Why and how the Sun and the stars shine

1 A bit of history

1.1 The solar neutrino problem

In the '70s of the last century, the solar neutrino saga, which at the beginning took the name of “solar neutrino problem”, began. The Sun produces 99% of its energy through a nuclear fusion chain called pp cycle (see fig. 1a); five of the nuclear fusion reactions in this chain emit electron-flavored neutrinos. The first experiment (Homestake) that studied solar neutrinos using a radiochemical technique, which dates back to the '70s of the last century, found a deficit in the solar neutrinos flux by comparing the measured flux with the prediction by the John N. Bahcall’s Standard Solar Model (SSM). A similar result was obtained later by the Gallex experiment, again using a radiochemical technique, at the Gran Sasso laboratory.

The problem of solar neutrinos was solved a few decades later by the Canadian experiment SNO which showed that the deficit was due to the reactions used by the two radiochemical experiments, which were triggered only by electron-flavored neutrinos, while the solar neutrinos change flavor, as a consequence of the oscillation phenomenon, in the travel between the production point within the Sun and the detector placed on Earth. The detection mechanism of this experiment was based on the Cherenkov effect in heavy water; similarly a Japanese experiment (Super-Kamiokande) used the Cherenkov effect but in pure water.

Both these experiments were required to set a threshold at ≈ 5 MeV (later reduced a little), very high compared to the solar neutrinos’ energy, because the radiations from the natural radioactivity, present in the water or heavy water and in construction materials, conceal the neutrino events which are extremely rare: neutrinos can even cross the entire Universe without interacting. If we look at the energy spectrum of solar neutrinos (fig. 1b) we can observe that a 5 MeV threshold includes only a tail of the neutrinos produced by the reaction involving ⁸B and hep, and represents less than 0.01% of the total solar flux.

1.2 A story 80 years long

In 1988-1989 myself, with Frank Calaprice of Princeton and Raju Raghavan of Bell Lab, began to discuss the possibility of an experiment capable of measuring solar neutrinos from about 1 MeV of energy, in particular the neutrino spectrum from the nuclear fusion reaction which involves ⁷Be. To achieve this goal it was first necessary to reduce the radioactivity to levels never reached before: referring to the usual radioactivity of the detector materials, a gain of at least 10 orders of magnitude was needed.

However, I and some collaborators of mine, despite the skepticism of most people, would have liked to reach lower levels of radioactivity in order to study the solar flux from energies of few hundred keV: it was a very challenging goal but also an important milestone able to reveal the mechanism that powers the Sun.

Already in 1938 H. A. Bethe theorized, in a paper on Physical Review, that energy was generated in the stars by the fusion of light nuclei into heavier ones. In stars of similar mass to the Sun the nuclear fusion reactions follow a cycle called pp, which is shown in fig. 1, but in the most massive stars, that are the great majority in the Universe, the hydrogen fusion process proceeds through a cycle that is called CNO and whose reactions are catalyzed at higher temperatures than those of the Sun by heavier nuclei, namely carbon, nitrogen and oxygen (fig. 2). This process, which is dominant in the Universe, is also present in the Sun but only at 1%. A similar theory was also developed by C. F. Von Weizsäcker approximately in the same years as Bethe’s paper.

The density of the gas compressed in the Sun is about 10 times the Pb density and in order to avoid a solar gravitational collapse a very high temperature is needed; the pp cycle produces a temperature higher than 10 million degrees inside the Sun. For stars more massive than the Sun, with at least 30% greater mass, the CNO cycle triggers a much higher temperature which in turn prevents the star from imploding.

In the '90s of the last century the Gallex radiochemical experiment had measured the integrated flux of solar neutrinos, starting from a threshold of 233 keV, but the various reactions had never been measured individually and had never been identified through the energy spectra of the emitted neutrinos, except, as already mentioned, a tail of the reaction with ⁸B (Cherenkov experiments). For what concerns the CNO, this reaction, hypothesized by Bethe and Von Weizsäcker and taken up by the astrophysicists, never received an experimental evidence of its existence.

2 The Borexino detector

Our project was surrounded by the skepticism of many physicists of the sector; the collaboration in the meantime had expanded to include, in addition to Milan and Princeton, other groups: one from Dubna, previously my collaborators in Serpukhov, TUM Munich and four Italian groups, Genoa, Pavia, Perugia and Gran Sasso laboratory.

Between 1990 and 1995 we proceeded with an R&D effort to develop new radio-purification methods and to build a reduced and simplified version of the Borexino detector, called Counting Test Facility (CTF) (fig. 3) which allowed to test the effectiveness of the purification methods, which had just been developed. In 1995 in a seminar at the Gran Sasso laboratory I presented the results of tests in CTF demonstrating that our methods were able to achieve levels of radio-purification that saturated the CTF sensitivity (5×10^-16 grams of contaminants/grams of pure material). The INFN president at that time, Luciano Maiani, who was attending the seminar, approved the project.

2.1 The design

The Borexino detector is built like an onion, with concentric layers that follow the principle of graded shielding: the closer the layer is to the center, the greater its radio-purity. The active material, 278 t of liquid scintillator – a hydrocarbon (pseudocumene-PC) plus a fluorescent dye (PPO) – is positioned in the center, contained in a nylon vessel (Inner Vessel), 125 microns of wall thickness. Within it a Fiducial Volume (FV) is software defined to absorb the emissions of radio contaminants contained in the vessel wall. The inner vessel is surrounded by another nylon vessel, the Outer Vessel, which acts as a barrier against emissions from the 2212 photomultipliers, especially from glass and ceramics, mounted inside a Stainless Steel Sphere, which surrounds the entire inner part of the detector. The region between the two vessels and the sphere is filled with a buffer liquid, further ≈600 t of PC plus DMP, a quencher for the low luminescence produced by the pseudocumene. The whole structure is surrounded by a steel tank (Water Tank) containing about 4000 t of highly purified water, 16.9 m high and 9.0 m of radius (fig. 4).

Therefore the scintillator is shielded in total by ≈ 5.5 m water equivalent: 2.14 m of water, 1.25 m of buffer plus the thickness of the scintillator between the vessel and the virtual wall defined by the Fiducial Volume; all these layers shield the external radiation (rocks and environment of the underground Hall) as well as the emissions of the nylon walls, the photomultipliers and the sphere stainless steel. Ultra-dense polyethylene ropes hold the inner vessel in place, which is pushed up by the modest buoyancy due to the small density difference between the scintillator and the buffer liquid (0.01%, reduced later by an order of magnitude).

2.2 Installation

Nothing in the Borexino detector is standard. All materials of the detector and of the ancillary systems have been selected by means of Ge detectors selecting those with low radioactivity; the nylon has been extruded and the vessels assembled in radon-free clean rooms; all components have been carefully selected and some of them have been developed in collaboration with external companies; moreover all the surfaces also of the ancillary plants have been pickled, passivated and rinsed with strong detergents; all welds have been done in nitrogen atmosphere; everything inserted into the detector passed through one of the 5 specifically installed class 100 clean rooms and the stainless steel sphere itself was equipped as a class 10000 clean room; synthetic air has been used inside the sphere during the vessels installations (fig. 5). The PC, produced for Borexino according to a stringent quality control plan developed jointly with the company, has been procured via an ad hoc unloading station with lines assuring a direct connection between the production towers and special isotanks, moved from Sardinia to the underground laboratory in no more than 18 hours to minimize the nuclides production in it by the cosmic rays, and then transferred through a special unloading station into four big reservoir tanks in the so-called Storage Area installed underground.

2.3 Radiopurity

The radionuclides purification of the scintillator is based on four methods: distillation at 100 °C and 100 mbar, water extraction using a countercurrent packed column, sparging with nitrogen using a column in countercurrent through a structured packing material, and ultra-filtration with Teflon cartridge filters with nominal cutoff size of 0.05 μm. The distillation is effective for the less volatile components and therefore metal salts and dust particles, and for U, Th, K isotopes. In the water extraction the electrically charged impurities have a preference for polar liquids such as water: metal salts, silicate particles and again U, Th, K are efficiently removed. This method, unlike distillation, can be used directly on the already mixed scintillator (solvent + fluors). Gas stripping is used to remove gaseous impurities such as Ar, Kr and Rn; only nitrogen, with concentrations of ⁴⁰Ar and ⁸⁵Kr of the order of 10^-8 Bq/m³ was used and then purified from the Rn via cryogenic methods. The ultra-filtration stops dust particles as silicates and aluminates.

The first purification was very successful, achieving (1.67±0.06)·10^-17 and (4.6 ±0.8)·10^-18 g of contaminants/g of material for ²³⁸U and ²³²Th, respectively; ⁴⁰K is not observed and ²²²Rn is reduced to < 1 counts/day/100 t of PC, while ⁸⁵Kr is still present at the level of 10±5 counts/day/100 t. ⁸⁵Kr has been further reduced in the years 2010-2011 using water extraction in continuous: the effect of this second purification has been the reduction of Kr to >1.96 counts/day/100 t and ²³⁸U and ²³²Th to < 9.5·10^-20 and < 7.2·10^-19 g/g, respectively.

The ¹⁴C has been found ≈ 2·10^{-12 14}C/¹²C; the ¹⁴C cannot be purified and this low level is due to the crude oil procurement from particularly old and very deep layers. These unprecedented achievements have been the key for Borexino’s success.

3 The Borexino breakthroughs

In thirty years of activity Borexino has obtained real breakthroughs concerning the physics of the Sun and that of the stars, it has provided an important contribution to the physics of neutrino oscillation and it became one of the two experiments – the other is the Japanese KamLAND – which succeeded in detecting and studying geo-neutrinos.

Borexino detects the neutrinos interactions through the neutrino-electron scattering, while the antineutrinos are detected via the inverse beta decay reaction.

3.1 The physics of the Sun

Figure 1 reproduces the nuclear fusion chain of the so-called pp cycle and it is possible to observe that five of these reactions emit electron-flavored neutrinos. They are called pp, ⁷Be, pep, hep, ⁸B, following the nuclei involved. Measuring the individual fluxes of neutrinos, emitted by these nuclear fusion reactions, it is possible to identify, through their energy spectrum and shape, the reaction that produces them. The two reactions, ⁷Be and pep, are quite easily identifiable because their neutrinos are monochromatic and then the spectra of the recoiled electrons show the Compton shoulder. The other spectra can also be identified on the basis of their shape and energy.

One of the difficulties in measuring these fluxes is the low rate of neutrino interactions due to the very limited neutrino cross section: we can compare the solar flux that invests the Earth – 60 billion per cm² and per second – with the rates of the individual fluxes in about 100 t of Borexino scintillator; just as an example: ~48 counts/day for ⁷Be, ~2.5 counts/day for pep. Therefore, even the very low residues of radioactivity can interfere with the measurements of the neutrino signals.

The situation is well illustrated in fig. 6, where the results of the fit on signals and background are shown. Many radioactive residues interfere with every neutrino signal, but the most important – taking into account rates, energy and spectra shapes – are: ¹⁴C which interferes with the measurement of the neutrinos from the pp reaction (the first of the cycle), ²¹⁰Po and ⁸⁵Kr with ⁷Be, ¹¹C with pep and finally the external background with the ⁸B reaction.

It is worth dwelling on two of these backgrounds: the ²¹⁰Po and ¹¹C. The 210 Po is not in equilibrium with its ²³⁸U family, and increases in conjunction with operations carried out on the scintillator whether it be purification or topping up of the PC. Later it begins to decrease according to its life-time: 130.39 days. Nevertheless it is possible to well identify alpha particles, as produced by ²¹⁰Po in its decay, with respect to $\beta$'s and $\gamma$'s , which are related to the neutrinos interactions, via $\alpha / \beta$ discrimination by means of a Multilayer Perceptron machine learning algorithm.

¹¹C is of cosmogenic origin, produced on the ¹²C of the scintillator by the cosmic muons that invest the Hall C of the underground laboratory, where Borexino is installed. Hence it is continuously produced and cannot be purified. The ¹¹C decay rate compensates the production time and therefore the ¹¹C presence in the scintillator is practically constant; the only possible selection tool is to reduce and almost suppress it via a software 3-fold coincidence (TFC), and a following pulse shape analysis. The 3-fold coincidence is based on the incident muon tracking, on the ¹¹C decay – 29.4 minutes mean life – with the emission of a positron, that annihilates, and the capture of the neutron, produced in the muon interaction, which, after a delay due to the thermalization, is captured and emits a 2.2 MeV $\gamma$. This coincidence, with 2h veto in portions of the FV, suppress 90% of ¹¹C and at the same time halves the exposure; in addition this reduction is not enough for a pep measurement. But the positron, emitted in the ¹¹C decay, can form a positronium with an electron of the scintillator, which in its form of ortho- positronium has an average lifetime of 140 ns, reduced to 3 ns in the scintillator. This circumstance and the topology different from γ’s produced in the positron annihilation, can be exploited to reject another 5% of the total ¹¹C decays, via a boosted-decision-tree algorithm. What is plotted in fig. 6 is just the remaining 5%.

The ⁸B analysis follows a different approach because the energy spectrum, which is used in other cases, is distorted by the oscillation phenomenon (⁷Be and pep are monochromatic, and the pp spectrum shows a negligible distortion). The radial distributions of the contaminants, in particular ²⁰⁸Tl, which is the largest and almost only dangerous contaminant for ⁸B measurement, shows different spectrum than ⁸B. The ²⁰⁸Tl contaminants can be disentangled via the sequence ²¹²Bi-²¹²Po, easy to be identified, which decays to ²¹²Po with 64% B.R. and to ²⁰⁸Tl with 36% B.R. One source of the ²⁰⁸Tl, a ²³²Th daughter, is the IV nylon, and its decay products contribute to the surface contamination as well as the volatile progenitor ²²²Rn released into the scintillator.

In the ⁸B solar neutrinos analysis, a 3 MeV threshold is defined to eliminate part of the background. The strategy to disentangle the ⁸B neutrinos is based on an accurate subtraction of the contaminants radiation by exploiting the radial distribution of the background events once cuts are applied to muons, cosmogenic nuclides and the Bi-Po sequence. In order to increase the statistics, the entire IV is defined as FV.

Now we have to introduce a few information on the Standard Solar Model. The solar model describes the evolution of the Sun starting from the proto-star and considers it as a sphere of gas in hydrostatic equilibrium between the gravitational compression and the pressure gradient. The experimental data to which the model can refer are the Sun’s radius, its luminosity, i.e. the energy per time unit, and the so-called metallicity, i.e. the ratio Z/X between the mass of the elements present on the solar surface and that of hydrogen, obtained through the analysis of the observed absorption lines in the solar spectrum measurements. The intensity of the lines depends on the conditions of the solar atmosphere, namely the temperature, the electrons density and the pressure; therefore the atmosphere model is crucial. A first model dating back to 1998 (GS98) is a 1D model while more recently in 2009 (AGSS09) a 3D model was developed. These two models of the solar atmosphere give two different metallicities: $Z/X \sim 0.023$ for the first (HZ) and ~0.018 for the second (LZ). A fundamental test of the SSM model is the so-called helioseismology, that is the study of the oscillations present in the Sun, measured at its surface: these are due to stationary acoustic waves that are present below the photosphere. Helioseismology provides a measurement of the metallicity: while the comparison with the 1D model gives a good agreement, the 3D model disagrees. This problem, so-called solar abundance problem, is still open; it could be solved via the neutrino fluxes comparison between the model expectations, partially depending on the metallicity, and the experimental measurements.

In table 1 the solar neutrinos experimental rates and fluxes measured by Borexino are shown, and are compared with the SSM predicted fluxes calculated with the High and the Low multiplicity and with the most recent global fit, which includes all data related to various neutrinos sources: Sun, reactors, accelerators. All these fluxes are produced in the pp cycle.

Form these results it is possible to draw the following conclusions:

1) For the first time the nuclear reactions that produce neutrinos as part of the pp cycle have been individually measured; the only exception is the hep reaction which in Borexino has a very low rate and represents less than 10^-5 of the total flux. A good alignment is observed between the experimental data of Borexino and the SSM predictions, obviously within the uncertainties of both the experimental data and the model predictions.

2) The luminosity measured via the photons emitted by the Sun has been compared with the luminosity calculated from the neutrino fluxes taking into account that the pp cycle produces 99% of the entire solar energy. An excellent correspondence has been found: $L = (3.89^{+0.35}_{-0.42})\times 10^{33}$ erg s^-1 for neutrinos and $L = 3.846 \pm 0.015 \times 10^{33}$ erg s^-1 for photons. Since neutrinos take a few seconds to leave the Sun and eight minutes to reach the Earth, while photons take more than 100000 years, this result demonstrates that the Sun is in thermodynamic equilibrium over a 10⁵ years’ time scale and at the same time confirms the nuclear origin of solar energy with a precision never achieved until now by a single experiment studying solar neutrinos.

3) The ratio between the two branches of the pp cycle: pp I and pp II (see fig. 1) can be calculated using the Borexino data $RI/II = 2 \Phi ( {}^{7}Be ) / [ \Phi ( pp ) - \Phi ( {}^{7}Be ) ] = 0.178^{+0.027}_{-0.023}$ in agreement with the SSM expectations: $0.180 \pm 0.011$ for HZ and $0.161 \pm 0.010$ for LZ.

3.2 Seasonal modulation

Borexino also studied the seasonal modulation of solar neutrinos, which is due to the eccentricity of the Earth’s orbit and then the change of the Earth-Sun distance during the year. This modulation follows a sinusoidal shape with a period of one year and a modulation amplitude of about 7%. Although the solar origin of the neutrinos studied by Borexino is already implicitly demonstrated by the shape of the energy spectra of the neutrinos, the measure of seasonal modulation is a direct and unequivocal demonstration of the solar provenance of the neutrinos detected.

The measurement was made over a four-year period using the neutrinos of the ⁷Be reaction and selecting the events by α/β discrimination by means of neural networks. The results have been analyzed following two methods: fit on the distribution vs. time and the Lomb-Scargle method. The result gives a monthly rate of 33±3 counts/day/100 t with a modulation amplitude of 7.1 ±1.9% and a period of 367 ±10 days, in perfect agreement with what was expected, i.e. 6.7% of modulation amplitude and 365.25 days period.

4 The neutrino physics

The solar neutrinos are produced almost at the center of the Sun, and they have to cross several layers of solar matter to escape from it. Obviously, they are subjected to the neutrino oscillation phenomenon and in particular to the oscillation in matter. The paradigmatic model is the so-called MSW, from the initials of the physicists who developed it, which finds its origin from the consideration that the electron-flavored neutrinos scatter in matter on electrons through two channels, one of charged current and one of neutral current, while neutrinos with other flavors can interact only through the neutral current. Compared to the oscillation in vacuum, in the equations of the oscillation in matter, the term $A_{cc}$ which represents the charged current potential of the electron flavored neutrino depends on the matter electron density and on the neutrinos energy:

$ \frac{A_{\mathrm{cc}}}{\Delta m^{2}} = 1.526, e^{-7} \frac{n_{e}}{\frac{\mathrm {moles}}{\mathrm{cm^{3}}}} \frac{E}{\mathrm{MeV}} \frac{\mathrm{eV}^{2}}{\Delta m ^{2}} $ ,

where $n_e$ is the electrons density in solar matter and $E$ is the neutrinos energy. If this term is negligible, either because $n_e$ and/or $E$ are very small, the matter effect is negligible and the neutrino propagation behaves like in vacuum.

In the case of solar neutrinos, $n_e$ is almost constant in solar matter while the neutrino energy changes according to the fusion reaction that emits them. Borexino covers the energy range between 150 keV and 17 MeV. In fig. 7 the survival probability of the electron-flavored neutrino is shown as predicted by the MSW model; it consists of three parts, a first plateau at very low energies with a survival probability greater than the second plateau, that at higher energy shows a much lower value. The region between the two plateaus is called transition region. While the prediction for the plateau at higher energies has already been confirmed by the Cherenkov experiments (SNO and Super-Kamiokande), and later also by Borexino, the low-energy plateau was validated by Borexino with the pp neutrino flux measurement, while ⁷Be and pep, measured again by Borexino, fall in the transition region.

The most important achievements in the neutrino physics can be summarized as follows:
1) The survival probability of $\nu_e$ in vacuum regime has been measured for the first time by Borexino, which is the only experiment that succeeded in studying both the vacuum and the matter regime. In addition, the Borexino data allowed for the first time to probe the vacuum-matter transition.
2) The uncertainties of the individual measures do not allow to get a clear idea of the survival probability in the transition region. However, the experimental data do not seem to contradict the MSW model predictions within the experimental uncertainties.

4.1 The day/night effect

The day/night effect consists in the regeneration, in the Earth matter, of the electron-flavored neutrinos, that have oscillated in another flavor during the Sun-Earth journey, and cross the Earth during the night in order to reach the detector. We have to recall that the experimental results on the neutrino oscillation showed that in the tan $\theta$ vs. $\Delta m^{2}_{12}$ eV² plan the only allowed region is the so-called Large Mixing Angle (LMA) corresponding to a $\Delta m^{2}_{12}$ between 10^–4 and 10^–5 eV².

Borexino studied the day/night effect using the ⁷Be neutrinos, where this effect is expected to be null. However, using all and only the results of the solar neutrinos experiments before Borexino, in addition to the LMA region, also the so-called LOW region was found, corresponding to a $\Delta m^{2}_{12}$ between10^–6 and 10^-7 eV², that is allowed with lower probability (see the left part of fig. 8); for this region the day/night effect at the ⁷Be energy is possible. The LOW region is excluded if, in addition to solar experiments, the antineutrinos studied by the KamLAND experiment are considered.

The measurement carried out by Borexino gives an absolutely null value for the day/night effect in the ⁷Be neutrinos energy region: $A_{DN} = \frac{N-D}{(N+D)/2} = 0.0001 \pm 0.012$ (stat.) $\pm 0.007$ (sys.), which therefore excludes the region outlined in red in the plot on the right of fig 8. In this way, without needing the KamLAND antineutrinos and thus without using CPT symmetry, the results of Borexino are able to exclude the LOW region at more than $8.5 \sigma$.

4.2 Non-standard neutrino interactions

Theories beyond the standard model point to the existence of non-standard interactions for which flavor-changing NCs are possible. In particular, the Lagrangian for these non- standard interactions (NSI) is written as follows:

$ - \mathcal{L}_{NC - NSI} = \sum_{\alpha , \beta} 2\sqrt{2} G_{F} \epsilon^{ff'C}_{\alpha \beta} ( \bar{\nu}_{\alpha} \gamma^{\mu} P_{L}\nu\beta ) ( \bar{f} \gamma_{\mu} P_{C} f' ) $,

where $\epsilon^{ff'C}_{\alpha \beta}$ parametrizes the NSI strength normalized to the Fermi constant, $f$ and $f'$ are leptons or quarks, $\alpha , \beta = e, \mu , \tau$, and $C$ is the chirality of the $ff'$ current (L or R). In the analysis by Borexino only the flavor-diagonal $f = f'= e$ and $\alpha = \beta$ is taken into account; in this case $\epsilon_{\alpha}^{C} = \epsilon_{\alpha \alpha'}^{eeG}$ NSI influences the solar neutrino phenomenology because the neutrino propagation in matter undergoes changes. Borexino analyzed the elastic neutrino-electron scattering because it is the most sensitive to the NSI effects.

The procedure analysis is similar to that used for solar neutrinos but it includes in addition the NSI ε strength parameter, with a penalty factor in order to account for the assumed SSM metallicity (either HZ or LZ).

Hereafter the Borexino analysis results concerning the electron, and then the parameters $\epsilon_{e}^{R}$ vs. $\epsilon_{e}^{L}$ are shown (fig. 9), with the results of other experiments: the allowed region, defined by the intersection of all experiments, is a small area around zero.

5 The stars physics

The gravitational attraction inside the solar matter, which has a density of 1410 kg/m³, does not produce the implosion of the Sun because it is contrasted by the pressure caused by a temperature of about 10 MK produced by the pp cycle, which, as mentioned, is the source of 99% of the entire solar energy. In massive stars, with 1.30 times the Sun mass, this temperature is no longer sufficient because the gravitational forces are higher. It is theorized by the astrophysicists that the CNO be the primary channel for hydrogen burning in stars more massive than the Sun (fig. 2), and is in fact the primary channel for hydrogen burning in the Universe.

The CNO cycle provides an important opportunity for Borexino as it is present in the Sun, even if only at 1%. Therefore Borexino had the objective of measuring this cycle in the Sun. Although all the cuts used for the analysis of the other solar neutrinos were applied in order to reduce the events from radioactive residual impurities, cosmic muons, cosmogenic isotopes and external gammas, and a narrow FV was defined with a reduced radius and a vertical cut ( $r<2.8$ m and $-1.8$ m $< z < 2.2$ m ), this last in order to eliminate the background due to the reinforcements applied to the IV top and bottom, further problems were encountered. In the energy region considered for this study, i.e. in the range 320-2640 keV, the key background for the analysis is produced by ¹¹C and ²¹⁰Bi in addition to the solar neutrino signal of the pep reaction: that is to be compared with the signal of a few counts/day/100 t expected for the CNO (see fig. 6).

For ¹¹C the three-fold-coincidence (TFC) tagging has been applied as described in sect. 3.1. For the strong correlation with neutrinos from pep, a constraint of 1.4% was introduced into the fit, using solar luminosity and the robust assumption of the fixed ratio between the pp and pep rates, the existing data on the solar neutrinos and the latest parameters for the oscillation. These assumptions do not affect the measurement of the CNO because it is totally independent of them. Therefore the background to be understood is that connected with 210Bi; the energy distributions of the CNO and ²¹⁰Bi not only fall on the same energy window but also have a similar shape. 210Bi is a daughter of ²¹⁰Pb, which has a residual presence in the scintillator; ²¹⁰Bi decays in turn, with a $\beta$ emission, into ²¹⁰Po, which finally transits to the stable ²⁰⁶Pb nuclide with an $\alpha$ emission. ²¹⁰Pb cannot be detected because it decays $\beta$ with an energy of 63.5 keV, below the Borexino threshold.

To reach a reliable estimation of the ²¹⁰Bi rate, it is necessary to compare it with its daughter ²¹⁰Po, which however, in our case, is composed of two parts. In fact this nuclide increases when operations are performed on the detector such as purification and PC refilling, but, also when no operations are performed, it is fluctuating. This behavior can be explained if we admit that there are two components: one is a stable component produced by the residue of ²¹⁰Pb, in secular equilibrium, while the other is out of equilibrium, produced, after operations, by ²¹⁰Pb deposited on the internal surface of the IV, probably in the dust or particulate. If there are convective currents, they carry ²¹⁰Po into the fiducial volume (the spontaneous diffusion is negligible) and then the first requirement is to avoid these currents and to stabilize the temperature.

As a consequence, in 2016 the water tank was thermally insulated (fig. 10) and an active temperature control system was installed in the upper part of the detector. This system stabilized the temperature very well and only a residual seasonal modulation was observed at 0.3 °C/6 months, a sufficiently small effect that cannot trigger the convective currents. This effort paid off and the ²¹⁰Po decreased and reached its lowest value in a region above the equator of about 80 cm (restricted region), allowing to constrain the ²¹⁰Bi rate. Therefore, for this analysis only the data collected after the temperature stabilization have been considered, then between July 2016 and February 2020, corresponding to 1072 days live time.

The minimum value of ²¹⁰Po defines a stable plateau during the analysis period, with minimal fluctuations, which was determined with a 2D and 3D fit in the restricted region. This minimum value reasonably matches with the constant production of ²¹⁰Po in equilibrium with ²¹⁰Bi and with the ancestor ²¹⁰Pb. The two fits mentioned above give converging results at R ( ²¹⁰Po min ) = ($11.5 \pm 1.0$) counts/day/100 t. The error includes also the systematic uncertainties of the fit.

The ²¹⁰Bi rate has been extrapolated to the entire FV and at the same time a check was made to investigate if the ²¹⁰Bi was uniform over the entire FV. Therefore all the $\beta$-like events, in the energy interval showing the greatest Bi contribution, were selected and a uniform distribution was found with a systematic uncertainty $\le 0.78$ counts/day/100 t. A further verification has been carried out measuring the solar neutrino rate annual modulation and the result is aligned to the previous analysis (see sect. 3.2). In conclusion the ²¹⁰Bi rate is $\le (11.5\pm 1.3)$ counts/day/100 t.

The selected events were fitted in the 320–2640 keV energy window considering simultaneously the energy spectrum and the radial distribution. In the fit the pep rate was constrained to $(2.74\pm 0.04)$ counts/day/100 t while the above upper limit to the ²¹⁰Bi rate is enforced asymmetrically by multiplying the likelihood with a half-Gaussian term: the ²¹⁰Bi was left unconstrained between 0 and 11.5 counts/day/100 t cpd. The systematic errors were studied by varying the parameters of the fit and the theoretical shapes of the bismuth, and analyzing many others effects of lesser importance.

Finally the CNO rate has been determined to be:
$7.2^{+3.0}_{-1.7}$ counts/day/100 t; assuming oscillation in matter, its flux is $7.2 ^{+3.0}_{-2.0} \times 10^{8}$ cm^–2s^–1. A hypothesis test based on the profile likelihood and using million pseudo-data sets excludes the no-CNO observation with a significance $\ge 5.0 \sigma$ at 99% C.L. Therefore Borexino reached the first direct experimental demonstration of the CNO existence.

If all neutrino fluxes measured by Borexino (table I) are combined with the CNO measurement it is possible to conclude that the low metallicity hypothesis is excluded at $2.1 \sigma$, thus giving a good hint in favor of the high metallicity.

6 Geoneutrinos

Although geoneutrinos do not have a direct connection with the Sun and stars physics, I think it is worthwhile to give a hint of this research, because it provides information on the composition of the Earth interior.

Geoneutrinos are electron-flavored antineutrinos emitted by radioactive isotopes present inside the Earth; their radioactive families are ²³⁸U, ²³²Th, ²³⁵U and ⁴⁰K.

Their interactions are detected via the inverse $\beta$-decay, $\bar{\nu}_e +p \rightarrow e^{+} + n$, with a kinematic threshold at 1.806 MeV. This prevents the study of the antineutrinos from ⁴⁰K and ²³⁵U and allows the analysis of the 38% of those emitted by ²³⁸U and 15% by ²³²Th. The production of antineutrinos and the energy produced by these two radioactive chains are shown in the following expressions:

$ {}^{238}\mathrm{U} \rightarrow{}^{206}\mathrm{Pb} + 8\alpha + 6\mathrm{e}^{-} + 6\bar{\nu}_{e} + 51.7 $ MeV,

$ {}^{232}\mathrm{Th} \rightarrow{}^{208}\mathrm{Pb} + 6\alpha + 4\mathrm{e}^{-} + 4\bar{\nu}_{e} + 42.8$ MeV.

The geoneutrino flux is of the order of 10⁶ cm^–2s^–1, much smaller than that of solar neutrinos. In Borexino, about one event is measured every two months, and the interactions are very well tagged because the positron produced from inverse $\beta$-decay promptly annihilates, emitting two 511 keV $\gamma$ rays (prompt event, with a visible energy); the neutron emitted is captured on protons with the emission of a $\gamma$-ray, which provides a correlated delayed event. The characteristic time and spatial coincidence of the prompt and delayed events offer a clean signature for the anti-neutrino detection.

Due to the oscillation the flux that reaches the Earth is approximately halved; the effect of the oscillation is mediated due to the very large distance traveled: the average survival probability is more precisely 0.54.

The most important backgrounds for the geoneutrino study concern: unstable cosmogenic isotopes, $(\alpha , n)$ interactions, antineutrinos produced by nuclear reactors: the isotopes are easily rejected with appropriate cuts, while the extreme purity of Borexino makes negligible the background due to accidental coincidences and $(\alpha , n)$. Therefore, the greatest background is due to antineutrinos from nuclear power reactors whose energy spectrum overlaps the geoneutrinos window in the range 1.8–3.27 MeV.

In the world there are 440 nuclear power reactors that emit, for every 3 GW, $5.6 \times 10^{20}$ antineutrinos per second. The reactors signal in Borexino was evaluated taking into consideration for each reactor: the nominal thermal power, the detector-reactor distance, the thermal load factors for each month from December 2007 to April 2019 (the entire Borexino exposure), the different components of nuclear fuel and their characteristics, the produced spectrum of the antineutrinos, and finally the cross section for the inverse beta decay and, as already said, the survival probability of the electron-flavored neutrinos. These data are obtained from the data base of the European Atomic Agency IAEA.

There are only two experiments in the world capable of detecting geoneutrinos: Borexino and the Japanese KamLAND. Compared to KamLAND, Borexino has the disadvantage of a scintillator volume less than 1/3 of the KamLAND one, while has the advantage of a much higher radio purity and a lower reactor antineutrino background due to the lack of nuclear reactors in Italy.

The so-called 5 MeV excess due to the deviation of the reactor spectrum in the energy range 4–6 MeV, shown in recent experiments, compared to the paradigmatic spectrum of Mueller et al., has been taken into account; the difference between the two spectra, with and without the 5 MeV excess, is 6%.

Given the excellent tagging of the inverse beta decay interactions, it was possible to analyze all together the data collected by Borexino also during the operations, therefore from December 2007 to April 2019, i.e. in 3262.74 days corresponding to $(1.29 \pm 0.05)\times 10^{32}$ protons/year. In fig. 11a the result of an unbinned likelihood fit is plotted, including signal and background. The ratio between ²³²Th and ²³⁸U is assumed to be 3.9, corresponding to what is observed in the chondrules of the meteorites just called chondritic, which maintain the characteristics of the Solar System matter. In the fit of fig. 11b ²³²Th and ²³⁸U are left free.

The number of the geo-ν events corresponding to the entire Borexino exposure is $51.0^{+0.4+2.7}_{-8.6-2.1}$ (stat.-sys.) when the Th/U chondritic ratio is assumed, and $48.9^{+25.1}_{-20}$ when U and Th components are left free in the fit: this fit also finds $27.8^{+15.4}_{-11.8}$ events for uranium and $21.1^{+9.7}_{-8.4}$ events for thorium. The results of the two fits on the geo-$\nu$ number, with and without the chondritic ratio, are fully compatible, but there are larger errors when ²³⁸U and ²³²Th are left free: the ratio ²³²Th/²³⁸U obtained is not aligned with the chondritic ratio.

The expected reactor antineutrinos events have been evaluated to be $97.6^{+1.7}_{-1.6}$ without “5 MeV excess” and $91.9^{+1.6}_{-1.5}$ when the “5 MeV excess” has been assumed, to be compared with $92.5^{+13.3}_{-10.5}$ events obtained from the fit on the experimental data assuming the chondritic ratio and $95.8^{+14.3}_{-11.6}$ leaving Th and U as free parameters (what quoted includes both the statistical and systematic errors), showing a good alignment between evaluation and data.

What can be obtained from the study of geoneutrinos? Certainly two pieces of information: the first concerns the mantle and the second the contribution of the radiogenic heat to the total Earth heat.

We know that the Earth crust, by measuring the sediment and rocks, contains radioactive nuclides. To extract the mantle signal, the contribution of the local crust and of the far crust must be subtracted from the total signal. In Borexino the signal due to the crust in the region around the Gran Sasso (LOC) was first calculated, considering an area of ​​492 km × 444 km, together with the contribution of the rest of the crust (ROC), using the 1°×1° 3D model and integrating the contribution on the whole Earth. The two contributions give in total $28.8^{+5.5}_{-4.6}$ events; by subtracting the local contribution of the lithosphere from the total, we obtain: $N_{\mathrm{mantle}} = 23.1$ The statistical significance of the signal is studied with pseudo-MC experiments with and without a generated mantle signal. The $p$-value of the fit is $p= 9.796 \times 10^{–3}$ and the hypothesis of no signal in the mantle is rejected at 99.0% C.L., corresponding to $2.3\sigma$.

Direct measurements of the accessible lithosphere provide an estimate of the heat that it produces, namely $8.1^{+1.9}_{-1.4}$ TW corresponding to about 17% of the total Earth heat, which is currently estimated at $47 \pm 2$ TW. It is much more difficult to assess the radiogenic heat of the mantle because it is not known how the radioactive nuclides are distributed within it; the two extreme hypotheses range from a homogeneous distribution throughout the mantle to the radioactive activity concentrated at the boundary between the mantle and the Earth core. So we can only evaluate an interval, which is very large: 1.2–39.8 TW. Following a procedure on the scatter plot of the mantle signal vs. the radiogenic mantle heat we can obtain the median signal in the mantle and then the heat which, including also a ⁴⁰K contribution, is $30.0^{+13.5}_{-12.7}$ TW, and adding both mantle and lithosphere the total radiogenic heat is $38.2^{+13.6}_{-12.7}$ corresponds to 81% of the total terrestrial heat.

7 Some final comments

Borexino started thirty years ago and it is not over yet. Throughout these thirty years several members of the collaboration have come and gone and many young people even in the last 2-3 years joined, interested in the potential of the experiment and its results. However the collaboration has never exceeded 100 members at a given time, between researchers and technicians. Among them, about ten researchers and technicians started with me in 1990 and still continue to give their contribution to Borexino to this day. Unfortunately, during all these years 11 members of our collaboration have passed away.

Here I thank all collaborators who, in these thirty years, made the experiment possible. INFN has always supported this project and it was the major financier. I must thank particularly Luciano Maiani, at that time president of the INFN, who in 1995 has supported the approval and the Borexino funding, despite the skepticism of the physicists’ community who was judging the project too difficult and too challenging.

In recent years, Borexino has also received awards, such as the Bruno Pontecorvo International Prize and the Enrico Fermi Prize, both assigned to me, but in my mind to be awarded to the entire collaboration; Borexino’s results were ranked within the world top 10 in 2014 and in 2020 by the British Institute of Physics and in 2014 the Italian post office dedicated a commemorative stamp to the experiment. But the greatest recognition has been the continuous incoming flow of request we have been receiving for years, inviting to give talks to all the most important conferences in the world.

I want to close this summary of Borexino’s activity by saying that this project leaves a legacy to young researchers not to be afraid of engaging themselves in very challenging experiments because the important discoveries mainly come from endeavors. When engaging in such projects it is important to have what I like to call the “physical sense”, that is the ability to guess whether an experiment or a measurement, even if particularly difficult and apparently prohibitive, still has the possibility of succeeding. And what pushed us over the years in the Borexino project was this confidence in what we could achieve, despite being surrounded by skepticism since the beginning and met by increasingly complex measures, a skepticism that concerned even some internal members of the collaboration.

I would like to conclude with a statement of a great friend of Borexino, John N. Bahcall: “The most important discoveries will provide answers to questions that we do not yet know how to ask and will concern objects we have not yet imagined” (J. N. Bahcall in “How the Sun Shines”, 2000, Nobel e-Museum, astroph/0009259).