NPE), where the leakage of the $\alpha$ events from the 210Po right tail is
also negligible.

As anticipated, the MLP after the WE period, shows a performance degradation
because of PMT loss with corresponding degradation of event reconstruction
resolution, resulting in a time and space (radial) dependence. Furthermore,
the 214Po emits a mono-energetic alpha line about 50% higher than the 210Po
peak energy, that falls indeed in the region of interest for the solar
neutrino analysis. As a consequence of the energy dependence of the
scintillation temporal PDFs, the MLP efficiency evaluated at the 214Po line is
not directly applicable for example on the 210Po analysis. The correct
assessment of the space, time, and energy dependence of the MLP was studied
using also calibration data and Monte Carlo simulations. In the following
paragraph we will report the main MLP features investigated in the Borexino
analysis.

###  III.5 Different versions of $\alpha/\beta$ MLP

The most important MLP versions, which gave similar performances and were used
in the main Borexino analyses, are listed here:

  1. 1.

MLPv8: This version is the first showing a significant improvement with
respect to Gatti. The input variable are the ten t2t’s described above, in
addition with the RMS, the kurtosis and mean-time of the non-reconstructed
cluster.

  2. 2.

MLPv10: This version is similar to MLPv8, but t2t’s are replaced with 10
quantiles. In some cases, this version shows a slightly better performance as
compared with MLPv8, for the reasons described above, especially for low
energy events.

  3. 3.

MLPv12: This version was meant to solve problems coming from the energy
difference between the training 210Bi sample, and the low energy region where
the $\alpha/\beta$ is actually applied. The 210Bi sample energy is
artificially reduced by thinning out the number of photoelectron (randomly
removed), in a ratio of 1:2 for the 214Po and of 1:4 for 214Bi. Although this
method is assigning the correct statistical weight to the training samples
(and so similar to the low energy region), it cannot include real energy
dependence of the scintillation PDFs upon the energy.

  4. 4.

MLPv14: Finally this version, attempts to solve the problem of the low energy
extrapolation using 218Po events for $\alpha$’s (with lower energy and then
closer to 210Po). Since the 218Po precedes the 214Bi-214Po fast coincidence by
about 30 minutes, a space time cut (1 meter radius and 1.5 hour before) was
able to select a pure sample of about 1800 event candidates. Due to the very
low efficiency of the 218Po tagging, this sample contains on one hand a proper
representative set of low energy events, but on the other hand it has a
limited statistics.

In order to compare and contrast the different versions of MLPs, several
studies have been performed, especially in terms of efficiency, space
uniformity and time stability.

The TMVA package returns the normalised selector in the 0–1 interval, sharply
peaked at 0 for $\alpha$’s and at 1 for $\beta$’s in the Borexino choice.
Figure 8 shows the distribution of the 0–1 selector for the versions of
interest for $\alpha$’s (red) and $\beta$’s (blue) from test Sample-WEX. All
of them are basically comparable, even if MLPv8 shows in general a better
symmetry and sharper distributions. MLPv12, for the reasons discussed above,
shows a more smeared distribution, even though with a good separation. This
differences can be understood comparing the same discriminator with _Receiver
Operating Characteristic_ (ROC) plot as reported in Fig. 9. In particular,
assuming $\alpha$’s as signal and $\beta$’s as background (N.B.: opposite to
the typical TMVA convention in this particular analysis), the ROC curve
reports the _True Positive Rate_ (TPR) as function of the _False Positive
Rate_ (FPR) changing the selection threshold $m_{0}$ in the interval
$0<m_{0}<1$, that is

$\displaystyle{\rm TPR}$ $\displaystyle=$
$\displaystyle\int_{0}^{m_{0}}\mathcal{M}_{\alpha}(t)dt,$ (6)
$\displaystyle{\rm FPR}$ $\displaystyle=$
$\displaystyle\int_{0}^{m_{0}}\mathcal{M}_{\beta}(t)dt,$ (7)

where $\mathcal{M}_{\alpha,\beta}$ are the corresponding PDFs of the MLP
parameters, always from the test Sample-WEX. From Fig. 9, one can compare the
overall performance of different MLP versions (bluish and greenish curves) and
the Gatti parameter (red), as reported in the Figure legend. The better
discriminator approaches a right angle shape in the top-left corner. From this
Figure, we can conclude that all MLP have an overall good performance,
considerably better than Gatti for FRP $<0.2\%$, _e.g._ for 99.75% TPR one can
have a factor 2 less contamination from FPR.

###  III.6 MLP radial dependency

The radial dependence of the MLP selector, strictly related to the position
dependence of the reconstructed cluster, plays a crucial role in the 210Po
spatial analysis, as described at the end of Sec. II. It is therefore
important to study, through Monte Carlo simulations and through the test
samples, any possible feature of the MLP related to the position and its
possible bias in the 210Po activity determination. Figure 10 shows the radial
efficiency determined from the test Sample-Ph23 for MLPv8 (green) and Gatti
(red): the first shows a better behaviour in terms of spatial uniformity. If
one consider the CNO fiducial volume, located at about 21 $m^{3}$ on the
$x-$axis, the efficiency is pretty uniform. As discussed in [15], the non-
uniformity of the MLPv8 efficiency is indeed negligible, as compared with the
energy and time dependence, which will be discussed below.

Such studies have been performed with different methods and with different
values of the MLP selection threshold.

Figure 12: Example of energy dependence (npmt energy estimator) of the MLP
efficiency, performed over Monte Carlo data. The $\beta$-like event percentage
left after the MLP$<0.015$ cut are reported for the versions MLPv12 (orange)
and MLPv10 (cyan). Both curves are fitted to exponential functions, namely the
red and the blue lines in the logarithmic plot.

###  III.7 Stability and energy dependence of MLPs

The time dependency of the MLP, for a given selection cut, in the 210Po energy
region has been carefully studied for the time stability of the 210Bi and
210Po activity in the context of the CNO neutrino analysis, see Sec. IV. In
order to obtain the selection efficiency of 210Po and the corresponding
leakage of $\beta$ events by the cut itself, events in the fiducial volume
analysis are fitted with the so-called _MLP-complementary_ method. In the
latter, the data-set, year by year from 2011, are split into two histograms
depending on events passed or not the MLP cut, named “MLP-subtracted” and
“MLP-complementary”, respectively, as reported in Fig. 11.

For typical analysis, a PSD threshold of $m_{0}<0.05$ and energy in 150-300
NPE interval are used. Under these conditions, the fitted spectra, as a
function of the energy $E$ [NPE], are defined as

$\displaystyle S_{\beta}(E)$ $\displaystyle=$ $\displaystyle S_{\rm
bx}(E)(1-Ae^{-E/E_{0}}),$ (8) $\displaystyle S_{\alpha}(E)$ $\displaystyle=$
$\displaystyle S_{\rm bx}(E)(Ae^{-E/E_{0}}),$ (9)

where $S_{\rm bx}(E)$ is the typical Borexino spectrum with all fitted species
(see _e.g._ [13]), $S_{\alpha,\beta}$ are the resulting $\alpha$ and $\beta$
selected spectra and, finally, $A$ and $E_{0}$ are two free parameters. Notice
that the ansatz that the energy dependence of the MLP cut is exponential is
suggested by Monte Carlo simulations and calibration data, and also by general
considerations about the statistical nature of the neural network output.
Figures 12 and 13 show the exponential energy dependence of the MLP cut from
Monte Carlo simulation and from calibration data, respectively. In both
Figures, the energy estimator is npmt, i.e. the number of hit PMTs during a
scintillation event without double counting piled-up events, see [1] for
further details. Either in calibration data and in MC events, the percentages
left after the MLP cut show an exponential behaviour, supporting the choice of
the energy dependence of the efficiency assumed in the MLP-complementary fit.

Figure 13: Energy dependence (npmt estimator [1]) of the fraction of $\beta$
events left after the MLPv10 cut on real $\beta$-like calibration data.

Finally, Fig. 14 shows the time evolution of the MLP efficiency for the 210Po
energy range year by year, resulting from the model in Eq. (8). The slightly
decreasing linear trend is compatible with expectation of the event
reconstruction degradation, mainly related to the linear PMT loss. This
dependence is used to correct the measurement of the 210Po activity and to
determine the corresponding systematic uncertainty for the final result on the
CNO neutrino interaction rate.

Figure 14: MLP time dependence efficiency from MLP-complementary fit for the
combined Phase-II and Phase-III period for MLPv8, performed over one-year time
intervals (blue histogram). A fitted linear trend (red) shows, at leading
order, the degradation of the MLP efficiency due to the PMTs loss
($p$-value$=0.48$ ). The uncertainty increases over time because the reduced
210Po statistics due to its decay.

##  IV Event-by-event MLP tagging in the main Borexino analyses.

###  IV.1 Polonium-210 studies for the CNO quest

The possibility of tagging $\alpha$ events with high efficiency in space and
time was of crucial importance for the first measurement of neutrinos from the
CNO cycle [15] with Borexino and its subsequent update [16].

Given the degeneracy, and then the correlation, between 210Bi, $pep$ and CNO
spectra, the sensitivity to CNO neutrinos through the spectral analysis is
pretty poor, unless the 210Bi and $pep$-$\nu$ rates are independently
constrained in the spectral fit [31]. In particular, the $pep$-$\nu$ rate can
be constrained to 1.4% precision [31], using: solar luminosity along with
robust assumptions on the $pp$ to $pep$ neutrino rate ratio, global analysis
of existing solar neutrino data [32, 33], and the most recent oscillation
parameters [34]. The $pep$ constraint is essentially independent of any
reasonable assumption on the CNO rate, as the solar luminosity depends only
weakly on the contribution of the CNO cycle itself.

In practice, the only crucial element at play is the 210Bi rate, a $\beta$
emitter with a short half-life (5 days) coming from the 210Pb (present in the
scintillator at the beginning of Phase-II) through the decay chain:

$^{210}\mathrm{Pb}\xrightarrow[22.3\,\mathrm{years}]{\beta^{-}}{{}^{210}}\mathrm{Bi}\xrightarrow[5\,\mathrm{days}]{\beta^{-}}{{}^{210}}\mathrm{Po}\xrightarrow[138.4\,\mathrm{days}]{\alpha}{{}^{206}}\mathrm{Pb}\,.$
(10)

Assuming the secular equilibrium, the 210Bi rate can be determined from the
210Po activity [35, 36, 31]. Since the 210Po activity can be measured
precisely through the MLP high efficiency $\alpha/\beta$ tagging, this
strategy provided the key solution to tackle the species correlation in the
spectral fit and then to lead the Borexino Collaboration to the first
observation of the CNO neutrino interaction rate and its subsequent upgrade
[15, 16]. Anyway, the story was not that easy: at the beginning of Borexino
Phase-II (early 2013), it was clear that presence of convection motions,
caused by the seasonal change of the temperature in the Gran Sasso
experimental hall, made it impossible to apply the 210Bi-210Po link, as
suggested by the sequence in Eq. (10).

In order to solve this problem, a long and challenging thermal stabilisation
program was undertaken by the Collaboration to prevent the scintillation
convctive motion, and the consequent contaminant mixing in the scintillator.
This program, started in mid-2014, consisted of different phases: (i)
installation of high precision temperature probes inside and outside the
detector, (ii) thermal insulation of the detector with different layers of
rock wool, (iii) active temperature control systems of the detector and (iv)
of the experimental room. This longstanding effort worked properly and allowed
one to set an upper limit on the 210Bi rate, a crucial ingredient for the
final extraction of the CNO neutrino interaction rate from the spectral
analysis.

It is worth mentioning that the MLP tagging, with its high efficiency,
uniformity and stability, helped in all the stages of this enterprise: from
the understating of the 210Po migration in the scintillator, through study of
the effects of the different phases of the thermal insulation program, until
the determination of the 210Bi upper limit rate (see for details the appendix
of [15]).

For the CNO analysis the space and time dependence of the $\alpha/\beta$ tag
in the 210Po region was studied carefully using the MLP complementary analysis
and Monte Carlo simulations. In particular, the latter was crucial for the
optimisation of the cut and for the efficiency dependency upon the radial
position and time. In particular the best cut was defined by maximising the
standard signal-to-background (S/B) figure of merit (FoM):

${\rm FoM}=\frac{S}{\sqrt{S+B}}.$ (11)

In this case $S$ can be assumed as true positive events (real $\alpha$’s), and
$B$ as false positive ($\beta$-like events leaking out from the distribution
tail). For MLPv8 the best $\alpha$ cut, corresponding to the Phase-III data
set, was found at $m_{0}<0.3$.

###  IV.2 MLP in other analyses

Besides the CNO analysis, the MLP $\alpha/\beta$ tagging was used in many
other analyses published by the Borexino Collaboration. In particular, it
played an important role in the high significance detection of the seasonal
modulation of 7Be neutrinos due to the Earth orbit eccentricity [37], for the
reduction of the 210Po component in the region of interest of the 7Be
spectrum. This analysis was updated including the entire Phase-II and Phase-
III data set, leading to the first independent measurement of the Earth orbit
eccentricity with only solar neutrinos [38].

In the geo-neutrino analysis, the MLP was used in the event selection with a
high performance even at large radii ($\sim$4m) close to the Nylon vessel [24]
and for the 210Po background estimation for the neutron background induced by
alpha decays. In addition, the MLP selection was used for the space and time
selection of the 210Po data events for the accurate tuning of Monte Carlo used
for simulating the 210Po spectrum in Phase-II [39] and Phase-III. In
particular, this study played an important role in the comprehensive analysis
of the $pp$ chain [12, 13]. This study has provided a measurement of the most
important solar neutrino fluxes, which is in favour of the MSW-LMA neutrino
oscillation scenario at 98% CL. (see _e.g._ [40] and Refs. therein).

## Conclusions

In this paper, we offer a detailed review of the $\alpha/\beta$ pulse shape
discrimination adopted in Borexino. We present various implementations used
during more than a decade of data taking, starting with the Gatti optimal
filter and the corresponding statistical subtraction of the $\alpha$ component
from the energy spectrum, ending with the more sophisticated PSDs based on
ANNs, specifically exploiting MLP. The latter, with its high efficiency,
spatial uniformity and time stability, allowed us to event-by-event select the
210Po events, a crucially important background reduction which made possible
the observation of CNO solar neutrinos in Borexino.

Compared to the Gatti parameter approach, ANNs single out parameters relevant
to PSD in a highly non factorizable way. In the case of Borexino, the Gatti
parameter was limited by information loss in the photon arrival times after
position reconstruction correction. The integration of variables in the MLP
before event reconstruction improved the performance of the $\alpha/\beta$
selection. The MLP implementation required careful calibration to select the
best input parameters, tune the algorithm, and evaluate its performance. Its
spatial and time efficiency were monitored and used for the evaluation of the
global systematic uncertainty of some of the most important Borexino results
for which the method was used.

The $\alpha/\beta$ pulse shape discrimination allowed by intrinsic properties
of the scintillator, was proven to be fully exploitable in an ultra-pure,
large-volume detector such as Borexino. In particular, it played an essential
role in the neutrino spectroscopy of the entire $pp$ chain and the first
observation of neutrinos from the CNO cycle.

## Acknowledgements

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-2013/10/E/ST2/00180) (Poland). We gratefully acknowledge the
computing services of Bologna INFN-CNAF data centre and U-Lite Computing
Center and Network Service at LNGS (Italy). This research was supported in
part by PLGrid Infrastructure (Poland).

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