Final results of Borexino Phase-I on low-energy solar neutrino spectroscopy

G. Bellini et al. (Borexino Collaboration)

Phys. Rev. D 89, 112007 – Published 25 June 2014

Abstract

Borexino has been running since May 2007 at the Laboratori Nazionali del Gran Sasso laboratory in Italy with the primary goal of detecting solar neutrinos. The detector, a large, unsegmented liquid scintillator calorimeter characterized by unprecedented low levels of intrinsic radioactivity, is optimized for the study of the lower energy part of the spectrum. During Phase-I (2007–2010), Borexino first detected and then precisely measured the flux of the ${}^{7}{Be}$ solar neutrinos, ruled out any significant day-night asymmetry of their interaction rate, made the first direct observation of the pep neutrinos, and set the tightest upper limit on the flux of solar neutrinos produced in the CNO cycle (carbon, nitrogen, oxigen) where carbon, nitrogen, and oxygen serve as catalysts in the fusion process. In this paper we discuss the signal signature and provide a comprehensive description of the backgrounds, quantify their event rates, describe the methods for their identification, selection, or subtraction, and describe data analysis. Key features are an extensive in situ calibration program using radioactive sources, the detailed modeling of the detector response, the ability to define an innermost fiducial volume with extremely low background via software cuts, and the excellent pulse-shape discrimination capability of the scintillator that allows particle identification. We report a measurement of the annual modulation of the ${}^{7}{Be}$ neutrino interaction rate. The period, the amplitude, and the phase of the observed modulation are consistent with the solar origin of these events, and the absence of their annual modulation is rejected with higher than 99% C.L. The physics implications of Phase-I results in the context of the neutrino oscillation physics and solar models are presented.

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Received 1 August 2013

DOI:https://doi.org/10.1103/PhysRevD.89.112007

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Vol. 89, Iss. 11 — 1 June 2014

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Figure 1
Energy spectrum of solar neutrinos. The spectral shapes are taken from [1] while the flux normalization from [2]. The vertical axis reports the flux in ${cm}^{- 2}$ $s^{- 1}$ $(10^{3} keV)^{- 1}$ for the continuous neutrino spectra, while in ${cm}^{- 2}$ $s^{- 1}$ for the monochromatic lines ( ${}^{7}{Be} − ν$ at 384 and 862 keV, shown as red dotted lines, and pep- $ν$ at 1440 keV, shown as a blue continuous line). The numbers in parentheses represent the theoretical uncertainties on the expected fluxes.
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Figure 2
The schematic view of the Borexino detector.
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Figure 3
Neutrino-electron elastic scattering cross section as a function of the neutrino energy for $ν_{e}$ (solid line) and for $ν_{μ}$ or $ν_{τ}$ (dashed line).
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Figure 4
Mean dark-noise rate per PMT in counts per second as a function of time starting from May 16, 2007 (day 0).
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Figure 5
Top: examples of the distributions of the number of PMT hits for three laser intensities. Bottom: relation between the average number of PMT hits detected by Borexino inner detector and the laser intensity. These calibration values, which are approximately, but not exactly, linear, were used to compute the detection efficiency shown in Fig. 6.
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Figure 6
The trigger efficiency as a function of the average number of detected PMT hits. For each point, the average number of hits was obtained from the calibration shown in Fig. 5. The fit function is an error function with standard deviation computed assuming exact Poisson statistics. The fit mean value is 25.7, in very good agreement with the nominal triggering threshold of $K = 25$ .
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Figure 7
An example of a single data acquisition gate (so-called event) containing two well-separated clusters, which are due to two different interactions inside the scintillator.
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Figure 8
The quenching factor $Q_{β} (E)$ calculated from Eq. (7) with the Borexino quenching parameter $k B = 0.011 cm / MeV$ .
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Figure 9
Position of the various radioactive sources deployed in the scintillator and used to calibrate the Borexino detector.
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Figure 10
The number $N^{'}$ (black) and $N^{''}$ (red) of available channels for the computation of $N_{h}$ / $N_{p}$ and $N_{pe}$ energy estimators, respectively, as a function of time starting from May 16, 2007 (day 0). The slow decrease is due to the PMT mortality, while the sudden changes are due to the failure/repair of the electronics.
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Figure 11
Probability density functions for detected hits as a function of time elapsed from the emission of scintillation light. Different curves are for increasing values of the hit charge $q_{i}^{j}$ collected at the phototube: from 1 p.e. (red solid curve) to 10 p.e. (red dashed curve).
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Figure 12
Resolution ( $σ$ ) of the reconstructed $x$ (solid red line), $y$ (dashed black line), and $z$ (dotted blue line) coordinates as a function of energy (in number of photoelectrons) for events from calibration sources placed in the center of the detector.
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Figure 13
Distribution of differences between the reconstructed and nominal (CCD) coordinates for the radon source data ( ${}^{214}{Po}$ ) measured in 182 different positions in the scintillator: $x$ (top), $y$ (center), and $z$ (bottom).
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Figure 14
Spatial distribution (in the $x − z$ plane, $| y | < 0.5 m$ ) of all reconstructed events (besides muons) in different energy regions. The color axis represents the number of events per $0.0144 m^{3}$ in a pixel of $0.12 m \times 1.00 m \times 0.12 m$ ( $x \times y \times z$ ). (a) ${}^{210}{Po}$ peak region: 145–300 $N_{pe}$ (290–600 keV). (b) ${}^{7}{Be}$ shoulder: 300–375 $N_{pe}$ (650–850 keV). (c) ${}^{11}C$ energy region: 425–650 $N_{pe}$ (850–1300 keV). (d) ${}^{208}{Tl}$ peak region: 900–1500 $N_{pe}$ (1800–3000 keV). Events occurring outside the IV ( $R = 4.25 m$ ), near the top end cap ( $z = 4.25 m$ ), most clearly seen in the low-energy region shown in (a), are due to the small leak of scintillator from the IV to the buffer region (Sec. 2a).
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Figure 15
Comparison between the energy spectra ( $N_{h}$ energy estimator) for events from the external ${}^{228}{Th} / {}^{208}{Tl}$ source calibration data (black solid line) and from the 2610 keV simulated external $γ$ rays (red dotted line), reconstructed within 3 m from the detector center. The $χ^{2} / NDF$ (where NDF indicates number of degree of freedom) between the two histograms is $≃ 1.2$ . The source is positioned in the upper hemisphere; a similar result is obtained for the lower one.
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Figure 16
Comparison between the reconstructed radius of events with energy $N_{h} > 400$ from the external ${}^{228}{Th} / {}^{208}{Tl}$ source calibration data (black solid line) and from the 2610 keV simulated external $γ$ rays (red dotted line), reconstructed within 3 m from the detector center. The left and right plots show examples for a source position in the upper/north and lower/south hemispheres, respectively. The $χ^{2} / NDF$ between the two histograms (Monte Carlo and data) is in the range 0.8–0.9.
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Figure 17
$ρ − z$ projection of the IV as of May 3, 2009 (black curve). The blue curve shows the shape of the 75 ton FV used for the measurement of the ${}^{7}{Be} − ν$ interaction rate. The red curve illustrates the profile of the FV used for the ${}^{7}{Be} − ν$ annual modulation analysis.
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Figure 18
$z − x$ distribution ( $| y | < 0.5 m$ ) of the events in the energy region 800–900 keV, which are mainly due to ${}^{210}{Bi}$ contaminating the IV surface. The color axis represents the number of events per $0.0144 m^{3}$ in a pixel of $0.12 m \times 1.00 m \times 0.12 m$ ( $x \times y \times z$ ). This spatial distribution reveals the IV shift and deformation with respect to its nominal spherical position shown in solid black line.
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Figure 19
$R$ – $\cos (θ)$ distribution of the events in the energy region 800–900 keV from November 2007 used for the IV shape reconstruction. The best-fit vessel shape is shown in a solid red line. The dotted red line represents the nominal spherical vessel with $R = 4.25 m$ .
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Figure 20
Calculated energy spectra due to solar neutrinos (shown as continuous red line) and of the main background components (dotted lines). The realistic Borexino energy resolution is included. The rates are fixed at the values shown in parenthesis and given in units of $cpd / 100 ton$ .
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Figure 21
Number (counts/week) of ${}^{214}{Bi} − {}^{214}{Po}$ coincidences detected in the whole scintillator volume as a function of time starting from May 16, 2007 (day 0). The spikes are due to the filling operations.
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Figure 22
Spatial distribution of the ${}^{214}{Po}$ events (from ${}^{214}{Bi} − {}^{214}{Po}$ coincidences) in the $z − ρ$ plane from May 2007 to May 2010. The solid red line indicates the FV used in the ${}^{7}{Be} − ν$ analysis. The layer of events in the upper hemisphere is outside this volume and corresponds to the increased rate of radon at the beginning of data taking (see Fig. 21).
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Figure 23
Count rate $R (t)$ in the energy region from $390 < N_{pe}^{d} < 450$ in the FV used for the ${}^{7}{Be} − ν$ annual modulation analysis from the beginning of the year 2008 until the middle of year 2010. This region is dominated by the ${}^{210}{Bi}$ contribution. The red line is a fit according to Eq. (22).
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Figure 24
${}^{210}{Po}$ count rate in the FV used in the ${}^{7}{Be} − ν$ rate analysis as a function of time. The various increases are due to the IV-filling operations shown by the three vertical lines (Sec. 2a) and due to the tests of the purification methods (details in text).
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Figure 25
${}^{210}{Po}$ count rate as a function of the vertical position $z$ . The four plots represent the four periods separated by the IV-filling campaigns shown by the vertical lines in Fig. 24.
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Figure 26
Distribution of the ${}^{212}{Po}$ events (from ${}^{212}{Bi} − {}^{212}{Po}$ coincidences) in the $z − ρ$ plane from May 2007 to May 2010. The solid red line indicates the FV used in the ${}^{7}{Be} − ν$ analysis.
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Figure 27
The reference $r_{α} (t_{n})$ (red line) and $r_{β} (t_{n})$ (black line) pulse shapes obtained by tagging the radon-correlated ${}^{214}{Bi} − {}^{214}{Po}$ coincidences. The dip at 180 ns is due to the dead time on every individual electronic channel applied after each detected hit (see Sec. 6). The small shoulder around 60 ns is due to the light reflected on the SSS surface and on the PMTs’ cathodes.
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Figure 28
The distribution of $G_{α}$ (red line) and $G_{β}$ (black line) [see Eq. (28)] for events obtained by tagging the radon correlated ${}^{214}{Bi} − {}^{214}{Po}$ coincidences.
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Figure 29
The distribution of the $G_{β}^{-}$ parameter (black line) obtained from ${}^{214}{Bi} (β^{-})$ events and of the $G_{β}^{+}$ parameter (red line) obtained from ${}^{11}C (β^{+})$ events.
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Figure 30
Example of an event with $N_{peak} = 2$ . The horizontal axis shows the time difference of each hit with respect to trigger signal, so the absolute values have no physical meaning.
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Figure 31
An example of the application of the $S_{p}$ energy-dependent cut [see Eq. (39)] indicated by the solid red line on a sample of events which are not tagged as muons and which are reconstructed in the ${}^{7}{Be}$ -FV. The events having the $S_{p}$ variable above the value indicated by the line are excluded from the data set. Clearly, the cluster at energies of $N_{p} \sim 130$ with $S_{p} > 1.7$ is due to anisotropic noise events.
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Figure 32
Effect of selection cuts on the raw spectrum in the $N_{p}$ variable (marked 1 and shown in blue). The black spectrum (marked 2) shows the effect of the muon and muon daughter cut. The shape of the final $N_{p}$ spectrum (marked 3 and shown in red) is dominated by the effect of the ${}^{7}{Be} − ν$ FV cut. The effect of other cuts described in Sec. 13a is important at the level of fit, but cannot be appreciated visually.
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Figure 33
Final spectra for all events in the ${}^{7}{Be} − ν$ FV passing all selection cuts, shown in three different energy estimators: $N_{pe}$ (1-dotted blue), $N_{p}$ (2-solid black), and $N_{h}$ (3-dashed red).
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Figure 34
Final spectrum for all events passing all selection cuts is shown in $N_{p}$ variable (black) with respect to the $γ$ sources calibration spectra: ${}^{203}{Hg}$ (red), ${}^{85}{Sr}$ (green), ${}^{54}{Mn}$ (magenta), ${}^{65}{Zn}$ (blue), and ${}^{40}K$ (cyan). The calibration peaks are all normalized to an area of 500.
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Figure 35
The distribution of the Gatti $G_{α β}$ parameter as a function of energy ( $N_{p}$ variable). The continuous (blue) line shows the energy-dependent cut; it removes all the events with $G_{α β}$ above the line. The (red) points enclosed within the dashed line show the $G_{α β}$ distribution for MC-simulated pep- $ν$ ’s; these events are an example of true $β$ events with a spectrum extending over a sufficiently large energy range. The $β$ events are not affected by this cut, while, on the contrary, this cut removes a large fraction of the $α$ events, as those from the ${}^{210}{Po}$ decay.
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Figure 36
Example of $α$ - $β$ statistical subtraction with the analytical method for events in the energy range $200 < N_{p e}^{d} < 205$ . The blue and red lines show the individual Gaussian fits to the Gatti parameter distributions for the $β$ and $α$ components, respectively, while the green line is the total fit.
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Figure 37
Example of $α$ - $β$ statistical subtraction with the Monte Carlo method for events in the energy range $168 < N_{h} < 170$ . The plot shows the Gatti parameter of the data (black) and the fit (green) aiming to separate the $α$ (red) and $β$ (blue) contributions. In comparison to the analytical method demonstrated in Fig. 36, the blue $β$ and $α$ shapes are Monte Carlo shapes which are not Gaussian. See text for details.
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Figure 38
Prediction of the sensitivity to the pep- $ν$ interaction rate measurement ( $z$ axis) as a function of the residual ${}^{11}C$ rate ( $y$ axis) and the effective live time ( $x$ axis) obtained by fitting the MC-simulated data. The color $z$ axis is expressed as the uncertainty (%) on the pep- $ν$ interaction rate returned by the fit. We recall that the ${}^{11}C$ rate measured without applying the TFC subtraction is ( $28.5 \pm 0.2 (stat) \pm 0.7 (sys)) cpd / 100 ton$ . The black diamond corresponds to the TFC-subtracted spectrum used in the analysis (see Fig. 40).
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Figure 39
The spatial regions vetoed in the TFC method: a cylinder around the muon track (blue) and some examples of spheres centered around the point where the $γ$ following the neutron capture is reconstructed (areas with horizontal lines around the stars) and their projections along the muon track (green areas).
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Figure 40
Energy spectra ( $N_{p e}^{d}$ energy estimator) before (black) and after (red) the application of the TFC technique for ${}^{11}C$ removal. Both spectra are normalized to the same exposure.
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Figure 41
Hit-emission time profile of a single event due to $β^{+}$ decay, where the positron deposits its kinetic energy (first peak) and then forms orthopositronium. The orthopositronium exists for $\sim 10 ns$ before the positron annihilates with a bulk electron to produce $γ$ rays (second peak). The PS-BDT value (Fig. 45) of this cluster is $- 0.44$ .
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Figure 42
Reconstructed photon emission times relative to the start time of the cluster: ${}^{214}{Bi}$ ( $β^{-}$ ) events with $425 < N_{h} < 475$ identified by a ${}^{214}{Bi} − {}^{214}{Po}$ fast coincidence tag (black) and ${}^{11}C$ ( $β^{+}$ ) events tagged by the TFC (red).
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Figure 43
Reconstructed photon emission times for ${}^{11}C$ events. The data (black points) have been fitted to the ${}^{11}C$ Monte Carlo shapes without (1-blue) and with (2-green) orthopositronium formation (mean life of 3.1 ns); the relative weights of these two shapes were left free in the fit. The best fit, the sum of the two MC shapes, is shown in red.
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Figure 44
Average fraction of the energy deposited by ${}^{214}{Bi}$ decays in the form of $γ$ rays. For deposited energy below 1400 keV, only less than 5% of the energy is due to $γ$ rays. Therefore, low-energy ${}^{214}{Bi}$ decays are a sample of mostly pure $β^{-}$ decays.
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Figure 45
Distributions of the PS-BDT parameter for the test samples of $β^{-}$ (black) and $β^{+}$ (red) events as described in the text.
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Figure 46
Data points for six different $γ$ -ray lines: ${}^{203}{Hg}$ , ${}^{85}{Sr}$ , ${}^{54}{Mn}$ , ${}^{65}{Zn}$ , ${}^{40}K$ , and the 2230 keV $γ$ from the neutron capture: $N_{p e}^{ideal} / MeV$ expressed as a function of $E_{γ}$ . The data is fit with the function corresponding to Eq. (42) which is obtained by a dedicated “ad hoc” Monte Carlo using the Birk’s quenching model (details in text). The best fit values are $k B = 0.00115 \pm 0.0007$ and $Y_{0}^{p e} = 489 \pm 2 p . e . / MeV$ .
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Figure 47
Energy spectra ( $N_{p}$ , $N_{h}$ , and $N_{p e}^{d}$ variables) of the calibration sources placed in the detector’s center: measured data (black lines) versus the Monte Carlo simulation (areas dashed with red lines). The peaks represent (from the left to the right) the total $γ$ decay energy of ${}^{57}{Co}$ , ${}^{139}{Ce}$ , ${}^{203}{Hg}$ , ${}^{85}{Sr}$ , ${}^{54}{Mn}$ , ${}^{65}{Zn}$ , ${}^{40}K$ , and ${}^{60}{Co}$ .
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Figure 48
The $N_{h}$ peak position vs $z$ coordinate of the ${}^{214}{Po}$ $α$ peak from the radon calibration source, shown for the data (black circles) and the Monte Carlo simulation (red stars). The various points at fixed $z$ position correspond to different $x$ and $y$ coordinates. The reduction of the collected light for negative $z$ is due to the concentration of broken PMTs close to the detector’s “South pole.”
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Figure 49
The relative difference $\frac{N_{h} (MC) - N_{h} (data)}{N_{h} (data)}$ as a function of the radial position $R$ of the ${}^{214}{Po}$ $α$ peak from the radon calibration source. Blue triangles: ${}^{7}{Be} − ν$ FV; green circles: pep- $ν$ FV; red stars: outside both FVs.
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Figure 50
The energy estimators $N_{p}$ , $N_{h}$ , and $N_{p e}$ versus energy for $β$ events uniformly generated in the ${}^{7}{Be}$ -FV as obtained with the Monte Carlo simulation.
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Figure 51
The $G_{α β}$ variable for the ${}^{85}{Sr}$ $γ$ -calibration source placed at the location $(x, y, z) = (0, 0, 3) m$ , for measured (black line) and Monte Carlo simulated (red filled area) events.
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Figure 52
Example of fit of the energy spectrum obtained using the Monte Carlo method without $α$ - $β$ statistical subtraction. The fit was done using the $N_{h}$ energy estimator. After the fit, the horizontal axis was converted into an energy scale in keV. The values of the best-fit parameters, the rates of individual species, are given in $cpd / 100 ton$ .
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Figure 53
Example of fit of the energy spectrum obtained using the analytical method with $α$ - $β$ statistical subtraction. The fit was done using the $N_{p e}^{d}$ energy estimator. After the fit, the horizontal axis was converted into an energy scale in keV. The values of the best-fit parameters, the rates of individual species, are given in $cpd / 100 ton$ .
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Figure 54
Example of fit of the energy spectrum obtained using the analytical method without $α$ - $β$ statistical subtraction. The fit was done using the $N_{p}$ energy estimator. After the fit, the horizontal axis was converted into an energy scale in keV. The values of the best-fit parameters, the rates of individual species, are given in $cpd / 100 ton$ .
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Figure 55
Typical example of the distribution of the residuals of the fit. This plot corresponds to the fit shown in Fig. 54.
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Figure 56
The experimental exposure function (black continuous line) and the ideal exposure function (red dotted line) corresponding to 3 years of data taking without interruptions at LNGS, as functions of the $θ_{z}$ angle ( $1 ° / bin$ ). The interval from $- 180 °$ to $- 90 °$ corresponds to day (360.25 days) and the one from $- 90 °$ to 0° to night time (380.63 days). We recall that at LNGS latitude the Sun is never at the zenith.
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Figure 57
Energy spectra ( $N_{h}$ ) during the day time in different volumes. The two curves showing the lowest count rate and almost superimposed have been obtained selecting events within the standard FV used for the ${}^{7}{Be} − ν$ rate analysis (1: black dotted line) and within a sphere of 3 m radius (2: red solid line). The curve with the highest count rate (3: green) shows the events selected within a 3.5 m radius sphere while the curve with the intermediate count rate (4: blue) refers to a 3.3 m radius sphere.
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Figure 58
Normalized $θ_{z}$ -angle distribution of the events in the ${}^{7}{Be} − ν$ energy window and reconstructed within the enlarged FV. The effect of the Earth elliptical orbit has been removed. The fit with a flat straight line yields $χ^{2} / NDF = 141.1 / 139$ . The blue solid line shows the expected effect with the LOW (large mixing angle at small mass differences) solution $Δ m_{12}^{2} = 1.0 \times 1 0^{- 7} {eV}^{2}$ and tan $θ_{12}^{2} = 0.955$ .
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Figure 59
Difference of the night and day spectra in the enlarged FV. The top panel shows an extended energy range including the region dominated by the ${}^{11}C$ background while the bottom panel is a zoom in the ${}^{7}{Be} − ν$ energy window. The fit is performed in the ${}^{7}{Be} − ν$ energy region (between 250 and 800 keV) with the residual ${}^{210}{Po}$ spectrum and the electron recoil spectrum due to the ${}^{7}{Be}$ solar neutrino interaction. The blue curve plots the ${}^{7}{Be} − ν$ spectrum used in the fit with the amplitude $⟨ R ⟩ = 46 cpd / 100 ton$ . The residual ${}^{7}{Be} − ν$ resulting from the fit is too small to be shown.
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Figure 60
$G_{α β}$ parameter in the FV used in the seasonal modulation analysis shown as a function of energy ( $N_{p e}^{d}$ estimator). The red line indicates where the Gatti-cut was placed: in the window of (105,380) $N_{p e}^{d}$ , 66% of $β$ ’s survive while almost 100% of $α$ ’s are rejected. Black bars represent, for different energy bins, the $G_{β}$ interval covering 99.9% of $β$ events.
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Figure 61
Energy distribution of the ${}^{210}{Po} (α)$ peak (expressed in $N_{p e}^{d}$ estimator) in the FV used for the ${}^{7}{Be} − ν$ annual modulation analysis.
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Figure 62
Absolute distance between ${}^{214}{Bi} − {}^{214}{Po}$ fast coincidence events in three periods: May 2007 (solid red line), July 2009 (solid green line), and April 2010 (dotted blue line).
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Figure 63
Results obtained with rate analysis. The continuous line is the curve of Eq. (84). The data are grouped in bins of 60 days.
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Figure 64
The results obtained via the fit of the rate-versus-time distribution shown in Fig. 63: a $Δ χ^{2}$ ( $Δ χ_{R}^{2} = χ_{R_{\min}}^{2} − χ_{R_{x y}}^{2}$ ) map (vertical axis) as a function of eccentricity $ϵ$ and period. The yellow star indicates the best-fit results, while the yellow triangle the expected values. Confidence contours of 1, 2, and $3 σ$ are indicated with black solid lines.
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Figure 65
Rate of events in the energy region $105 < N_{p e}^{d} < 380$ and in the FV used in the seasonal modulation analysis as a function of time shown with 10-day binning: the red data points (a) were scaled by a constant factor. For comparison, the black plot (b) shows the count rate due to external $γ$ s which is stable in time since it is not correlated with the changes of the IV shape.
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Figure 66
Lomb-Scargle periodogram for the red data points marked (a) in Fig. 65 after the subtraction of the exponentail trend due to the ${}^{210}{Bi}$ contamination. The spectral power density at 1 year is 7.961, as indicated by the vertical line.
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Figure 67
Distributions of the Lomb-Scargle spectral power density (SPD) at frequency corresponding to 1 year for $1 0^{4}$ simulations of a 7% solar neutrino annual flux modulation with realistic background (solid black line) and the same number of white-noise simulations of background without any seasonally modulated signal (red area). Indicated with vertical lines are the sensitivity thresholds of $1 σ$ (solid), $2 σ$ (dashed), and $3 σ$ (dotted) C.L. with corresponding detection probabilities of 81.62, 43.54, and 11.68%, respectively. The detected SPD $(1 year) = 7.961$ (see Fig. 66) represents an evidence of the annual-modulation signal with a significance higher than $3 σ$ ; our chance of detecting the annual modulation at this level of significance is 11.68%.
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Figure 68
A sequence of the IMF extracted from the data (black circles) plus white noise (red diamonds) by means of a sifting algorithm using 20 iterations for each one. The ${IMF}_{8}$ is the one used to extract the results about the seasonal modulation. The last ${IMF}_{9}$ , also called “trend,” is the best representation of the background variation in time.
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Figure 69
Set of 100 intrinsic mode functions ${IMF}_{8}$ extracted after the addition of dithering (grey lines). The black solid line is their average and the dashed red line is the annual modulation from Eq. (84). The number of the IMF is the expected one for the annual frequency of $ν_{year} = 0.00274 [d^{- 1}]$ .
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Figure 70
Blue circles show the eccentricities and periods obtained with the EMD method with 100 ${IMF}_{8}$ ’s (see Fig. 69). The projections shown on the vertical and horizontal axis are well described by Gaussian curves. The solid black line of Fig. 69 is represented by the yellow star in the middle of 1, 2, and $3 σ$ C.L. contours. The black star represents the expectations. The measured period is in perfect agreement with 1 year, while the eccentricity is compatible within $3 σ$ .
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Figure 71
Energy spectra and the best-fit results performed on the $N_{h}$ energy estimator in the pep and CNO neutrino analysis. The top panel shows the best fit to the TFC-subtracted spectrum, while the bottom one presents the fit to the complementary, TFC-tagged events. The best-fit values for the rates of the species included in the fit are shown in the legend. Units are $cpd / 100 ton$ .
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Figure 72
Energy spectra and the best-fit results performed on the $N_{p e}^{d}$ energy estimator in the pep and CNO neutrino analysis. The top panel shows the best fit to the TFC-subtracted spectrum, while the bottom one presents the fit to the complementary, TFC-tagged events. The best-fit values for the rates of the species included in the fit are shown in the legend. Units are $cpd / 100 ton$ .
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Figure 73
Left: radial distribution of events in the energy range $N_{h} = 500 - 900$ (black data points) and the best fit (solid red line) obtained performing the multivariate fit with the $N_{h}$ energy estimator. The $y$ -axis scale is logarithmic. A similar and statistically consistent result is obtained using the variable $N_{p e}^{d}$ . Right: distribution of the PS-BDT parameter for events in the energy range 450–900 $N_{p e}^{d}$ (black data points) and the best fit (solid red line) obtained performing the multivariate fit with the variable $N_{p e}^{d}$ . The scale is logarithmic. Similar and consistent results are obtained with $N_{h}$ variable.
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Figure 74
Correlation between CNO- $ν$ and ${}^{210}{Bi}$ rates. Left: $Δ χ^{2}$ map obtained from a likelihood ratio test between the likelihood of the best-fit result and the maximum likelihood returned by the fit when the ${}^{210}{Bi}$ and CNO- $ν$ interaction rates are fixed to different values. The right column gives the colors corresponding to $Δ χ^{2}$ values; note that the same red color has been used to plot all $Δ χ^{2} \geq 14$ values to allow to visualize color variations in the relevant region of the plot. The plot has been obtained using $N_{h}$ energy estimator. The pep- $ν$ rate is fixed to the standard solar model prediction. Right: ${}^{210}{Bi}$ interaction rate returned by the fit for different (fixed) CNO- $ν$ interaction rates. The shaded area is the statistical uncertainty.
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Figure 75
Correlation between the pep and CNO neutrino rates. Left: $Δ χ^{2}$ map obtained from a likelihood ratio test between the likelihood of the best-fit result and the maximum likelihood returned by the fit when pep and CNO neutrino interaction rates are fixed to different values. The right column gives the colors corresponding to $Δ χ^{2}$ values; note that the same red color has been used to plot all $Δ χ^{2} \geq 14$ values to allow one to visualize color variations in the relevant region of the plot. The numerical values of this map are given in Table 23. The plot has been obtained using $N_{p e}^{d}$ energy estimator; we get similar and consistent numbers using the $N_{h}$ variable. Right: pep- $ν$ interaction rate returned by the fit for different (fixed) CNO neutrino interaction rates. The shaded area is the statistical uncertainty.
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Figure 76
Residuals after the best-fit values of all species except the signal from pep and CNO solar neutrinos are subtracted from the data energy spectrum (black data points). The $e^{-}$ recoil spectrum due to pep- $ν$ is shown by the solid red line. A second fit has been performed excluding the pep and CNO signals. The dashed blue line gives the resulting best-fit spectrum, after the subtraction of the best estimates for the backgrounds, as obtained from the fit used for the signal extraction. The two isotopes showing a significant difference in the output of the background only fit and in the one with the solar neutrinos are ${}^{210}{Bi}$ and ${}^{234 m}{Pa}$ .
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Figure 77
Comparison between the ${}^{210}{Bi}$ energy spectrum measured using the magnetic spectrometer [77] [crosses, labeled as Langer (1954)] with spectra obtained using different correction factors: those calculated in [78] [red curve labeled as Daniel (1962)] and in [79] [blues curve labeled as Grau (2005)]. Also shown are the fit to the Langer data according to Eq. (88) with all three parameters and with $b = 0$ .
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Figure 78
The Borexino data analysis in the $\tan^{2} θ_{12} - Δ m_{21}^{2}$ space. Allowed regions ( $NDF = 2$ ) at 68.27% C.L. (pink), 95.45% C.L. (green), and 99.73% C.L. (blue). (a) Impact of the Borexino ${}^{7}{Be}$ - $ν$ rate measurement. (b) Combined analysis of Borexino measurements of ${}^{7}{Be}$ - and ${}^{8}B$ - $ν$ rates. (c) Impact of ${}^{7}{Be}$ - and ${}^{8}B$ - $ν$ rates together with the ${}^{7}{Be} − ν$ day-night asymmetry results. (d) Global impact of all the Borexino measurements to date, including the pep- $ν$ rate.
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Figure 79
$χ^{2}$ profile of the $Δ m_{21}^{2}$ parameter in the Borexino results analysis. Dashed lines indicate the $1 σ$ [ $χ^{2} (Δ m_{21}^{2}) = 1$ ], $2 σ$ [ $χ^{2} (Δ m_{21}^{2}) = 4$ ], and $3 σ$ [ $χ^{2} (Δ m_{21}^{2}) = 9$ ] levels. The MSW-LOW region ( $10^{- 8} {eV}^{2} < Δ m_{21}^{2} < 10^{- 6} {eV}^{2}$ ) is ruled out at more than $8.5 σ$ .
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Figure 80
Allowed regions ( $NDF = 3$ ) of the space of parameters at 68.27% C.L. (pink), 95.45% C.L. (green), and 99.73% C.L. (blue) by the solar-without Borexino (left panel) and solar-with Borexino (right panel) data set.
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Figure 81
Comparison of the $χ^{2}$ profile for $Δ m_{21}^{2}$ obtained by the analysis of all available solar data without (left) and with (right) the Borexino contribution, after marginalization over $\tan^{2} θ_{12}$ and $\sin^{2} θ_{13}$ .
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Figure 82
The $1 σ$ theoretical range of high (red) and low (blue) metallicity standard solar model for $f_{Be}$ and $f_{B}$ , compared to the $1 σ$ (light pink), $2 σ$ (light green), and $3 σ$ (light blue) allowed regions by the global analysis of solar-with Borexino plus KamLAND results. The theoretical correlation factors are taken from [85].
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Figure 83
Electron neutrino survival probability as a function of neutrino energy according to the MSW-LMA model; see Eq. (94). The pink band is for ${}^{8}B$ solar neutrinos, considering their production region in the Sun. The other points represent other solar neutrino fluxes, considering their proper production regions, and reported for monoenergetic ${}^{7}{Be}$ (862 keV) and $p e p$ (1444 keV) neutrinos and for the mean energies of fluxes with continuous energy spectrum: $p p$ (267 keV), ${}^{13}N$ (707 keV), ${}^{15}O$ (997 keV), and $h e p$ (9625 keV).
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Figure 84
Electron neutrino survival probability as a function of neutrino energy according to the MSW-LMA model. The band is the same as in Fig. 83, calculated for the production region of ${}^{8}B$ solar neutrinos which represents well also other species of solar neutrinos. The points represent the solar neutrino experimental data for ${}^{7}{Be}$ and $p e p$ monoenergetic neutrinos (Borexino data), for ${}^{8}B$ neutrinos detected above 5000 keV of scattered-electron energy $T$ (SNO and Super-Kamiokande data) and for $T > 3000 keV$ ( $SNO LETA + Borexino data$ ), and for $p p$ neutrinos considering all solar neutrino data, including radiochemical experiments.
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Physical Review D

covering particles, fields, gravitation, and cosmology