# A Search for Electron Neutrino Transitions to Sterile States in the BEST
Experiment

V.V. Barinov  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow 117312, Russia  B.T. Cleveland  SNOLAB, Sudbury, ON P3Y 1N2,
Canada  S.N. Danshin  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow 117312, Russia  H. Ejiri  Research Center for Nuclear
Physics, Osaka University, Osaka, Japan  S.R. Elliott  Los Alamos National
Laboratory, Los Alamos, NM, USA  D. Frekers  Institut für Kernphysik,
Westfälische Wilhelms-Universität Munster, D-48149 Munster, Germany  V.N.
Gavrin<EMAIL_ADDRESS>Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow 117312, Russia  V.V. Gorbachev  Institute for Nuclear
Research of the Russian Academy of Sciences, Moscow 117312, Russia  D.S.
Gorbunov  Institute for Nuclear Research of the Russian Academy of Sciences,
Moscow 117312, Russia  W.C. Haxton  Nuclear Science Division, Lawrence
Berkeley National Laboratory, Berkeley, CA 94720, USA  Department of Physics,
University of California, Berkeley, CA 94720, USA  T.V. Ibragimova  Institute
for Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia
I. Kim  Los Alamos National Laboratory, Los Alamos, NM, USA  Yu.P. Kozlova
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow
117312, Russia  L.V. Kravchuk  Institute for Nuclear Research of the Russian
Academy of Sciences, Moscow 117312, Russia  V.V. Kuzminov  Institute for
Nuclear Research of the Russian Academy of Sciences, Moscow 117312, Russia
B.K. Lubsandorzhiev  Institute for Nuclear Research of the Russian Academy of
Sciences, Moscow 117312, Russia  Yu.M. Malyshkin  Institute for Nuclear
Research of the Russian Academy of Sciences, Moscow 117312, Russia  R.
Massarczyk  Los Alamos National Laboratory, Los Alamos, NM, USA  V.A. Matveev
Joint Institute for Nuclear Research (JINR) Joliot-Curie 6, 141980, Dubna,
Moscow Region, Russia  I.N. Mirmov  Institute for Nuclear Research of the
Russian Academy of Sciences, Moscow 117312, Russia  J.S. Nico  National
Institute of Standards and Technology, 100 Bureau Dr, Gaithersburg, MD 20899,
USA  A.L. Petelin  JSC ‘State Scientific Center Research Institute of Atomic
Reactors’, Dimitrovgrad, 433510, Russia  R.G.H. Robertson  Center for
Experimental Nuclear Physics and Astrophysics, and Department of Physics,
University of Washington, Seattle, WA 98195, USA  D. Sinclair  Carleton
University 1125 Colonel By Drive Ottawa, K1S 5B6, Canada  A.A. Shikhin
Institute for Nuclear Research of the Russian Academy of Sciences, Moscow
117312, Russia  V.A. Tarasov  JSC ‘State Scientific Center Research Institute
of Atomic Reactors’, Dimitrovgrad, 433510, Russia  G.V. Trubnikov  Joint
Institute for Nuclear Research (JINR) Joliot-Curie 6, 141980, Dubna, Moscow
Region, Russia  E.P. Veretenkin  Institute for Nuclear Research of the Russian
Academy of Sciences, Moscow 117312, Russia  J.F. Wilkerson  Department of
Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599,
USA  Triangle Universities Nuclear Laboratory, Durham, NC 27708, USA  A.I.
Zvir  JSC ‘State Scientific Center Research Institute of Atomic Reactors’,
Dimitrovgrad, 433510, Russia

###### Abstract

The Baksan Experiment on Sterile Transitions (BEST) probes the gallium anomaly
and its possible connections to oscillations between active and sterile
neutrinos. Based on the Gallium-Germanium Neutrino Telescope (GGNT) technology
of the SAGE experiment, BEST employs two zones of liquid Ga target to explore
neutrino oscillations on the meter scale. Oscillations on this short scale
could produce deficits in the 71Ge production rates within the two zones, as
well as a possible rate difference between the zones.

From July 5th to October 13th 2019, the two-zone target was exposed to a
primarily monoenergetic, 3.4-MCi 51Cr neutrino source 10 times for a total of
20 independent 71Ge extractions from the two Ga targets. The 71Ge production
rates from the neutrino source were measured from July 2019 to March 2020. At
the end of these measurements, the counters were filled with 71Ge doped gas
and calibrated during November 2020. In this paper, results from the BEST
sterile neutrino oscillation experiment are presented in details. The ratio of
the measured 71Ge production rates to the predicted rates for the inner and
the outer target volumes are calculated from the known neutrino capture cross
section. Comparable deficits in the measured ratios relative to predicted
values are found for both zones, with the $4\sigma$ deviations from unity
consistent with the previously reported gallium anomaly. If interpreted in the
context of neutrino oscillations, the deficits give best fit oscillation
parameters of $\Delta m^{2}=3.3^{+\infty}_{-2.3}$ eV2 and
sin${}^{2}2\theta=0.42^{+0.15}_{-0.17}$, consistent with
$\nu_{e}\rightarrow\nu_{s}$ oscillations governed by a surprisingly large
mixing angle.

††preprint: APS/123-QED

##  I Introduction

Figure 1: The Ga target and extraction piping diagram also indicating the
source handling apparatus.

Sterile neutrinos ($\nu_{s}$) are hypothetical fermions which are singlets
with respect to the Standard Model (SM) gauge groups [1, 2, 3, 4, 5, 6, 7].
Lacking electric, color, or hypercharges, they would not participate in
electromagnetic, strong or weak interactions, but could thorough their
gravitational interactions have consequences for astrophysics. In particular,
sterile neutrinos could play a role in the formation of large-scale structure
while evading the strong three-flavor constraints imposed from multiple
experiments [8, 9, 10, 11, 12, 13, 14]. Sterile neutrinos naturally arise in
many extensions of the Standard Model and have been invoked to account for
various anomalies, including (for specific parameters) dark matter.

The existence of singlet state neutrinos is theoretically well-motivated.
Neutrinos are the only fermions known to have an intrinsic left-handed
chirality. When their right-handed counterparts are considered, the simplest
neutrino mass generation model of the type-I seesaw mechanism is allowed [15,
16, 17, 18]. The inactive right-handed neutrinos have so-called Majorana
masses independent of the Higgs mechanism and can generate neutrino masses at
any scale. Assuming the existence of sterile neutrinos can therefore not only
explain the non-zero mass of neutrinos but also account for neutrino masses
being at least 5 orders of magnitude smaller than the electron rest mass.

All plausible couplings of sterile neutrinos are weak and are thought to be
beyond the detection capability of the current detectors. Hence, the only
possibility for detecting sterile neutrinos is to observe the mixing with
active neutrinos. In general, extensions of the SM to include sterile
neutrinos do not impose strong constraints on their masses. While sterile
neutrinos have been proposed with masses ranging from sub-eV to GUT scales,
the 1 eV scale is particularly interesting as multiple physical phenomena can
be explained by a single hypothesis. Such indications include unexplained
excesses in the electron neutrino fluxes in LSND [19, 20] and MiniBooNE
experiments; reactor antineutrino anomalies at 10-100 m oscillation lengths
[21, 22, 23]; and the anomalous deficits in the radiochemical-source
measurements of the SAGE [24, 25] and the GALLEX [26, 27] experiments. Recent
results from MicroBooNE indicate a discrepancy with the MiniBooNE result [28],
but do not rule out the sterile neutrino hypothesis [29, 30].

Figure 2: Flowchart showing the experimental steps of the BEST experiment.
Time spent on each step and relevant efficiencies are also presented.

The Baksan Experiment on Sterile Transitions (BEST) is a two-distance
oscillation experiment designed to explore the deficit of electron neutrinos
($\nu_{e}$) previously reported in the SAGE and GALLEX source experiments,
commonly known as the gallium anomaly. The experimental design is presented in
Fig. 1, with the flowchart showing the experimental steps in Fig. 2. About 47
tonnes of liquid Ga metal, divided into two concentric zones, serve as the
target for the charged-current reaction 71Ga$(\nu,e)$ 71Ge. The decay of the
mainly monoenergetic neutrino source of the 51Cr isotope is via electron
capture with the emission of $\nu_{e}$ in four spectral lines. The $\nu_{e}$
energies and their branching ratios are: 747 keV (81.63%), 427 keV (8.95%),
752 keV (8.49%) and 432 keV (0.93%) [31]. The source was placed at the center
of the two zones of the Ga target, irradiating the two volumes simultaneously.
This allows the production rates of 71Ge to be measured at two different
distances. As the sizes of the two target zones are roughly at a meter scale,
BEST has a high sensitivity to the oscillatory behaviour in the $\nu_{e}$ flux
at the same scale, which corresponds to $\Delta m^{2}\approx 1$ eV2. Neutrino
oscillations at this short scale would be indicated by deficits in the 71Ge
production rates within the two zones, as well as the differences between
them. The use of an electron capture $\nu_{e}$ source is a powerful technique
to search for neutrino oscillations. The $\nu_{e}$ spectrum is simple, being
comprised of a dominant monoenergetic component and 3 sub-dominant lines. It
is a well-understood spectrum relying on well-known nuclear and atomic physics
parameters.

In this article, we present the results from the BEST $\nu_{s}$ oscillation
search experiment. Section II describes the geometry of the two-zone gallium
target which is irradiated by a 51Cr $\nu_{e}$ source. Section III reviews how
the 71Ge atoms are extracted from the target volumes and transferred to the
proportional counters. Section IV is devoted to a description of the counter
systems which measure the 71Ge decay rates. In Section V, techniques of
candidate event selection and analysis to set the exclusion within the
$(\Delta m^{2},\textrm{sin}^{2}(2\theta))$ oscillation parameter space are
introduced. We discuss the first result from the BEST experiment in Section
VI, and conclude in Section VII that our result is consistent with the
neutrino oscillation hypothesis with $\Delta m^{2}>1$ eV2 sterile neutrinos.

##  II Target Geometry

The Baksan Neutrino Observatory (BNO) is located in a dedicated underground
laboratory in the northern Caucasus mountains of Russia. Located 3.5 km below
the surface, the deep underground laboratory has an overburden of 4700 meters-
water-equivalent (mwe), resulting in muon flux in the laboratory of $(3.03\pm
0.10)\times 10^{-9}$/(cm2 s) [25]. The entirety of the laboratory is lined
with 60 cm of radiopure concrete and 6 mm steel shell to reduce $\gamma$ and
neutron backgrounds from rocks.

The concentric two-zone gallium target of the BEST experiment is located in
the main hall of the laboratory. The inner spherical volume with inner
diameter 133.5 cm contains (7.4691$\pm$0.0631) t of liquid Ga, and the outer
cylindrical volume with inner diameter 218 cm and maximum Ga target height
211.2 cm contains (39.9593$\pm$0.0024) t of Ga. The Ga metal, heated to 30.0
°C, remains molten inside the volumes. The carrier Ge and any produced 71Ge
was extracted at the end of each exposure for the calculation of production
rates. The extraction procedures are explained in Sec.V.7.1, and discussed
more thoroughly in Ref. [32].

The 51Cr source with known activity was contained in a cylindrical cartridge
with a radius of 4.3 cm and height of 10.8 cm. This source assembly was placed
in a specially constructed tungsten radiation shield with a thickness of
$\approx$30 mm and a weight of 42.8 kg, which provides radiation safety from
gamma activity from the source, and a steel shell with a special cap for
capturing the source by a manipulator. The assembly was placed at the center
of the two zones to irradiate both volumes simultaneously.

The source was delivered to BNO on July 5, 2019 and was placed into the two-
zone target at 14:02 local time (UTC+3) that same day, and this is our chosen
reference time for the source strength. The source was manufactured by
irradiating 4 kg of 50Cr-enriched metal for 100 d in a reactor at the State
Scientific Center Research Institute of Atomic Reactors, Dimitrovgrad, Russia.
The activity ($A$) of the source is measured by calorimetric method based on
the heat released by the source [33, 34]. At the end of each extraction, the
source is moved into a lead container for the activity measurement. The
$\gamma$ spectrum is measured at 21.65 m distance with a Ge detector for an
hour. After the spectral measurement, the source is moved to the calorimeter
to measure the heat emitted by the source for 20-21 hours. The tungsten shield
of the source does not affect calorimetry as the shield is included within the
calorimeter. Moreover, it helps by absorbing gamma radiation from impurities
in the source, increasing the accuracy of determining the heat release from
51Cr. Ten calorimetric measurements were performed after nine days of
irradiation of gallium targets. The decay of 51Cr ($\approx$90% to the ground
state and $\approx$10% to the excited state) releases on average ($36.750\pm
0.084$) keV per decay event. From spectrometric measurements of gamma
radiation of the source, the heat contribution from radioactive impurities is
found to be ($3.7\pm 0.5$) mW, which is $5\times 10^{-6}$ of the initial 51Cr
source power. The measured activity of the source at the reference time is
(3.414$\pm$0.008) MCi. The measured half-life of the 51Cr source is
(27.710$\pm$0.017) days, which is in a good agreement with the known 51Cr
half-life of (27.704$\pm$0.017) days [35]. A full description of the source
and the calorimetric measurements of its intensity can be found in Refs. [33,
34].

##  III 71Ge Extraction

Table 1: Extraction and exposure details for the inner target. The times of exposure are given in days of year 2019. A mass of (7.4691$\pm$0.0631) t of liquid Ga was irradiated by the neutrino source in the inner target. Extraction | Source exposure | Solar exposure | Extraction efficiency
---|---|---|---
Name | Date (2019) | Begin | End | Begin | End | Ga mass | from Ga | into GeH4
(day in year) | (day in year) | (day in year) | (day in year) | (tonnes)
Inner-1 | 15 Jul 16:01 | 186.585 | 196.376 | 183.667 | 196.667 | 7.594 | 0.9747(97) | 0.9460(123)
Inner-2 | 25 Jul 16:32 | 197.362 | 206.372 | 196.689 | 206.689 | 7.586 | 0.9814(98) | 0.9559(124)
Inner-3 | 04 Aug 16:37 | 207.282 | 216.374 | 206.792 | 216.692 | 7.578 | 0.9795(98) | 0.9673(126)
Inner-4 | 14 Aug 15:35 | 217.286 | 226.371 | 216.749 | 226.649 | 7.57 | 0.9801(98) | 0.9515(124)
Inner-5 | 24 Aug 17:17 | 227.258 | 236.458 | 226.620 | 236.720 | 7.562 | 0.9808(98) | 0.9554(124)
Inner-6 | 03 Sep 15:18 | 237.342 | 246.369 | 236.738 | 246.638 | 7.554 | 0.9818(98) | 0.9548(124)
Inner-7 | 13 Sep 15:11 | 247.243 | 256.368 | 246.733 | 246.733 | 7.546 | 0.9813(98) | 0.9381(122)
Inner-8 | 23 Sep 15:17 | 257.241 | 266.369 | 256.737 | 266.637 | 7.538 | 0.9835(98) | 0.9789(127)
Inner-9 | 03 Oct 15:00 | 267.24 | 276.369 | 266.725 | 276.625 | 7.529 | 0.9824(98) | 0.9545(123)
Inner-10 | 13 Oct 14:59 | 277.201 | 286.367 | 276.724 | 286.624 | 7.521 | 0.9806(98) | 0.9372(122)
Table 2: Extraction and exposure details for the outer target. The times of exposure are given in days of year 2019. A mass of (39.9593$\pm$0.0024) t of liquid Ga was irradiated by the neutrino source in the outer target. Extraction | Source exposure | Solar exposure | Extraction efficiency
---|---|---|---
Name | Date (2019) | Begin | End | Begin | End | Ga mass | from Ga | into GeH4
(day in year) | (day in year) | (day in year) | (day in year) | (tonnes)
Outer-1 | 15 Jul 13:59 | 186.585 | 196.376 | 183.783 | 196.583 | 44.237 | 0.9868(99) | 0.9503(124)
Outer-2 | 25 Jul 13:51 | 197.362 | 206.372 | 196.877 | 206.577 | 44.191 | 0.9841(98) | 0.9581(125)
Outer-3 | 04 Aug 12:47 | 207.282 | 216.374 | 206.833 | 216.533 | 44.145 | 0.9881(99) | 0.9668(126)
Outer-4 | 14 Aug 12:51 | 217.286 | 226.371 | 216.835 | 226.535 | 44.098 | 0.9858(99) | 0.9622(125)
Outer-5 | 24 Aug 14:35 | 227.258 | 236.458 | 226.808 | 236.608 | 44.052 | 0.9871(99) | 0.9609(125)
Outer-6 | 03 Sep 12:35 | 237.342 | 246.369 | 236.924 | 246.524 | 44.004 | 0.9893(99) | 0.9253(120)
Outer-7 | 13 Sep 12:29 | 247.243 | 256.368 | 246.82 | 256.520 | 43.954 | 0.9904(99) | 0.9514(124)
Outer-8 | 23 Sep 12:32 | 257.241 | 266.369 | 256.822 | 266.522 | 43.906 | 0.9897(99) | 0.9897(129)
Outer-9 | 03 Oct 12:27 | 267.24 | 276.369 | 266.819 | 276.519 | 43.857 | 0.9881(99) | 0.9664(126)
Outer-10 | 13 Oct 12:26 | 277.201 | 286.367 | 276.818 | 286.518 | 43.807 | 0.9877(99) | 0.9538(124)

Between July 5th and October 13th 2019, the two-zone target was exposed to the
source 10 times with an average exposure time of 9.18 d. The irradiation
schedule was devised to maximize the number of extracted 71Ge atoms. The
extraction was carried out at the end of each exposure. The first extraction
counting periods were shorter in time due to the limited number of available
counters. The shorter counting time has almost no effect on the number of
calculated 71Ge decays but increases the statistical uncertainty due to the
lower statistics of the measured counter background. After each exposure
period, the Ga targets were transferred to chemical reactor vessels, and the
produced 71Ge atoms were extracted from Ga with the technique almost identical
to the one used for the SAGE experiment [32, 25]. Here we provide an overview.

The Ge carrier and the 71Ge atoms produced by the neutrino capture reaction
are extracted from the Ga targets into an aqueous solution by adding a weak
acidic solution of H2O2 [36, 37], ensuring independent extractions of 71Ge
atoms from each zone of the Ga target. The acid oxidizes the Ge metal and GeH4
is synthesized in gaseous form. Gallium is pumped out of the vessels until the
aqueous acidic solution on the surface of gallium begins to be pumped out. The
GeH4 is then mixed with Xe gas and introduced into a low-background
proportional counter.

The extraction efficiency of Ge from a large mass of Ga is determined with
high accuracy using mass spectrometry [38]. For BEST, the process of the
efficiency estimation follows the same procedure as was used in SAGE. A known
amount of stable 72Ge (2.4 $\mu$mol, 92%) and 76Ge (2.4 $\mu$mol, 95%)
carriers were added to the volume in form of Ga-Ge alloy as the stable Ge
carrier, and the molten Ga was stirred thoroughly to disperse the Ge
throughout the volumes [32]. The total extraction efficiency is given by the
ratio of the amount of Ge in the synthesized GeH4 to the initial amount of Ge
in the Ga targets at the beginning of the exposure. Taking into account all
the factors that can affect the extraction efficiency including the
temperature, the amount of oxidizing agent (H2O2), and the volume of the
aqueous phase which defines the time of later concentration of Ge, the
extraction procedure reached (98$\pm$0.2)% efficiency from Ga, and the total
efficiency including synthesis reached ($95\pm 1.6$) %. The extraction and
exposure details for the inner and outer targets are summarized in Tables 1
and 2, respectively. Due to the inefficiency of the extraction, a small
proportion of the carrier Ge is still present within the Ga. The numbers of
these carryover atoms were estimated and accounted for in the production rate
calculation.

At each source exposure, gallium was pumped with the same levels into the two
volumes, which were monitored by level sensors in the zones. Therefore, a
larger mass of gallium was irradiated in solar neutrino exposures than in
exposures with a source. This contribution was also accounted for.

##  IV Counting of 71Ge

Table 3: Counting parameters for the inner target. $\Delta$ is the exponentially weighted live time. The live time and $\Delta$ include all time cuts. The counter efficiency for each extraction has the same fractional uncertainty of $-2.1/+2.3$%, as explained in details in Sec. V.7. Extraction | Counter filling |  Counter efficiency (after event selection) |  Day counting began (day in year) | Live time (days) | $\Delta$
---|---|---|---|---|---
Run |  Counter name |  Pressure (mmHg) |  GeH4 fraction (%) | K-peak | L-peak | K-peak | L-peak | K-peak | L-peak
Inner-1 | YCT92 | 630 | 8.8 | 0.3663 | 0.3803 | 197.66 | 54.478 | 34.364 | 0.8102 | 0.7450
Inner-2 | YCT2 | 640 | 9.5 | 0.3647 | 0.3785 | 207.623 | 53.706 | 29.834 | 0.7839 | 0.6542
Inner-3 | YCN43 | 650 | 9.3 | 0.3605 | 0.3599 | 217.693 | 50.525 | 50.525 | 0.7143 | 0.7143
Inner-4 | YCT97 | 640 | 9.2 | 0.3679 | 0.3769 | 227.644 | 52.808 | 29.884 | 0.7872 | 0.3672
Inner-5 | YCN46 | 650 | 9.5 | 0.3649 | 0.3654 | 237.790 | 150.436 | 150.436 | 0.7470 | 0.7470
Inner-6 | YCN42 | 640 | 9.8 | 0.3577 | 0.3604 | 247.597 | 140.143 | 133.113 | 0.7717 | 0.3892
Inner-7 | YCT92 | 640 | 9.3 | 0.3676 | 0.3793 | 257.617 | 129.483 | 130.843 | 0.7493 | 0.6776
Inner-8 | YCT2 | 645 | 9.5 | 0.3656 | 0.3779 | 267.634 | 129.060 | 131.764 | 0.7754 | 0.7855
Inner-9 | YCN43 | 640 | 9.1 | 0.359 | 0.3610 | 277.678 | 152.034 | 152.034 | 0.8019 | 0.8019
Inner-10 | YCT97 | 650 | 9.1 | 0.3698 | 0.3755 | 287.625 | 144.446 | 147.014 | 0.7629 | 0.7955
Table 4: Counting parameters for the outer target. $\Delta$ is the exponentially weighted live time. The live time and $\Delta$ include all time cuts. The counter efficiency for each extraction has the same fractional uncertainty of $-2.1/+2.3$%, as explained in details in Sec. V.7. Extraction | Counter filling |  Counter efficiency (after event selection) |  Day counting began (day in Year) | Live time (days) | $\Delta$
---|---|---|---|---|---
Run |  Counter Name |  Pressure (mmHg) |  GeH4 fraction (%) | K-peak | L-peak | K-peak | L-peak | K-peak | L-peak
Outer-1 | YCN113 | 635 | 9.5 | 0.3422 | 0.3596 | 197.66 | 53.788 | 33.662 | 0.7648 | 0.6996
Outer-2 | YCT3 | 635 | 9.5 | 0.3707 | 0.3792 | 207.623 | 54.376 | 30.640 | 0.8043 | 0.6755
Outer-3 | YCNA9 | 640 | 10.5 | 0.2933 | 0.3358 | 217.693 | 51.070 | 51.070 | 0.7650 | 0.7650
Outer-4 | YCT9 | 635 | 9.6 | 0.3658 | 0.381 | 227.644 | 52.981 | 30.423 | 0.7820 | 0.3755
Outer-5 | YCN41 | 635 | 10.0 | 0.3568 | 0.3727 | 237.790 | 147.774 | 147.774 | 0.8025 | 0.8025
Outer-6 | YCT4 | 630 | 9.0 | 0.3585 | 0.3577 | 247.597 | 139.382 | 131.148 | 0.8012 | 0.3843
Outer-7 | YCN113 | 630 | 10.3 | 0.3407 | 0.3607 | 257.617 | 134.985 | 136.161 | 0.7977 | 0.7108
Outer-8 | YCT3 | 640 | 9.5 | 0.3716 | 0.3785 | 267.634 | 129.098 | 131.802 | 0.8298 | 0.8398
Outer-9 | YCNA9 | 635 | 9.9 | 0.293 | 0.3360 | 277.678 | 155.439 | 155.439 | 0.7865 | 0.7865
Outer-10 | YCT9 | 645 | 9.5 | 0.3677 | 0.3797 | 287.625 | 143.604 | 146.307 | 0.7567 | 0.7905

The decay process 71Ge + $e^{-}$ $\rightarrow^{71}$Ga +$\nu$ has a half-life
of 11.4 d [39]. Two peaks at 10.4 keV (K peak) and 1.2 keV (L peak) are
observable in the proportional counters. The K-capture, which constitutes
$\approx$88% of all decays, can release 10.4 keV Auger electrons (41.5%), 1.2
keV Auger electrons with 9.2 keV X rays (41.2%), or 0.1 keV Auger electrons
with 10.3 keV X rays (5.3%) [32]. On the other hand, L and M captures give
almost entirely Auger electrons of energies 1.2 keV and 0.12 keV respectively
[32]. Since the proportional counters have much higher efficiency for Auger
electrons than for the X rays, the number of K and L peak counts are almost
equal. The pulse shapes recorded from the counters are analyzed to suppress
the contribution from unwanted backgrounds.

The design of the proportional counters used is described in detail in Refs.
[40, 32, 41]. The proportional counters, filled with GeH4 from each
extraction, are placed in the well of a NaI veto detector within a large
passive shield. To suppress the 222Rn background, the shield volume is purged
with boil-off gas from the liquid nitrogen. The average counting time for the
first four runs is approximately 50 d and was extended to about 140 d in the
later six runs to account for the decreased source strength. Calibration data
with an 55Fe source were taken once every 2 weeks throughout the measurements
to ensure the stability of counting. Mean variations of the positions of the
calibration peak during full counting period were 1.4% and 1.3% for the inner
and the outer volumes and taken into account in analysis. Two periods during
the counting were excluded from the analysis due to identified issues:

  1. 1.

The time response of all slots in high-gain channel (L-peak) was slow between
2019-08-23 07:03 and 2019-09-14 14:49.

  2. 2.

There was a failure in the low gain channel (K-peak) between 2019-11-06 21:10
and 2019-11-09 20:45.

The livetime of some runs were affected accordingly.

After the counting of the extraction samples from the Cr experiment was
completed in the spring-fall of 2020, the counting efficiency was directly
measured for each counter. The volume efficiencies, accounting for the dead
volume near the cathode, were directly measured with 37Ar. The event selection
efficiencies of the waveform analysis were measured with 71Ge calibrations.
The rise-time ($T_{N}$) values from the 71Ge calibration data were arranged in
ascending order and an upper limit set such that 4% of the calibration events
were excluded [25]. Details of the event selection and the 71Ge calibration
runs to verify the selection scheme are summarized in Sec. V.2 and Appx. A
respectively.

The calculated counting efficiency using the measured pressure in the counter,
GeH4 fraction, and 37Ar volume efficiency was determined for each extraction.
They are summarized in Tables 3 and 4 with other counting parameters. Values
of the exponentially weighted livetime $\Delta$ are also presented for each
extraction

$\Delta=\sum_{k=1}^{n}(e^{-\lambda t_{bk}}-e^{-\lambda t_{ek}})~{},$ (1)

where the sum is over $n$ counting intervals, each of which has a starting
time $t_{bk}$ and an ending time $t_{ek}$. $\lambda$ is the decay constant of
71Ge.

##  V Analysis

A primary analysis on the BEST data was performed by the analysis group in the
Institute for Nuclear Research, Russia, and was verified by an independent
analysis carried out by a separate analysis group in Los Alamos National
Laboratory. Both of the analyses obtained similar results to within about 2%.
This difference is due to minor event-selection differences at the edges of
the selection borders in energy and rise time. This difference is accounted
for by the estimated systematic uncertainties in the efficiencies for those
cuts.

Pulses from the proportional counters are digitized at two different gains.
The higher gain channel is chosen for the 71Ge L-peak, and the lower gain
channel is used for the K-peak analysis. The pulse shape analysis is performed
to differentiate the 71Ge signals from backgrounds.

The analytic form of pulse shapes used in the BEST analysis is derived from
the model studied in Ref. [42] where

$\begin{split}V&(0<t<T_{N})=V_{0}\Big{[}\frac{t+t_{0}}{T_{N}}\mathrm{ln}\Big{(}1+\frac{t}{t_{0}}\Big{)}-\frac{t}{T_{N}}\Big{]}~{},\\\
V&(t>T_{N})=V_{0}\Big{[}\mathrm{ln}\Big{(}1+\frac{t-T_{N}}{t_{0}}\Big{)}-1-\frac{t+t_{0}}{T_{N}}\mathrm{ln}\Big{(}1-\frac{T_{N}}{t+t_{0}}\Big{)}\Big{]}~{},\end{split}$
(2)

with $V(t<0)=0$. Here, $T_{N}$ is the rise-time, which is the duration over
which the ionization arrives at the anode, $t_{0}$ is the time inversely
proportional to the ion mobility, and $V_{0}$ is the voltage proportional to
the total amount of ionization. We fit every pulse that is not identified as
saturation or breakdown to Eq. 2. The fit is made from 40 ns before the time
of pulse onset to 400 ns after onset. Five parameters are determined by the
fit: $t_{\textrm{onset}},V_{\textrm{onset}},V_{0},t_{0}$, and $T_{N}$. The
integral of the fit function over pulse waveform for 800 ns after the onset is
the energy parameter [24]. The $T_{N}$ value is used as a background rejection
parameter.

###  V.1 Calibration

Following the procedure described in Ref. [24], we first find the position and
the width of the 5.895 keV calibration peak in each calibration run. The
expected locations for the 71Ge K and L peaks are then calculated from the
energy ratios with corrections for the non-linearity factors as described in
Ref. [32]. For the counters with the GeH4 fraction $G$, counter pressure $P$,
and operating voltage $V$, the non-linearity factors for the location $P_{K}$
and the resolution $R_{K}$ of the K peak follow the empirical formulas
reported in Ref. [32] as

$\footnotesize\begin{split}\frac{P_{K}(^{71}\mathrm{Ge})}{P(^{55}\mathrm{Fe})}&=\frac{10.367}{5.895}[1-(4.5G+2.78)(V-V_{\mathrm{crit1}})\times
10^{-6}]~{},\\\
\frac{R_{K}(^{71}\mathrm{Ge})}{R(^{55}\mathrm{Fe})}&=\sqrt{\frac{5.895}{10.367}}[1+1.5\times
10^{-3}(V-V_{\mathrm{crit2}})]~{},\end{split}$ (3)

where the corrections are applied if $V$ is above the respective critical
voltages $V_{\mathrm{crit1}}=10.5G+0.6P+588$ and
$V_{\mathrm{crit2}}=6G+P/3+824$. The typical corrections for the K-peak
position and the resolution are 2% and 15%, respectively, and no corrections
are required for the L peak as the critical voltages are much higher than for
the K peak.

Table 5: The cut values on $T_{N}$ of the pulses for 71Ge K and L peak for each counting in the inner target. Events with $T_{N}$ values greater than each cut value are selected out. Counter Filling | $T_{N}$
---|---
Extraction name |  Counter name |  Pressure (mmHg) |  GeH4 fraction (%) |  K-peak (ns) |  L-peak (ns)
Inner-1 | YCT92 | 630 | 8.8 | 17.6 | 13.0
Inner-2 | YCT2 | 640 | 9.5 | 16.6 | 10.1
Inner-3 | YCN43 | 650 | 9.3 | 13.2 | 10.0
Inner-4 | YCT97 | 640 | 9.2 | 17.3 | 11.4
Inner-5 | YCN46 | 650 | 9.5 | 15.2 | 11.3
Inner-6 | YCN42 | 640 | 9.8 | 13.2 | 9.1
Inner-7 | YCT92 | 640 | 9.3 | 17.6 | 13.0
Inner-8 | YCT2 | 645 | 9.5 | 16.6 | 10.1
Inner-9 | YCN43 | 640 | 9.1 | 13.2 | 10.0
Inner-10 | YCT97 | 650 | 9.1 | 17.3 | 11.4
Table 6: The cut values on $T_{N}$ of the pulses for 71Ge K and L peak for each counting in the outer target. Events with $T_{N}$ values greater than each cut value are selected out. Counter Filling | $T_{N}$
---|---
Extraction name |  Counter name |  Pressure (mmHg) |  GeH4 fraction (%) |  K-peak (ns) |  L-peak (ns)
Outer-1 | YCN113 | 635 | 9.5 | 13.6 | 9.1
Outer-2 | YCT3 | 635 | 9.5 | 16.4 | 10.3
Outer-3 | YCNA9 | 640 | 10.5 | 18.8 | 13.2
Outer-4 | YCT9 | 635 | 9.6 | 14.9 | 9.1
Outer-5 | YCN41 | 635 | 10.0 | 13.4 | 10.3
Outer-6 | YCT4 | 630 | 9.0 | 13.2 | 10.2
Outer-7 | YCN113 | 630 | 10.3 | 13.6 | 9.1
Outer-8 | YCT3 | 640 | 9.5 | 16.4 | 10.3
Outer-9 | YCNA9 | 635 | 9.9 | 18.8 | 13.2
Outer-10 | YCT9 | 645 | 9.5 | 14.9 | 9.1

We determine the proper energy scaling for each extraction from 55Fe
calibrations. The extrapolations were verified with 71Ge filled counters
operated after the BEST extraction counting ended. This additional study
enabled the comparison between the 71Ge K and L peak 2 FWHM regions predicted
by extrapolation and the true peak positions. All counters used in the
experiment underwent the performance study. Details of the counter calibration
using 71Ge isotope are described in Appx. A.

###  V.2 Event selection

Events are further selected to reduce the background that mimics 71Ge-induced
events. The event selection algorithm uses the SAGE analysis [24], which has
been continuously updated throughout the last 20 years. Some of the updates
were reported in [41].

We implement five event selection cuts to remove non-physical signals: 1)
flatness cut, 2) shield-open cut, 3) NaI veto cut, 4) setting the energy
windows for the Ge L and K peaks and 5) rise-time cut. These cuts effectively
remove the background events. The first three types of cuts affects only the
background level of the counters, and do not affect the selection of real
events from 71Ge decays. Therefore the efficiency of such cuts can be
considered equal to 100%. The use of such cuts off reduces the background of
the counters and, accordingly, the magnitude of the statistical error.

####  V.2.1 Flatness cut

The first step of event selection is to identify two types of background
events: saturation waveforms that are mostly from high energy
$\alpha$-particles and those that originate from high-voltage breakdown. The
$\alpha$-particles, either from the decay of internal 222Rn or the natural
radioactivity in the counter material, can easily be identified by examining
the end of the waveforms. Saturated pulses have amplitudes greater than 16 keV
and relatively flat shape at the end of the pulse. The events from high-
voltage breakdown have a characteristic waveform that rises steeply and stays
flat at the end. While the positive ions from the real 71Ge events are
collected smoothly over time with resulting extended rise time, events from
high-voltage breakdown have a characteristic waveform that rises steeply and
stays flat at the end after the rising edge. Because both types of background
events have a characteristic shape at the end of the waveform, we analyze the
pulses between 500 and 1000 ns after the digitization begins. Typical
waveforms of an $\alpha$-induced saturated pulse and a background that
originates from high-voltage breakdown are shown in Fig. 3. They are easily
distinguishable from the typical pulse shape of true 71Ge candidate event
shown in Fig. 4.

Figure 3: Flat-top pulses measured in proportional counters. Top: Saturated
event induced by high-energy $\alpha$-particles. Bottom: Background candidate
event that originates from high-voltage breakdown. Figure 4: Typical pulse
shapes measured in proportional counters. Top: 71Ge K peak candidate event
with $T_{N}=$ 7.44 ns. Bottom: Background candidate event induced by either
Compton scattering or high-energy $\beta$-particles in K peak energy region.
The slow background event with extended ionization has much slower fall time
($T_{N}=$ 30.67 ns) when the pulse begins at $\approx$200 ns than the true
71Ge candidate event.

####  V.2.2 Shield-open cut

Next, we apply a cut to remove data during periods of expected high
backgrounds. When counters were calibrated, they were exposed to the
laboratory atmosphere with an average Rn content of 2 pCi/liter. Events
produced by 222Rn daughters can produce false 71Ge signals that mimic our
signal, and hence the effect of external Rn is minimized by making a time cut
on the data for 2.6 h after any shield opening. The estimated effectiveness of
this time cut is nearly 100%. Any rejected period are considered in our
exposure calculation, and there is no efficiency associated with it. The
uncertainty is found to be negligible [43].

####  V.2.3 Time cut of Radon chains

We apply an additional time cut to minimize the number of false 71Ge signals
from internal radon and its daughters [32]. The first two decays in the 222Rn
chain are $\alpha$ decays producing off-scale pulses that saturate the
detector. Since 222Rn decays to 210Pb ($T_{1/2}-22$-y) typically in about an
hour, removing a few hours around each saturated event effectively rejects the
false 71Ge events from the radon chain. We remove all data from 15 minutes
before each saturation event to 3 hours after it [32, 41]. The probability
that the decays of the elements of this chain will be registered after such
cut off is 1.1% for 214Pb and 3.5% for 214Bi and 214Po [44] We treat them as
the systematic uncertainties related to internal radon.

####  V.2.4 NaI veto

Background events from $\beta$ decays from the counter walls, which are
accompanied by $\gamma$ rays are tagged and vetoed by the surrounding NaI
detector. Since 71Ge has no $\gamma$ rays associated with its decay other than
internal bremsstrahlung, all events coincident with a NaI detector response
are eliminated.

####  V.2.5 Setting the energy windows for the Ge L and K peaks

The measure of energy is the integral of the pulse waveform for 800 ns after
the pulse onset. The locations of the energy acceptance windows for 71Ge
events in the L and K peaks were set from the 55Fe calibration as described in
Sec. V.1. If the peak position changes from one calibration to the next, then
the energy window for event selection is slid linearly in time between the two
calibrations. The resolution at each peak is held constant and is set to be
the average of the resolutions of 55Fe for all counter calibrations, scaled to
the L- or K-peak energy as described in Sec. V.1. Events are then accepted as
candidates only if their energy is within $\pm$ 1 FWHM ((98% acceptance) of
the central peak energy.

####  V.2.6 Rise-time cut

The electron capture process produces point-like ionization in the counter.
Pulses from the 71Ge electron capture decay should therefore be fast, and
$T_{N}$ values should be near zero. On the other hand, backgrounds from
Compton scattering or from high-energy $\beta$-particles traversing the
counter produce extended ionizations, leading to slow pulses with large
$T_{N}$. Figure 4 illustrates the difference in pulse shapes of the point-like
71Ge candidate event with $T_{N}=7.44$ ns and the background showing the
characteristic of extended ionization with $T_{N}=30.67$ ns. Both events fall
into the K peak energy region with 9.64 keV, but the background candidate
event has much slower fall time when the pulse begins at $\approx$200 ns than
the true 71Ge candidate.

Figure 5:  Upper panel: energy vs rise-time histogram of all events of the
inner target after the shield-open cut observed in all ten exposures during
the first 30 days after extraction. The live time is 245 days, and 1364 events
are shown. Lower panel: the same histogram for the 481 events that occurred
during an equal live-time interval beginning at 40 days after extraction.
Figure 6:  Upper panel: energy vs rise-time histogram of all events of the
outer target after the shield-open cut observed in all ten exposures during
the first 30 days after extraction. The live time is 249 days, and 1387 events
are shown. Lower panel: the same histogram for the 504 events that occurred
during an equal live-time interval beginning at 40 days after extraction.

The cut value for the $T_{N}$ for each counter is obtained from separate
measurements by filling counters with a typical gas mixture (see Tables 5 and
6) and adding a trace of active 71GeH4. The counters were filled with a
mixture (%GeH4 $\sim$10% and pressure $\approx$650 mmHg) very similar to the
mixture used in the BEST extractions. Each proportional counter was installed
into the same channel of the counting system with approximately the same high
voltage as used in the BEST measurements. The cut values for the K and L peaks
are set such that 4% of the slowest events in 71Ge calibration data are
excluded. The derived typical cut values for K and L peaks were 13.2 ns and
9.1 ns respectively. However, the cut values vary by detector, and individual
$T_{N}$ values were used for each counter. The cut values on $T_{N}$ windows
for K and L peak for each counter are given in Tables 5 and 6.

Table 7: A summary of event selection cuts and their individual acceptances. Cut | Efficiency (%)
---|---
Energy window | 98$\pm 1.3$
Rise-time | 96

Table 7 summarizes the event selection cuts applied in the BEST analysis and
their individual acceptances. Figures 5 and 6 show the histograms of all
events from the inner and the outer zones, after the first four cuts, for the
earlier and the later counting times. The number of events outside the peaks
is about the same in both panels as these are mainly due to background. The
expected location of the 71Ge K and L peaks as predicted by the 55Fe
calibrations are shown as darkened.

Figure 7: Plots with energy spectra of selected events for whole dataset
showing K and L peaks. Top: Energy spectrum of the inner target. Bottom:
Energy spectrum of the outer target.

Figure 7 shows the energy spectra of the selected K and L peak candidate
events for the inner and the outer targets. We note that the event selection
windows were determined from post-experiment calibrations, and hence the cut
boundary parameters were effectively blind during the analysis. No additional
blindness scheme was implemented.

###  V.3 Neutrino-Nucleon Cross Section

The neutrino-nucleon cross section ($\sigma$) has to be calculated from
nuclear physics input. The bulk of the cross section is in the ground-state to
ground-state transition determined from the matrix element for the decay of
71Ge. When the original gallium anomaly was observed, there was concern that
the transition strengths to excited states were not fully understood. The
cross section was first estimated in the seminal work of Bahcall [45]. He
derived the ground state contribution from the 71Ge half-life, but the excited
state contributions were estimated from charge exchange (i.e. (p,n))
reactions. Bahcall considered a number of possible uncertainties, the largest
coming from the excited state contributions due to the quality of the (p,n)
data available at that time. For the central value, Bahcall used the best
estimate of the transition strength values to the first two excited states and
a Q-value of (232.69$\pm$0.15) keV. Other excited states are too high in
energy to contribute for 51Cr $\nu$’s energy. Bahcall estimated the
uncertainty to be the change in $\sigma$ (-1.6/+2.8%) if one ignores the
excited states. Other uncertainties came from forbidden corrections to the
beta decay matrix elements (2.3%) with small uncertainties associated with the
$Q_{val}$ (0.05%), the reaction threshold (0.2%), and the 71Ge half life
(0.3%). He considered part of the excited state uncertainty as one-sided,
hence the asymmetric uncertainties. The charge exchange data has been improved
[46] by recent work [47, 48, 49] indicating that they are not the cause of the
discrepancy. However, the excited-state contribution uncertainty is critical
because the (p,n) measurements have a significant cancellation between the
Gamow-Teller and tensor matrix elements resulting in an underestimate of the
transition strengths [50]. The (p,n) cross section could be entirely due to
the tensor interaction, in which case (either) excited state contribution to
neutrino absorption might be zero. Unfortunately, there is no direct
measurement of the cross section at this energy that is not subject to the
caveats of neutrino oscillation physics.

Table 8: A summary of the published neutrino-nucleon cross section estimates for 71Ga in units of $10^{-45}$cm2. Reference | Cross Section | Q-value (keV)
---|---|---
[45] | $5.81^{+0.21}_{-0.16}$ | 232.69(15)
[49] | 5.93$\pm$0.14 | 232.69(15)
[51] | $5.910\pm 0.114$ | 233.5(1.2)
[52] | 5.67$\pm$0.06 | 232.49(22)
[53] | 5.938$\pm$0.116 | 232.443(93)

Kostensalo et al. [52] used a more recent $Q_{val}=(232.49\pm 0.22)$ keV [54]
and a nuclear shell model calculation with wave functions obtained using
recent two-nucleon interactions to calculate the transition strengths. The
shell model calculations thus avoid the drawback of using the (p,n)
measurements but need to be experimentally confirmed by indirect means, such
as state energies and electromagnetic properties. The paper of Semenov et al.
[53] reproduces Bahcall’s approach but uses modern values for the transition
strengths and a Q-value of (232.443$\pm$0.093) keV [55]. This small difference
in Q-value between the estimates generates a small uncertainty compared to the
other uncertainties in this work and we ignore it.

The Semenov et al. and Kostensalo et al. results differ by about 4%, which is
about 2-3 times larger than the uncertainty estimated for each. Interestingly,
the original Bahcall number is half way between these two results with a
$\pm$4% uncertainty. We therefore use the Bahcall $\sigma$ value and the
associated conservative uncertainties from his estimate of
$[(5.81^{+0.21}_{-0.16})\times 10^{-45}$] cm2. The recent cross section
estimates are summarized in Table 8.

###  V.4 Energy-weighted Likelihood Fits

The time sequence of the candidate 71Ge events are analyzed with a maximum
likelihood method to separate the 71Ge decay with 11.4 day half-life from
remaining background, which is constant over time. The likelihood fit takes
into account the decay of 51Cr during the exposure period.

In general, the likelihood function ($\mathcal{L}$) used in the BEST analysis
can be written for a single run as

$\mathcal{L}=e^{-\int_{\mathrm{counting~{}time}}P(t,E)dt}\prod_{i}^{l}P(t_{i},E_{i})dt_{i}~{},$
(4)

where $l$ is the total number of observed events, $t_{i}$ is the time of event
$i$, and $P(t,E)$ is the total probability that an event with energy $E$ will
occur at time $t$. If the events that pass all selection criteria are
classified as either the signal or the background, the function $P(t,E)$ can
be generalized to the form

$P(t,E)=w_{p}(E)p\epsilon e^{-\lambda t}+w_{b}(E)b~{},$ (5)

where $p$ and $b$ are the production rate and the background rate,
respectively; $\epsilon$ is the overall efficiency; and the terms $w_{p}(E)$
and $w_{b}(E)$, called the weight factors, are the energy-dependent
probabilities that the event is due to signal or to background, respectively.
The functional form of $w_{p}(E)$ must be obtained from an experiment in which
only signal is present, and the form of $w_{b}(E)$ is measured when only
background is present. It is assumed that both weight factors are independent
of time and are normalized such that

$\int_{E_{lo}}^{E_{hi}}w_{p}(E)dE=\int_{E_{lo}}^{E_{hi}}w_{b}(E)dE=1~{},$ (6)

where the integrals extend over the range $E_{lo}$ to $E_{hi}$ used in event
selection ($\pm$1 FWHM for this analysis).

Combining Eq. 4, 5 and 6, the likelihood function is generalized to

$\footnotesize\mathcal{L}=e^{-p\epsilon\Delta/\lambda-b\tau}\prod_{i}^{l}\big{[}w_{p}(E_{i})p\epsilon
e^{-\lambda t_{i}}+w_{b}(E_{i})b\big{]}dt_{i}dE_{i}~{},$ (7)

where $\Delta$ is the probability an event will occur while counting is in
progress and $\tau$ is the total counting time. For maximization purposes, the
likelihood function can be arbitrary up to a constant. Hence, Eq. 7 can be
divided by $\prod_{i}^{l}w_{b}(E_{i})dt_{i}dE_{i}$ and written as

$\mathcal{L}=e^{-p\epsilon\Delta/\lambda-b\tau}\prod_{i}^{l}\big{[}\frac{w_{p}(E_{i})}{w_{b}(E_{i})}p\epsilon
e^{-\lambda t_{i}}+b\big{]}~{},$ (8)

which is the standard function that is used by all radiochemical solar
neutrino experiments to analyze their counting data where the usual term
$e^{-\lambda t_{i}}$ has been replaced by

$\frac{w_{p}(E_{i})}{w_{b}(E_{i})}e^{-\lambda t_{i}}~{}.$ (9)

The signal and the background weight factors were determined by examining the
energy spectra for each counter. The use of the energy weight factors shifts
the results by values that are quite compatible with the statistical
uncertainty.

###  V.5 Predicted 71Ge Production Rates

Table 9: Values and uncertainties of the terms that enter the calculation of the predicted production rate. All uncertainties are symmetric except for the cross section. |  | Uncertainty
---|---|---
| Value | Magnitude | %
Ga density $\rho$ (g Ga/cm3) | 6.095 | 0.002 | 0.033
Avogadro’s number $N_{0}$ ($10^{23}$ atoms Ga/mol) | 6.0221 | 0.0 | 0.0
Ga molecular weight $M$ (g Ga/mol) | 69.72307 | 0.00013 | 0.0002
71Ga Atomic abundance $f_{1}$ (atoms 71Ga/100 atoms Ga) [56] | 39.8921 | 0.0062 | 0.016
Atomic density $D=\rho N_{0}f_{1}/M$ ($10^{22}$ atoms 71Ga/cm3) | 2.1001 | 0.0008 | 0.037
Source activity at reference time $A$, MCi | 3.414 | 0.008 | 0.23
Cross section $\sigma$ ($10^{-45}$ cm2/ (71Ga atom 51Cr decay)), Bahcall | 5.81 | +0.21,-0.16 | +3.6,-2.8
Path length in Ga $<L_{in}>$ (cm) | 52.03 | 0.18 | 0.3
Path length in Ga $<L_{out}>$ (cm) | 54.41 | 0.18 | 0.3
Predicted production rate (71Ge atoms/d), r${}_{\textrm{In}}$ | 69.4 | +2.5,-2.0 | +3.6,-2.8
Predicted production rate (71Ge atoms/d), r${}_{\textrm{Out}}$ | 72.6 | +2.6,-2.1 | +3.6,-2.8
Figure 8: The geometrical dimensions of the Ga targets. All dimensions are in
centimeters. There is an empty segment in the spherical vessel due to the
thickened wall and the bottom of the re-entrant tube.

The expected 71Ge production rates depend on various physical constants
presented in Table 9 and the geometries of the targets, illustrated in Fig. 8.
To account for the non-trivial shapes of the Ga targets, a Monte Carlo
integration technique was adapted for the neutrino capture rate calculations
in the target zones. Calculations with the nominal geometry described in Fig.
8 yield 1.22545 m3 and 6.5561 m3, which correspond to 7.4691 t and 39.9593 t
of liquid Ga. The measured masses of the targets are 7.4060 t and 39.9617 t
for the inner and outer zones, respectively, with negligible uncertainty for
both masses.

We attribute the discrepancy of the measured and the calculated mass values to
the uncertainty in the dimensional measurements, especially in the shell
thickness of the spherical volume. In a study to explore how sensitive the
Monte Carlo results are to various alterations in geometry, modifying the
spherical shell thickness from the nominal value of 0.50 cm to 0.68 cm was
found to neutralize the difference in the mass values. We include the error of
inaccurate knowledge of the sphere shell thickness as a systematic uncertainty
of the path length, which for the sphere amounts to 0.4% and for the cylinder
is negligible. Accounting also for the uncertainties in gallium weighing, we
use (7.4691$\pm$0.0631) t and (39.9593$\pm$0.0024) t as the nominal masses of
the inner and the outer targets.

The effective path length in a volume, or the average path length the
$\nu_{e}$ takes through the Ga target, is calculated by the integral

$<L>=\int_{V}\frac{1}{4\pi d^{2}}~{}d\vec{x}~{},$ (10)

where $d$ is the distance between emission in the source and absorption in the
Ga targets

We carry out the Monte Carlo integration to yield the path length values for
the inner and the outer volumes. Taking into account both the statistical
uncertainty due to the $10^{7}$ sample Monte Carlo integration and the
systematic uncertainty from the dimensional uncertainties of the apparatus,
the average path lengths for the two volumes are

$\begin{split}<L>_{in}&=(52.03\pm 0.18)~{}\mathrm{cm~{},and}\\\
<L>_{out}&=(54.41\pm 0.18)~{}\mathrm{cm}~{}.\end{split}$ (11)

The neutrino capture rate $r_{j}$ in the volume $V_{j}$ can be written as

$\begin{split}r_{j}&=\sum_{k}^{4}\int_{V_{j}}FP_{ee}(\vec{x},E_{\nu_{k}})\sigma_{k}nd\vec{x}\\\
&\approx
V_{0}\frac{1}{N}\sum_{k}^{4}\sum_{i=1}^{N}\frac{p_{k}P_{ee}(d,E_{\nu_{k}})}{d^{2}}\Theta_{j}(\vec{x}_{i})~{},\end{split}$
(12)

where $F$ is the flux of $\nu_{e}$, $P_{ee}(\vec{x},E_{\nu_{k}})$ is the
oscillation survival probability of $\nu_{e}$ with energy $E_{\nu_{k}}$,
$p_{k}$ is the branching ratio for the $\nu_{e}$ with energy $E_{\nu_{k}}$,
$\sigma_{k}$ is the cross section, $n$ is the 71Ga number density, and
$\Theta(\vec{x}_{i})=1$ if $\vec{x}_{i}$ is within $V$ and 0 otherwise. The
summation over $k$ is for the four neutrino energies and their branching
ratios 747 keV (81.63%), 427 keV (8.95%), 752 keV (8.49%) and 432 keV (0.93%)
[31]. The summation over $i$ is for the Monte Carlo integration over $N$
samples. For the no oscillation case, the survival probability at a short
distance is simply $P_{ee}(d)=1$. In the case where only one sterile mass
eigenstate $m_{4}$ is introduced to the standard three active mass states and
$m_{4}\gg m_{1},m_{2},m_{3}$, the survival probability of any active neutrino
at a short distance $d$ is governed by the two-neutrino oscillation model

$P_{ee}(E_{\nu},d)=1-\mbox{sin}^{2}2\theta\mbox{sin}^{2}\left(1.27\frac{\Delta
m^{2}[\mbox{eV}^{2}]d[\mbox{m}]}{E_{\nu}[\mbox{MeV}]}\right)~{},$ (13)

where $\Delta m^{2}$ is the mass squared difference between the active and the
inactive state, $\theta$ is the mixing angle, and $E_{\nu}$ is the neutrino
energy.

Using the source activity of (3.414$\pm$0.008) MCi, the neutrino-nucleus cross
section, and combining the uncertainty in quadrature, Eq. 12 gives the
predicted production rates of

$\begin{split}r&{}_{\textrm{In-
predic}}=(69.4^{+2.5}_{-2.0})~{}\textrm{atom/d}~{},\\\ r&{}_{\textrm{Out-
predic}}=(72.6^{+2.6}_{-2.1})~{}\textrm{atom/d}~{}.\end{split}$ (14)

The uncertainties are dominated by the cross-section uncertainty, and thus are
strongly correlated in these predictions for the two volumes.

###  V.6 Measured 71Ge Production Rates

The results of the likelihood fits for individual runs and the combined fits
are presented in Tables 10 and 11. For each run and also for the combined
runs, the likelihood fits to the time distribution of the candidate events
were performed as described in Refs. [24, 25].

The background from solar neutrino capture conforms to (66.1$\pm$3.1) solar
neutrino unit (SNU 1111 SNU = 1 interaction per 1036 target atoms per second)
[41] as explained below, and was taken into account when determining the
production rate from the source. In general, the contribution from the solar
neutrino capture is 0.6 and 3.2 counts for the inner and the outer targets.
These are 0.8% and 4.6% of the total numbers fit to 71Ge, respectively. Due to
the inefficiency of extractions, there are also residual 71Ge atoms that
carryover from one extraction to the next. The contribution from the carryover
atom was typically less than 1 count for each volume and was also taken into
account.

Table 10: A summary of the likelihood fits for the production rate for the inner target from each extraction and the combined fit of all extractions, in K and L peak independently and combined. The production rates for each exposure for the K and L-peaks are referred to its starting time, taking into account the total detection efficiency. The production rate for their combined result is referred to the reference time. All production rates for the K+L-peak are referred to the reference time. The stated uncertainties are statistical and given in 68% confidence. The Cramér-von Mises statistics $Nw^{2}$ measures the goodness of fit of the sequence of event times. The probability was inferred from $Nw^{2}$ by simulation. K-peak |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ |  Probability (%)
Inner-1 | 88 | 88.0 | 87.6 | 0.4 | 0.0 | 48.2${}^{+6.3}_{-4.2}$ | 0.110 | 49
Inner-2 | 81 | 70.2 | 68.1 | 0.4 | 1.7 | 40.5${}^{+5.6}_{-6.8}$ | 0.081 | 58
Inner-3 | 69 | 61.2 | 60.1 | 0.3 | 0.8 | 39.0${}^{+5.9}_{-6.1}$ | 0.124 | 35
Inner-4 | 65 | 65.0 | 63.9 | 0.3 | 0.8 | 37.5${}^{+5.6}_{-4.2}$ | 0.057 | 82
Inner-5 | 57 | 32.5 | 31.6 | 0.3 | 0.6 | 19.5${}^{+3.6}_{-4.6}$ | 0.107 | 23
Inner-6 | 44 | 36.4 | 35.7 | 0.2 | 0.5 | 22.0${}^{+3.7}_{-4.4}$ | 0.026 | 96
Inner-7 | 43 | 23.0 | 22.4 | 0.3 | 0.3 | 14.0${}^{+3.1}_{-3.9}$ | 0.048 | 70
Inner-8 | 28 | 15.9 | 15.3 | 0.3 | 0.3 | 8.9${}^{+2.6}_{-3.3}$ | 0.160 | 11
Inner-9 | 40 | 6.3 | 5.8 | 0.3 | 0.2 | 3.4${}^{+2.0}_{-2.9}$ | 0.071 | 54
Inner-10 | 28 | 13.3 | 12.9 | 0.3 | 0.1 | 7.8${}^{+2.3}_{-3.0}$ | 0.031 | 91
Comb. K | 543 | 411.6 | 403.2 | 3.0 | 5.4 | 58.5${}^{+3.4}_{-3.5}$ | 0.095 | 44
L-peak |  |  |  |  |  |  |  |
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ |  Probability (%)
Inner-1 | 92 | 89.5 | 89.2 | 0.4 | 0.0 | 51.4${}^{+8.3}_{-4.1}$ | 0.353 | 6
Inner-2 | 48 | 42.5 | 40.8 | 0.4 | 1.4 | 28.0${}^{+5.2}_{-5.7}$ | 0.023 | 99
Inner-3 | 63 | 56.2 | 55.1 | 0.3 | 0.8 | 35.8${}^{+6.1}_{-5.8}$ | 0.079 | 58
Inner-4 | 28 | 19.5 | 19.0 | 0.2 | 0.3 | 23.3${}^{+6.2}_{-7.4}$ | 0.101 | 36
Inner-5 | 77 | 27.5 | 26.6 | 0.3 | 0.6 | 16.4${}^{+3.4}_{-4.6}$ | 0.046 | 74
Inner-6 | 37 | 11.6 | 11.2 | 0.2 | 0.2 | 13.7${}^{+4.6}_{-6.0}$ | 0.115 | 20
Inner-7 | 48 | 22.1 | 21.5 | 0.3 | 0.3 | 14.4${}^{+3.5}_{-4.5}$ | 0.142 | 9
Inner-8 | 31 | 17.6 | 17.0 | 0.3 | 0.3 | 9.4${}^{+2.3}_{-3.1}$ | 0.066 | 53
Inner-9 | 66 | 17.4 | 16.9 | 0.3 | 0.2 | 9.9${}^{+2.9}_{-3.7}$ | 0.209 | 5
Inner-10 | 60 | 11.4 | 11.0 | 0.3 | 0.2 | 6.3${}^{+2.4}_{-3.2}$ | 0.085 | 44
Comb. L | 550 | 311.3 | 304.0 | 3.0 | 4.3 | 50.7${}^{+3.7}_{-3.7}$ | 0.07 | 48
K+L-peak |  |  |  |  |  |  |  |
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ |  Probability (%)
Inner-1 | 180 | 176.3 | 175.5 | 0.8 | 0.0 | 49.4${}^{+4.0}_{-4.2}$ | 0.398 | 6
Inner-2 | 129 | 111.5 | 107.7 | 0.8 | 3.1 | 44.9${}^{+5.6}_{-5.9}$ | 0.065 | 70
Inner-3 | 132 | 117.6 | 115.3 | 0.7 | 1.6 | 62.9${}^{+7.1}_{-7.4}$ | 0.056 | 76
Inner-4 | 93 | 87.3 | 85.6 | 0.6 | 1.1 | 73.3${}^{+8.0}_{-8.6}$ | 0.077 | 59
Inner-5 | 134 | 60.2 | 58.4 | 0.6 | 1.2 | 49.8${}^{+7.7}_{-8.2}$ | 0.033 | 92
Inner-6 | 81 | 48.8 | 47.7 | 0.4 | 0.7 | 69.5${}^{+11.0}_{-12.0}$ | 0.067 | 49
Inner-7 | 91 | 45.0 | 43.9 | 0.5 | 0.6 | 64.6${}^{+11.6}_{-12.6}$ | 0.127 | 13
Inner-8 | 59 | 33.6 | 32.4 | 0.6 | 0.6 | 53.8${}^{+11.0}_{-12.2}$ | 0.037 | 87
Inner-9 | 106 | 23.7 | 22.7 | 0.6 | 0.4 | 49.9${}^{+14.9}_{-16.5}$ | 0.164 | 10
Inner-10 | 88 | 25.2 | 24.3 | 0.6 | 0.3 | 69.1${}^{+17.3}_{-19.4}$ | 0.108 | 26
Comb. K+L | 1093 | 724.0 | 708.2 | 6.1 | 9.7 | 54.9${}^{+2.4}_{-2.5}$ | 0.099 | 28
Table 11: A summary of the likelihood fits for the production rate for the outer target from each extraction and the combined fit of all extractions, in K and L peak independently and combined. The production rates for each exposure for the K and L-peaks are referred to its starting time, taking into account the total detection efficiency. The production rate for their combined result is referred to the reference time. All production rates for the K+L-peak are referred to the reference time. The stated uncertainties are statistical and given in 68% confidence. The Cramér-von Mises statistics $Nw^{2}$ measures the goodness of fit of the sequence of event times. The probability was inferred from $Nw^{2}$ by simulation. K-peak |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ |  Probability (%)
Outer-1 | 89 | 77.3 | 75.4 | 1.8 | 0.1 | 46.8${}^{+6.1}_{-7.5}$ | 0.051 | 83
Outer-2 | 99 | 89.1 | 86.3 | 1.8 | 1.0 | 49.1${}^{+5.1}_{-7.5}$ | 0.070 | 67
Outer-3 | 62 | 48.1 | 46.1 | 1.3 | 0.7 | 34.3${}^{+5.7}_{-8.2}$ | 0.158 | 20
Outer-4 | 64 | 55.2 | 53.0 | 1.7 | 0.5 | 31.2${}^{+4.6}_{-6.1}$ | 0.037 | 90
Outer-5 | 50 | 28.4 | 26.2 | 1.7 | 0.5 | 15.3${}^{+2.5}_{-5.0}$ | 0.148 | 9
Outer-6 | 50 | 43.6 | 41.6 | 1.6 | 0.4 | 25.4${}^{+3.1}_{-5.4}$ | 0.085 | 53
Outer-7 | 28 | 20.5 | 18.7 | 1.6 | 0.2 | 11.6${}^{+1.7}_{-4.2}$ | 0.227 | 6
Outer-8 | 33 | 25.0 | 22.9 | 1.9 | 0.2 | 12.1${}^{+1.7}_{-4.0}$ | 0.130 | 25
Outer-9 | 23 | 10.6 | 9.1 | 1.4 | 0.1 | 6.6${}^{+0.0}_{-4.0}$ | 0.055 | 63
Outer-10 | 27 | 16.7 | 14.9 | 1.7 | 0.1 | 9.0${}^{+1.6}_{-4.0}$ | 0.043 | 78
Comb. K | 524 | 415.4 | 395.1 | 16.6 | 3.7 | 59.4${}^{+3.6}_{-3.7}$ | 0.066 | 58
L-peak |  |  |  |  |  |  |  |
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ |  Probability (%)
Outer-1 | 92 | 51.5 | 49.6 | 1.8 | 0.1 | 32.1${}^{+6.9}_{-9.1}$ | 0.216 | 11
Outer-2 | 75 | 75.0 | 72.6 | 1.5 | 0.9 | 48.1${}^{+7.5}_{-4.9}$ | 0.035 | 95
Outer-3 | 54 | 45.9 | 43.6 | 1.5 | 0.8 | 28.4${}^{+4.2}_{-6.2}$ | 0.202 | 15
Outer-4 | 34 | 27.0 | 25.9 | 0.8 | 0.3 | 30.4${}^{+6.0}_{-8.5}$ | 0.159 | 18
Outer-5 | 70 | 35.6 | 33.3 | 1.8 | 0.5 | 18.6${}^{+2.6}_{-5.3}$ | 0.045 | 74
Outer-6 | 47 | 18.3 | 17.3 | 0.8 | 0.2 | 22.1${}^{+5.1}_{-8.1}$ | 0.147 | 9
Outer-7 | 41 | 17.2 | 15.5 | 1.5 | 0.2 | 10.2${}^{+2.1}_{-4.6}$ | 0.071 | 47
Outer-8 | 35 | 18.0 | 15.9 | 1.9 | 0.2 | 8.1${}^{+1.4}_{-3.8}$ | 0.060 | 58
Outer-9 | 43 | 9.1 | 7.3 | 1.7 | 0.1 | 4.6${}^{+1.5}_{-4.0}$ | 0.080 | 44
Outer-10 | 54 | 14.6 | 12.6 | 1.9 | 0.1 | 7.1${}^{+1.6}_{-4.0}$ | 0.038 | 85
Comb. L | 545 | 320.6 | 302.0 | 15.3 | 3.3 | 50.9${}^{+3.9}_{-3.9}$ | 0.062 | 54
K+L-peak |  |  |  |  |  |  |  |
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ |  Probability (%)
Outer-1 | 181 | 133.4 | 129.6 | 3.7 | 0.1 | 41.1${}^{+5.2}_{-5.3}$ | 0.191 | 18
Outer-2 | 174 | 163.8 | 158.6 | 3.3 | 1.9 | 63.6${}^{+5.5}_{-5.7}$ | 0.065 | 73
Outer-3 | 116 | 92.5 | 88.2 | 2.8 | 1.5 | 51.4${}^{+6.9}_{-7.3}$ | 0.123 | 32
Outer-4 | 98 | 82.3 | 78.9 | 2.5 | 0.8 | 66.6${}^{+9.2}_{-9.8}$ | 0.045 | 84
Outer-5 | 120 | 64.0 | 59.5 | 3.5 | 1.0 | 46.9${}^{+7.2}_{-7.9}$ | 0.068 | 48
Outer-6 | 97 | 62.3 | 59.3 | 2.6 | 0.4 | 87.3${}^{+12.3}_{-13.2}$ | 0.095 | 30
Outer-7 | 69 | 38.0 | 34.4 | 3.2 | 0.4 | 50.4${}^{+9.6}_{-10.6}$ | 0.132 | 13
Outer-8 | 68 | 43.4 | 39.2 | 3.9 | 0.4 | 59.7${}^{+10.8}_{-11.7}$ | 0.072 | 50
Outer-9 | 66 | 20.2 | 17.0 | 3.0 | 0.2 | 43.0${}^{+13.5}_{-15.3}$ | 0.044 | 80
Outer-10 | 81 | 31.8 | 28.0 | 3.6 | 0.2 | 78.8${}^{+18.1}_{-20.0}$ | 0.028 | 96
Comb. K+L | 1069 | 738.8 | 699.8 | 32.2 | 6.8 | 55.6${}^{+2.6}_{-2.7}$ | 0.079 | 32

The best fit production rates of 71Ge from the combined data analysis for each
target at the reference time are

$\begin{split}r&{}_{\textrm{In-fit}}=(54.9\pm 2.5)~{}\textrm{atoms/d}~{},\\\
r&{}_{\textrm{Out-fit}}=(55.6\pm 2.7)~{}\textrm{atoms/d}~{}.\end{split}$ (15)

The stated uncertainty is statistical and is given with 68% confidence. The
fit values of 71Ge half-life from the inner and the outer targets are
($11.11\pm 0.69$) d and ($11.05\pm 0.72$) d. These are in good agreement with
the accepted half-life of ($11.43\pm 0.03$) d [39].

Table 12: Summary of the contributions to the systematic uncertainty in the measured neutrino capture rate. Origin of uncertainty | Uncertainty (%)
---|---
Chemical Extraction Efficiency |
Efficiency of extraction from Ga metal, $\delta_{G1}$ | $\pm 1.0$
Efficiency of synthesis into GeH4, $\delta_{G2}$ | $\pm 1.3$
Carrier carryover, $\delta_{G3}$ | negligible
Mass of Ga, $\delta_{G4}$ | negligible
Chemical Extraction Subtotal | $\pm 1.6$
Counting Efficiency |
Calculated Efficiency |
Volume Efficiency, $\delta_{C1}$ | -1.5, +1.8
Peak Efficiency, $\delta_{C2}$ | $\pm 1.3$
Simulations to adjust for counter filling, Monte Carlo interpolation, $\delta_{C3}$ | $\pm 0.6$
Calibration Statistics |
Gain Variation, $\delta_{C4}$ | $+0.4$
Resolution, $\delta_{C5}$ | $\pm 0.3$
Centroid, $\delta_{C6}$ | $\pm 0.1$
Rise time cut, $\delta_{C7}$ | negligible
Counting Efficiency Subtotal | -2.1, +2.3
Background Discrimination |
Residual Radon after Time Cuts, $\delta_{N1}$ | $-0.04$
Solar Neutrino Background, $\delta_{N2}$ | $\pm 0.20$
71Ge carryover, $\delta_{N3}$ | $\pm 0.05$
Background Discrimination Subtotal | $\pm 0.2$
Total Systematic Uncertainty | -2.7, +2.9

###  V.7 Systematic Studies

The systematic uncertainties are categorized into four basic types: 1)
uncertainties related to the chemical extraction efficiency ($\delta_{G}$), 2)
uncertainties in the counting efficiency ($\delta_{C}$), 3) uncertainties
arising from background phenomena ($\delta_{N}$), and the average path uength
uncertainty. The uncertainties related to the measurement are summarized in
Table 12, while item 4) is related to the prediction.

####  V.7.1 Extraction Efficiencies

i. Extraction efficiency from Ga metal, $\delta_{G1}$

Special Ge carriers were produced with a known number of 71Ge contents [32].
These carriers were added to the reactors, and the numbers of 71Ge atoms
extracted were counted and compared to the added numbers of 71Ge atoms. The
ratio of extracted and added Ge atoms determines the extraction efficiency.

Two types of carrier with different enrichment in Ge isotopes 72Ge and 76Ge
were used for the experiment. Both ligatures were made in 2015 and were used
in preliminary solar measurements. Ge extraction efficiency was determined by
a mass spectrometric method of extraction efficiency determination, which is
based on isotopic measurements [38]. In this method, the efficiency of
extraction of Ge from the Ga is calculated from the measured mass and isotopic
content of added and extracted Ge and all relevant uncertainties (uncertainty
in the total mass of Ge in the carrier, uncertainty in the Ge isotopic
fractions of the carrier, uncertainty in the mass of Ge in the extracted
sample, and uncertainty in the Ge isotopic fractions of the measured sample)
are take into account. The uncertainty of the extraction efficiency is
$\delta_{G1}=\pm 1.0\,\%$.

ii. Efficiency of synthesis into GeH4, $\delta_{G2}$

An uncertainty in how much extracted carrier has been synthesized into GeH4.
The amount of germanium entering the counter is calculated from the volume of
synthesized germane. The volume of germane is determined using the calibrated
stem of the Toepler pump, which is the main apparatus in the gas pumping
system and the preparation of the working mixture of the counter. In the
Topler pump, there are two calibrated stems with measured volumes. Their
volumes sizes were obtained by a large series of calibration measurements with
uncertainty not higher than 1%. The calculation is based on the difference in
mercury levels in the evacuated part of the Toepler pump, which is a
manometric tube with a linear scale, and a calibrated stems of a precisely
known volume, in which germane is located before being placed in the counter.
Division value of the vertical scale of the pressure gauge tube is 1 mm, and
the measurement error is half of 1 division, i.e. 0.5 mm. Measurement of the
volume of germane is carried out in two calibrated volumes of the Topler pump,
that reduces the measurement error to $\delta_{G2}=\pm 1.3$%.

iii. Carrier carryover, $\delta_{G3}$

The uncertainty of the amount of residual Ge carrier remaining from previous
extractions $\delta_{G3}$ with the mass spectrometric analysis, which is the
part of the uncertainty in $\delta_{G1}$. Therefore, we assign negligible
uncertainty to $\delta_{G3}$.

iv. Mass of Ga, $\delta_{G4}$

The gallium target mass affects the results of measurements of the neutrino
capture rate indirectly via neutrino path length from a radioactive source
through the gallium target in two zones. This uncertainty $\delta_{G4}$ is
taken into account in determining the average neutrino path length in the
gallium targets, which is given in Sec. V.5.

####  V.7.2 Counting Efficiencies

Uncertainties associated with the counting efficiencies of pulses from 71Ge
decays in proportional counters constitute the second group of uncertainties,
$\delta_{C}$.

i. Volume efficiency, $\delta_{C1}$

The volume efficiencies of the counters were measured at the end of the BEST
measurements using two isotopes, 37Ar and 71Ge. Here, the volume efficiency is
defined as the probability that the decay of a radioactive atom in the volume
of a proportional counter would produce a detectable signal [32]. This
efficiency was calculated for each detector by comparing the measured count
rate in each count to the count rate in the high-efficiency standardization
counter. A purified 37Ar gas was mixed with 90% Ar plus 10% CH4, and the
mixture is placed into counters under test. All counters were filled in turn
with the same gas mixture. Three spectra were taken from each counter with
each filling: the first with the voltage adjusted so that the 37Ar K peak is
set at the middle of the scale to see all pulses with energies above the K
peak, the second that both L and K peaks were on the scale, and the third with
the L peak, whose energy is 270 eV, at about mid-scale. After the
measurements, the mixture is transferred to the specially designed
standardization counter with a volume efficiency of (99.5$\pm$0.6)%, with a
very high transfer efficiency $E_{\mathrm{transfer}}>99.5$%. The
standardization counter is brought to about the same pressure as in the test
counter and the same spectra are measured. The ratio between the event rate
above a certain threshold in the test counter, $R_{\mathrm{test}}$, is then
compared with the event rate above the same threshold in the standardization
counter, $R_{\mathrm{test}}$. After minor corrections, the volume efficiency
of the counter is given by
$\epsilon_{V}=0.995R_{\mathrm{test}}E_{\mathrm{transfer}}D/R_{\mathrm{standard}}$
where $D$ is the decay factor of the 37 between the two measurements [32].
Additional measurements with 71GeH4 mixed with Xe-GeH4 were carried out using
a similar technique described above for verification. The uncertainty assigned
to volume efficiency is $\delta_{C1}=+1.5/-1.3$%. Considering that four
counters were used twice in the BEST measurements this counting systematic
uncertainty should be increased by 18%, $\delta_{C1}=+1.8/-1.5$%.

ii. Counting efficiency in peaks, $\delta_{C2}$

The peak counting efficiency is the ratio of the number of registered pulses
falling inside the $\pm$FWHM of the energy range of K and L peaks of 71Ge to
the total number of counts. The value was determined by additional measurement
using 71Ge for each counter. The details of the measurement are described in
Appx. A. The average uncertainty for peak efficiencies was $\delta_{C2}\pm
1.1$%. Considering that four counters were used twice in the BEST measurements
this counting systematic uncertainty should be increased by 18%,
$\delta_{C2}=\pm 1.3$%.

iii. Monte Carlo Interpolation, $\delta_{C3}$

The counting efficiency is closely related to the pressure ($P$) and the
percentage ($G$) of germane in the gas mixture [32]. The connection was
obtained in measurements of volume efficiencies in a wide range of parameters
for different counters. To obtain the counting efficiencies for arbitrary
parameters of the gas mixture, Monte Carlo calculations are used. Varying the
expression obtained in the analysis of measurements in accordance with the
uncertainties of the pressure and the percentage of germane ($\delta P=\delta
G=2$%), we obtain $\delta_{C3}=\pm 0.6$% for the K peak and $\pm 0.4$% for the
L peak.

iv. Gain variation, $\delta_{C4}$

During long-term measurements, the gain of the counting system may drift
resulting in uncertainty in energy determination. The drifts in the positions
of the 55Fe calibration peak were measured during the first month of each
counting. The average percentage of deviation from the mean values was $\pm
1.1$%. To measure the uncertainty due to the energy determination, the
calibration peak positions were shifted by $\pm 1.1$% and the capture rates in
the inner and the outer targets were obtained. This variation study yielded
the gain variation uncertainty of $\delta_{C4}=+0.4$%. This uncertainty is
one-sided because the gain drift can only reduce the number of selected
events.

v. Resolution and centroid, $\delta_{C5}$ and $\delta_{C6}$

The systematic uncertainty of the energy resolution of the calibrations was
determined similarly to item iv. The average percentage of deviation of the
resolution of the calibration peaks in the first month of measurements was
$\pm$ 2%, which gives the value $\delta_{C5}=\pm 0.3$%. Similarly, the
deviation of the centroid of the calibration peaks gives the value
$\delta_{C6}=\pm 0.1$%.

vi. Rise time cut, $\delta_{C7}$

At the end of the measurement, an additional study using 71Ge was carried out
to determine the rise time ($T_{N}$) values of pulses inside the 2 FWHM
regions of the L and K peaks of 71Ge. Therefore, in contrast to the previous
SAGE experiments where the average $T_{N}$ limits were used [32], there is no
uncertainty associated with the incorrect determination of the $T_{N}$ limit
interval. Hence, we assign negligible uncertainty to $\delta_{C7}$.

####  V.7.3 Background Discrimination

Uncertainties arising from background discrimination constitute the third
group of uncertainties, $\delta_{N}$.

i. Residual radon after the time cuts, $\delta_{N1}$

The number of false 71Ge events due to internal 222Rn is equal to the number
of detected saturated pulses induced by Rn decay ($N_{\textrm{Rn}}$)
multiplied by the survival probability of false events after the event
selection $\alpha_{K+L}$ [32]. The fraction of the false events to the total
number of events $N$ in K and L peaks is treated as a systematic uncertainty

$\begin{split}\delta_{N1}&=\frac{N_{\textrm{Rn}}\times\alpha_{K+L}}{N}\\\
&=\frac{425.9\times(4.3+7.8)\times 10^{-4}}{738.8+724.0}\\\
&=0.035\%~{}.\end{split}$ (16)

This uncertainty is one-sided negative because the effect of the residual Rn
is a reduction in the detected number of events.

ii. Solar neutrinos, $\delta_{N2}$ Solar neutrinos are the largest source of
background in the BEST experiment. They interact with the Ga targets at a
constant rate, and this background is taken into account in the analysis of
each measurement. During solar neutrino measurements, the number of solar-
neutrino-induced events in the inner and outer target zones was obtained to be
6.1 (see Table 10) and 32.2 (see Table 11). These values are compared with the
total number of pulses recorded in K and L peaks - 724.0 and 738.8. The solar
neutrino flux and thus the uncertainty associated with it are obtained from
previous gallium solar measurements as ($66.1\pm 3.1$) SNU [32], the
uncertainties are calculated to be $\delta_{N2}=6.1/724.0\times
3.1/66.1=0.04$% for the inner zone and $\delta_{N2}=32.2/738.8\times
3.1/66.1=0.2$% for the outer zone.

iii. 71Ge carryover, $\delta_{N3}$

The uncertainty associated with the 71Ge remaining after extractions is found
as the product of the statistical uncertainty and the ratio of the carryover
events ($N_{CO}$) to the total number of recorded events ($N$)

$\begin{split}\delta_{N3}&=\frac{N_{CO}}{N}\times\delta_{N}~{},\end{split}$
(17)

where $\delta_{N}=1/\sqrt{N}$ [32]. We obtain 0.03% and 0.05% uncertainties
for the inner and the outer zones.

####  V.7.4 Average Path Length Uncertainty

The mass of the internal target obtained from the considered geometry is 63.1
kg more gallium than compared with the mass measurement. Such a difference in
the target gallium mass will have the maximum effect on the change in the path
length if we assume a change in the inner radius of the sphere by 0.18 cm. We
take the resulting change in the radius as the radius uncertainty, which leads
to a systematic uncertainty in the neutrino path length in the inner target of
0.3%. For the external target, the corresponding uncertainty is 0.

The outer radius of the spherical vessel was measured in a direct way, with a
ruler, when the sphere vessel was available for measurements in the
underground laboratory of the GGNT. The surface of the sphere was covered with
a layer of epoxy resin. The maximum irregularities of the coating did not
exceed 1 mm, which varies the average neutrino path length in the outer target
of the cylindrical zone by 0.2%. Variations in the internal dimensions of the
cylindrical zone, as well as possible inaccuracies in the relative position of
the axes of the two zones of the target do not exceed 1 mm. A change in the
dimensions of the cylindrical zone within the indicated values leads to a
change in the average neutrino path length by approximately the same 0.18 cm.
Therefore, we take the total uncertainty of the average neutrino path length
in the cylindrical zone to be 0.3%, i.e., the uncertainties of the average
neutrino path length in both target zones are assumed to be equal.

The total systematic uncertainty, added in quadrature, is $-2.7/+2.9$%.
Including both the statistical and systematic uncertainties, we obtain 71Ge
production rates of

$\begin{split}r_{\mathrm{In-meas.}}&=\left[54.9\pm
2.5(\mathrm{stat.})^{+1.6}_{-1.5}(\mathrm{syst.})\right]\mathrm{atoms/d}\\\
&=\left[54.9^{+3.0}_{-2.9}\right]~{}\mathrm{atoms/d}~{},\\\ r_{\mathrm{Out-
meas.}}&=\left[55.6\pm
2.7(\mathrm{stat.})^{+1.6}_{-1.5}(\mathrm{syst.})\right]\mathrm{atoms/d}\\\
&=\left[55.6^{+3.1}_{-3.1}\right]\mathrm{atoms/d}~{}.\end{split}$ (18)

Figure 9: Top: the measured K+L peak rates of the inner target volume. Middle-top: the production rates of the inner target volume normalized to the reference time. The combined results for events in the L and K peaks are shown separately and compared to the predicted rate. Middle-bottom: the measured K+L peak rates of the outer target volume. Bottom: the production rates of the outer target volume normalized to the reference time. The combined results for events in the L and K peaks are shown separately and compared to the predicted rate. The blue (red) region represents the predicted (measured) production rate. The dotted lines enclose the $\pm 1\sigma$ uncertainty regions. Table 13: Summary of BEST detection efficiencies. The extraction efficiency values are taken from Tables 1 and 2. The counter efficiency values and $\Delta$ are taken from Tables 3 and 4. Same fractional uncertainties listed in Table 12 are assigned to each extraction. The saturation factor is obtained from Eq. 19 with negligible uncertainty.  Extraction |  Extraction efficiency (into GeH4) |  Saturation factor |  Counter efficiency (K peak) | $\Delta_{\mathrm{K}}$ |  Total detection efficiency (K peak) |  Counter efficiency (L peak) | $\Delta_{\mathrm{L}}$ |  Total detection efficiency (L peak)
---|---|---|---|---|---|---|---|---
Inner-1 | 0.946 | 0.3924 | 0.3663 | 0.8100 | 0.1101 | 0.3803 | 0.7450 | 0.1052
Inner-2 | 0.9559 | 0.3731 | 0.3647 | 0.7839 | 0.1020 | 0.3785 | 0.6542 | 0.0883
Inner-3 | 0.9673 | 0.3752 | 0.3605 | 0.7143 | 0.0935 | 0.3599 | 0.7143 | 0.0933
Inner-4 | 0.9515 | 0.3750 | 0.3679 | 0.7870 | 0.1033 | 0.3769 | 0.3669 | 0.0493
Inner-5 | 0.9554 | 0.3779 | 0.3649 | 0.7470 | 0.0984 | 0.3654 | 0.7470 | 0.0986
Inner-6 | 0.9548 | 0.3735 | 0.3577 | 0.7714 | 0.0984 | 0.3604 | 0.3891 | 0.0500
Inner-7 | 0.9381 | 0.3760 | 0.3676 | 0.7495 | 0.0972 | 0.3793 | 0.6739 | 0.0902
Inner-8 | 0.9789 | 0.3761 | 0.3656 | 0.7754 | 0.1044 | 0.3779 | 0.7865 | 0.1094
Inner-9 | 0.9545 | 0.3761 | 0.359 | 0.8015 | 0.1033 | 0.361 | 0.8015 | 0.1039
Inner-10 | 0.9372 | 0.3770 | 0.3698 | 0.7629 | 0.0997 | 0.3755 | 0.8002 | 0.1062
Outer-1 | 0.9503 | 0.3924 | 0.3422 | 0.7646 | 0.0976 | 0.3596 | 0.6996 | 0.0938
Outer-2 | 0.9581 | 0.3731 | 0.3707 | 0.8043 | 0.1066 | 0.3792 | 0.6755 | 0.0916
Outer-3 | 0.9668 | 0.3752 | 0.2933 | 0.7650 | 0.0814 | 0.3358 | 0.7650 | 0.0932
Outer-4 | 0.9622 | 0.3750 | 0.3658 | 0.7819 | 0.1032 | 0.381 | 0.3755 | 0.0516
Outer-5 | 0.9609 | 0.3779 | 0.3568 | 0.8025 | 0.1040 | 0.3727 | 0.8025 | 0.1086
Outer-6 | 0.9253 | 0.3735 | 0.3585 | 0.8009 | 0.0992 | 0.3577 | 0.3845 | 0.0475
Outer-7 | 0.9514 | 0.3760 | 0.3407 | 0.7976 | 0.0972 | 0.3607 | 0.7107 | 0.0917
Outer-8 | 0.9897 | 0.3761 | 0.3716 | 0.8295 | 0.1147 | 0.3785 | 0.8406 | 0.1184
Outer-9 | 0.9664 | 0.3761 | 0.293 | 0.7865 | 0.0838 | 0.336 | 0.7865 | 0.0961
Outer-10 | 0.9538 | 0.3770 | 0.3677 | 0.7567 | 0.1001 | 0.3797 | 0.7940 | 0.1084

Various efficiencies for individual extraction are collected and summarized in
Table 13. The fractional uncertainties are the same for every extractions and
given in Table 12. The saturation factor in the table represents the
saturation in production rate due to 71Ge and 51Cr activities and is given by

$\begin{split}f_{s}=&~{}\mathrm{exp}(-\lambda_{51}(t_{s}-T))\\\
&\times[\mathrm{exp}(-\lambda_{51}\theta_{\mathrm{Cr}})-\mathrm{exp}(-\lambda_{71}\theta_{\mathrm{Cr}})]/(1-\lambda_{51}/\lambda_{71})~{},\end{split}$
(19)

where $\lambda_{51}$ and $\lambda_{71}$ are the decay constants of 51Cr and
71Ge, $t_{s}$ is the start time of each source exposure, $T$ is the source
activity reference time and $\theta_{\mathrm{Cr}}$ is the time of exposure of
the Ga to the 51Cr source [24]222One should note that there is a typographical
error in Eq. 4 of Ref. [24]. There is a missing division sign in the
denominator, which is corrected in Eq. 19 here in this work. The correct
denominator is $(1-\lambda_{51}/\lambda_{71})$.. All parameters considered in
the calculation of $f_{s}$ are very well established, and therefore its
uncertainty is negligible. The total efficiency is the product of the chemical
extraction efficiency, the saturation factor, and the counting efficiency
[24]. The typical value of the total efficiency is (10.0$\pm$0.3%), where
systematic uncertainties are included.

Fig. 9 shows the K+L production rate fits for the inner and the outer zones.
The upper panel for each zone shows the production rate at the start of each
exposure, and the lower panel shows the rate normalized to the reference time.
The results are plotted at the start of each exposure where the production
rates are fit to. The vertical lines represent the uncertainty of the fits,
while horizontal lines only define the exposure period. The ratio of measured
to predicted production rates are

$\begin{split}R&{}_{\mathrm{In}}=\frac{54.9^{+3.0}_{-2.9}}{69.4^{+2.5}_{-2.0}}=0.79^{+0.05}_{-0.05}~{},\\\
R&{}_{\mathrm{Out}}=\frac{55.6^{+3.1}_{-3.1}}{72.6^{+2.6}_{-2.1}}=0.77^{+0.05}_{-0.05}~{}.\end{split}$
(20)

These are 4.2 $\sigma$ and 4.8 $\sigma$ less than unity, respectively. As a
cross-check, the 51Cr half-life was left free and fitted to the data. The
values of half-life from fits to the inner and the outer targets were
($30.97\pm 3.90$) d and ($31.55\pm 2.89$) d. They agree with the known 51Cr
half-life of (27.704$\pm$0.004) [35].

The ratio between the inner and the outer zones
${R}_{\textrm{out}}/{R}_{\textrm{in}}=(0.77\pm 0.05)/(0.79\pm 0.05)=0.97\pm
0.08$ is unity within uncertainty, and hence there is no difference in the
capture rates between the two zones.

Figure 10: Allowed regions for two BEST results. The best-fit point is
sin${}^{2}2\theta=0.42^{+0.15}_{-0.17}$, $\Delta m^{2}=3.3^{+\infty}_{-2.3}$
eV2 and is indicated by a point. Figure 11: Allowed regions for two GALLEX,
two SAGE and two BEST results. The best-fit point is
sin${}^{2}2\theta=0.34^{+0.14}_{-0.09}$, $\Delta m^{2}=1.25^{+\infty}_{-0.25}$
eV2 and is indicated by a point.

###  V.8 Oscillation Analysis

For $n=1,\dots,N$ experiments (the two BEST volumes are treated separately),
oscillation parameters are estimated by a global minimization of

$\small\chi^{2}(\Delta
m^{2},\sin^{2}2\theta)=({\bf{r}^{\mathrm{meas.}}-{r}^{\mathrm{calc.}}})^{\mathrm{T}}{\bf
V}^{-1}({\bf{r}^{\mathrm{meas.}}-{r}^{\mathrm{calc.}}})~{},$ (21)

where ${\bf{r}^{\mathrm{meas.}}}$ (${\bf{r}^{\mathrm{calc.}}}$) is the vector
of the measured (calculated) rates with ${r}_{i}^{\mathrm{calc.}}(\Delta
m^{2},\mathrm{sin}^{2}2\theta)$ and $V$ is the covariance matrix with

$V_{nk}=\delta_{nk}\varepsilon_{n}^{2}+\varepsilon_{CS}^{n}\times\varepsilon_{CS}^{k}~{},$
(22)

where $\varepsilon_{n}^{2}=\varepsilon_{n,stat}^{2}+\varepsilon_{n,syst}^{2}$
are uncorrelated uncertainties comprised of statistical and systematic
measurement uncertainties, and $\varepsilon_{CS}^{n}$ represent the correlated
uncertainties of $\sigma$ [57]. For Ga source experiments, the cross-section
uncertainties are the only significant contribution to the correlated
uncertainty. The uncertainties of the Bahcall cross-section for measurements
with a chromium source are equal to +3.6/–2.8%, and hence we use $\pm 3.6$%
for the calculation. In the measurement with an argon source, the cross-
section uncertainties do not change the values of the shown uncertainty due to
their relative smallness.

Fig. 10 illustrates the exclusion contours corresponding to $1\sigma,2\sigma$,
and $3\sigma$ confidence levels with the two BEST results assuming the
correlated cross section uncertainties. The best-fit result is at
sin${}^{2}2\theta=0.42^{+0.15}_{-0.17}$, $\Delta m^{2}=3.3^{+\infty}_{-2.3}$
eV2, and the contours refer to
$\chi^{2}=\chi^{2}_{\textrm{min}}+\Delta\chi^{2}$ for which
$\Delta\chi^{2}=2.30,6.18,11.83$ with two degrees of freedom have the coverage
probability of 68.27% (1$\sigma$), 95.45% (2$\sigma$) and 99.73% (3$\sigma$),
respectively [58]. This approximation is based on Wilks’ theorem. A concern
about the applicability of Wilks’ theorem in the oscillation analysis [59] was
investigated and found to have a small effect on the analysis [60].

Table 14: Results of all six Ga source experiments. Experiment | $R$
---|---
SAGE-Cr [24] | 0.95 $\pm$ 0.12
SAGE-Ar [25] | 0.79 $\pm$ 0.095 (+0.09 / -0.10)
GALLEX-Cr1 [27] | 0.953 $\pm$ 0.11
GALLEX-Cr2 [27] | 0.812 $\pm$ 0.11
BEST-Inner | 0.791 $\pm$ 0.05
BEST-Outer | 0.766 $\pm$ 0.05

We also combine the BEST results with the previous SAGE and GALLEX source
experiments to plot combined exclusion plots. Table 14 and Fig. 12 summarize
results from all gallium anomaly experiments, including the SAGE [24, 25] and
the GALLEX results [27].

Figure 12: Ratios of measured and predicted 71Ge production rates in all Ga
source experiments. The combined result is shown as a blue band.

If the cross-section uncertainties are considered to be completely
uncorrelated to each other and to the other uncertainties, they can be added
in quadrature as

$\sigma_{i}=\sqrt{\sigma_{i,\mathrm{others}}^{2}+(0.032\times R_{i})^{2}}~{},$
(23)

and the combined result $R_{0}$ is obtained by the sum
$R_{0}=\sum_{i}(w_{i}\cdot R_{i})$, where $w_{i}=(\sigma_{0}/\sigma_{i})^{2}$
and $\sigma_{0}=1/\sqrt{\sum_{i}(1/\sigma_{i}^{2})}$. The result is given as
$R_{0}\pm\sigma_{0}=0.81\pm 0.03$. The total uncertainty is 4.0%.

If we consider the correlation between systematic uncertainties, the average
value of $R_{0,\mathrm{Cr}}$ is obtained first, and then combined with the
SAGE-Ar experiment afterward. The uncertainty from the cross section
evaluation is the only significant contribution to the correlated uncertainty,
and hence the combined result of all six gallium anomaly experiments is given
as

$R_{0}=\Big{(}\frac{\sigma_{R}}{\sigma_{\mathrm{Cr}}}\Big{)}^{2}\cdot
R_{\mathrm{Cr}}+\Big{(}\frac{\sigma_{R}}{\sigma_{\mathrm{Ar}}}\Big{)}^{2}\cdot
R_{\mathrm{Ar}}\pm\sigma_{R},$ (24)

where
$\sigma_{R}=1/\sqrt{(1/\sigma_{\mathrm{Cr}}^{2}+1/\sigma_{\mathrm{Ar}}^{2})}$,
$R_{\mathrm{Cr}}=\sum_{i}^{\mathrm{Cr}}(w_{i}\cdot R_{i})$,
$w_{i}=(\sigma_{0}/\sigma_{i,\mathrm{others}})^{2}$,
$\sigma_{0}=1/\sqrt{\sum_{i}^{\mathrm{Cr}}(1/\sigma_{i,\mathrm{others}}^{2})}$
and $\sigma_{\mathrm{Cr}}=\sigma_{0}+0.032\cdot R_{\mathrm{Cr}}$, with the
uncertainty of the total Cr measurements is obtained by summing over. The
combined result of all six measurements is obtained as $R_{0}=0.80\pm 0.047$.
The total uncertainty is 6.1%, which is larger than the uncorrelated
estimation. The result is illustrated in Fig. 12 as a blue band.

Fig. 11 presents the combined result from all gallium source experiments;
SAGE, GALLEX and BEST, considering the correlated cross section uncertainties.
The best-fit result from the combined analysis of all Ga source experiments is
sin${}^{2}2\theta=0.34^{+0.14}_{-0.09}$, $\Delta m^{2}=1.25^{+\infty}_{-0.25}$
eV2.

Figure 13: Exclusion contours of all gallium anomaly experiments: two GALLEX,
two SAGE and two BEST results. The blue solid line and the blue dotted line
show the 2$\sigma$ and 3$\sigma$ confidence level, respectively. The figure
also presents the exclusion contours from Prospect [61], DANSS [62], Stéréo
[63], KATRIN [64], the combined analysis of RENO and NEOS data [65], reactor
anti-neutrino anomalies (RAA) [22], interpretations of the MicroBooNE result
for the oscillation hypothesis with fixed mixing angle (sin${}^{2}2\theta$)
and profiled over the angle [30], and the model-independent 95% upper bound on
sin${}^{2}2\theta$ from all solar neutrino experiments [66]. The 2$\sigma$
allowed region of Neutrino-4 [67] is also presented and the grey shading
represents the merged exclusion of the very short baseline (VSBL) null
results.

Fig. 13 compares the combined result from all gallium anomaly experiments to
some other sterile neutrino search experiments. The exclusion curves of
Prospect [61], DANSS [62], Stéréo [63], KATRIN [64], the combined analysis of
RENO and NEOS data [65]. One can see that the gallium anomaly result is still
in a strong tension with these experiments except a tiny region above 8 eV2.
The interpretations of the MicroBooNE result for the oscillation hypothesis
either fixed or profiled over the mixing angle (sin${}^{2}2\theta$) [30] are
also presented. These results do not either favor or exclude the allowed
region for the gallium anomaly experiments. The 95% allowed region from the
reactor antineutrino anomalies (RAA) [22] is also illustrated in the figure.
One can see that the tension between the Ga anomalies and the RAA still
persists. The figure also shows the 2$\sigma$ allowed region of Neutrino-4
[67] and the model-independent 95% solar neutrino upper bound on
sin${}^{2}2\theta$ [66]. The Neutrino-4 allowed region is within the 3$\sigma$
bound, and therefore is not excluded by the gallium anomaly experiments. The
model-independent 95% solar neutrino bound on sin${}^{2}2\theta$ excludes most
of the 2$\sigma$ allowed region, but the parameter space near $\Delta
m^{2}\approx$ 1 eV2 is not entirely excluded.

##  VI Discussion

Table 15: A summary of the 71Ge counter calibration.  Counter |  55Fe | L-peak | K-peak
---|---|---|---
|  |  | Position | Resolution | Position | Resolution
Pos. (a.u.) |  Resol. (%) |  En non-linearity factor |  Pred. from 55Fe (a.u.) |  True (a.u.) |  Ratio |  Pred. from 55Fe (%) |  True (%) |  Ratio |  Pred. from 55Fe (a.u.) |  True (a.u.) |  Ratio |  Pred. from 55Fe (%) |  True (%) |  Ratio
sys2z |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
YCN43 | 437.39 | 18.2 | 0.982 | 86.81 | 87.16 | 1.00 | 40.92 | 40.29 | 0.98 | 755.35 | 725.50 | 0.96 | 15.88 | 12.98 | 0.82
YCNA9 | 431.86 | 19.7 | 0.938 | 85.71 | 86.32 | 1.01 | 44.29 | 42.57 | 0.96 | 712.39 | 728.84 | 1.02 | 17.19 | 16.02 | 0.93
YCN41 | 442.80 | 18.0 | 0.967 | 87.88 | 87.00 | 0.99 | 40.40 | 37.24 | 0.92 | 753.01 | 732.49 | 0.97 | 15.68 | 13.34 | 0.85
YCN46 | 448.93 | 18.5 | 0.915 | 89.10 | 90.61 | 1.02 | 41.48 | 39.62 | 1.04 | 722.39 | 748.55 | 0.96 | 16.10 | 14.16 | 0.88
Mean sys2z std |  |  |  |  |  |  1.00 0.01 |  |  |  0.96 0.03 |  |  |  1.00 0.04 |  |  |  0.87 0.05
sys3 |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
YCN113 | 337.36 | 18.8 | 0.982 | 66.96 | 57.71 | 0.86 | 42.09 | 42.13 | 1.00 | 582.61 | 564.38 | 0.97 | 16.34 | 14.13 | 0.86
YCT92 | 334.67 | 18.9 | 0.964 | 66.42 | 56.72 | 0.85 | 42.40 | 41.88 | 0.99 | 567.37 | 563.97 | 0.99 | 16.46 | 13.90 | 0.84
YCT3 | 342.65 | 19.3 | 0.932 | 68.01 | 58.24 | 0.86 | 43.23 | 40.61 | 0.94 | 561.61 | 576.97 | 1.03 | 16.78 | 14.19 | 0.85
YCT2 | 337.00 | 18.8 | 0.982 | 66.89 | 57.77 | 0.86 | 42.24 | 41.62 | 0.99 | 581.98 | 563.33 | 0.97 | 16.40 | 14.05 | 0.86
YCT9 | 334.80 | 19.1 | 0.978 | 66.45 | 57.97 | 0.87 | 42.90 | 39.42 | 0.92 | 575.83 | 569.62 | 0.99 | 16.65 | 13.89 | 0.83
YCT97 | 332.81 | 18.6 | 0.932 | 66.05 | 57.11 | 0.86 | 41.64 | 38.75 | 0.93 | 545.48 | 559.25 | 1.03 | 16.16 | 13.51 | 0.84
Mean sys3 std |  |  |  |  |  |  0.86 0.01 |  |  |  0.96 0.03 |  |  |  1.00 0.03 |  |  |  0.85 0.01

The deficits in the production rates observed in both volumes are consistent
with the results from previous gallium source experiments, confirming the
gallium anomaly. While no difference in the production rates in the two target
zones is observed and the oscillation length cannot be well constrained, the
large deviation of R’s from unity indicates that the oscillation hypothesis is
favored over no-oscillation hypothesis to the extent of current knowledge.
Effects from systematic errors in the extraction efficiency, the counting
efficiency, Ga target mass, the geometry of the system, the source strength
and the counting system operation have been thoroughly considered and
confirmed by additional measurements to be small (see Tables 9 and 12)
compared to the observed deficits. In addition, GALLEX’s 71As experiment [68]
ruled out any chemical effects of ’hot atoms’ that would make it difficult to
extract 71Ge atoms formed during neutrino capture. Since the physics processes
involved in the Ga experiments are simple and understood very well, it is hard
to attribute the result to some unaccounted effects. The 71Ge decay rate is
very well known, and the neutrino-nucleon cross-section $\sigma$ to the ground
state in 71Ga is hence also well determined. The $\sigma$ cannot be reduced
below the ground state value by considering the excited state contribution to
be zero. The transition to excited states cannot decrease the overall value of
the total $\sigma$ and hence cannot be the origin of the reported deficits.
When the solar neutrino flux measured by the SAGE [41] is compared to the
Borexino [69] result, the two experiments claim similar $pp$ flux. SAGE
measured the $pp$ flux produced in the Sun to be (6.0$\pm$0.84)$\times
10^{10}$ / (cm2 s), while Borexino yielded $6.1\pm 0.05^{+0.3}_{-0.5}\times
10^{10}$ / (cm2 s). However, the uncertainties are too large to either confirm
or exclude the overall efficiency as explanation of the gallium anomaly. Other
explanations without sterile neutrinos are possible but would inevitably
require some fundamental misunderstanding of nuclear or atomic physics.

After the BEST measurements the gallium anomaly is more significant; the
weighted average value of the neutrino capture rate relative to the expected
value for all Ga experiments is $0.80\pm 0.05$. The result indicates that the
oscillation length is similar to, or smaller than, the volume of the BEST
targets. Hence, $\Delta m^{2}$ is unbounded from above by the BEST experiment.
A future Ga experiment with shorter baseline would help constraining the value
of $\Delta m^{2}$, but it would require the use of stronger neutrino source
which would be challenging. The use of neutrino source with larger neutrino
energy could also be considered.

##  VII Conclusion

The first result from the BEST sterile neutrino oscillation experiment was
presented. We report 21% and 23% deficits of the 71Ge production rates based
on the cross section [45] in the inner and the outer zones of the BEST Ga
target. The values are consistent with the previously reported gallium anomaly
from SAGE and GALLEX experiments, with higher significance. The weighted
average value of the neutrino capture rate relative to the expected value for
all Ga experiments is $0.80\pm 0.05$, which corresponds to a 4$\sigma$
deviation from unity. If attributed to the neutrino oscillation to sterile
state, the result corresponds to the best fit of $\Delta
m^{2}=3.3^{+\infty}_{-2.3}$ eV2 and sin${}^{2}2\theta=0.42^{+0.15}_{-0.17}$.
No difference in the capture rates from the two zones at different distances
is observed. This indicates a neutrino oscillation at a scale shorter than the
BEST dimensions.

The allowed region in the ($\Delta m^{2}$, sin${}^{2}2\theta$) parameter space
for all combined Ga source experiments shows that the Ga anomalies are still
in tension with most other sterile neutrino search experiments. On the other
hand, the Neutrino-4 allowed region coincides with the gallium anomaly result,
and therefore is not rejected by the Ga source experiments. The value of
$\Delta m^{2}$ should be better constrained to conclusively determine whether
the Neutrino-4 result is accepted or rejected by the Ga source experiments.

Ga source experiments are unique in their capability of searching for the very
short range neutrino oscillation at a meter scale, and the simplicity of the
physics involved. They can provide critical information on the field of the
sterile neutrino search experiments, providing a key to detect the neutrino
oscillation to sterile states and the opportunity of resolving the existing
tension between the reactor antineutrino experiments and other experiments.
Further study with other neutrino sources will enhance our knowledge on the
gallium anomaly, and hence lead to an irreplaceable contribution to the
pursuit of the physics beyond the Standard Model.

###### Acknowledgements.

We thank V. A. Rubakov for constant stimulation of our interest and for
fruitful discussions. This work is supported by Federal Agency for Scientific
Organizations (FANO), Ministry of Education and Science of Russian Federation
under agreement no. 14.619.21.0009 (unique project identifier no.
RFMEFI61917X0009), State Atomic Energy Corporation Rosatom, and the Office of
Nuclear Physics of the US Department of Energy.

Table 16: A summary of the $T_{N}$ event selection efficiency test using 71Ge calibration data. Counter |  Num. Ev. L- all |  L-peak $T_{N}$ cut (ns) |  Num. Ev L- selected | ratio |  Num. Ev. k- all |  K-peak $T_{N}$ cut (ns) |  Num. Ev k- selected | ratio
---|---|---|---|---|---|---|---|---
sys2z
YCN43 | 495 | 10 | 463 | 0.94 | 489 | 13.2 | 468 | 0.96
YCNA9 | 1281 | 13.2 | 1244 | 0.97 | 1167 | 18.8 | 1142 | 0.98
YCN41 | 1353 | 10.3 | 1299 | 0.96 | 1434 | 13.4 | 1374 | 0.96
YCN46 | 941 | 11.3 | 897 | 0.95 | 865 | 15.2 | 837 | 0.97
mean 2z std |  |  |  |  0.95 0.01 |  |  |  |  0.97 0.01
sys3
YCN113 | 1643 | 9.1 | 1626 | 0.99 | 1488 | 13.6 | 1426 | 0.96
YCT92 | 265 | 13.0 | 250 | 0.94 | 243 | 17.6 | 237 | 0.98
YCT4 | 508 | 10.2 | 497 | 0.98 | 328 | 13.2 | 313 | 0.95
YCT3 | 314 | 10.3 | 297 | 0.95 | 258 | 16.4 | 252 | 0.98
YCT2 | 1475 | 10.1 | 1415 | 0.96 | 1483 | 16.6 | 1427 | 0.96
YCT9 | 397 | 9.1 | 388 | 0.98 | 341 | 14.9 | 322 | 0.94
YCT97 | 1622 | 11.4 | 1551 | 0.96 | 1607 | 17.3 | 1562 | 0.97
mean sys3 std |  |  |  |  0.96 0.02 |  |  |  |  0.96 0.02
Table 17: BEST results without the first Cr extractions | Number of events assigned to |
---|---|---
Extraction |  Number of candidate events |  Number fit to 71Ge |  51Cr source production |  Solar $\nu$ production | Carryover |  71Ge Production decay rate (atoms/day) | $Nw^{2}$ | Probability (%)
Inner 9 Ex. | 913 | 553.9 | 538.9 | 5.3 | 9.7 | 57.7${}^{+3.2}_{-3.1}$ | 0.093 | 26
Outer 9 Ex. | 889 | 599.7 | 564.5 | 28.5 | 6.6 | 59.8${}^{+3.1}_{-3.2}$ | 0.058 | 59

##  Appendix A 71Ge Calibration

###  A.1 Correction for Energy Calibration

To verify the extrapolation for the K and L peak regions from the 55Fe
calibration peak, counters were filled with 71Ge for additional analysis.
Positions and resolutions of the peaks from extrapolation were compared to the
true 71Ge peaks. This analysis also enables to select events with real $T_{N}$
limits and verify the accepted value of 96% acceptance.

Table 15 summarizes the positions and the resolutions of the 55Fe calibration
and the 71Ge K and L peaks. The values predicted from the calibration peak,
taking into account the counter non-linearity factors, are presented with the
true location of the X-ray peaks. Means and standard deviations of the two
types of counter systems: sys2z (new) and sys3 (old), are also presented for
each peak. These values are used as correction factors for the event
selection.

###  A.2 Rise time window $T_{N}$ Selection Test

The 71Ge calibration data is also used as a test to the $T_{N}$ limits event
selection windows. The cutoff values, determined as the ratio of $T_{N}$
selected events to all events without selection, is expected to be close to
96%. The values presented in Table 16 agrees with 96% within uncertainty.
Hence the $T_{N}$ event selection efficiency is verified.

##  Appendix B Results without the First Data Points

Figure 14: Top: Allowed regions for two BEST results without the first
extractions. The best-fit point is sin${}^{2}2\theta$=0.35, $\Delta m^{2}$
=6.7 eV2 and is indicated by a point. Bottom: Allowed regions for two GALLEX,
two SAGE and two BEST results without the first extractions. The best-fit
point is sin${}^{2}2\theta$=0.31, $\Delta m^{2}$ = 2.85 eV2 and is indicated
by a point.

In Fig. 9, it seems that the most significant contributions to the deficit for
both the inner and the outer volumes originate from the first extractions. To
examine the effect of the first extraction, we carried out an analysis leaving
out the first extraction. For the 9 runs of each target without the first
extractions, the combined best fit rates are:

$\begin{split}r_{\mathrm{In-meas.}}&=57.7\pm
3.2~{}\mathrm{atoms}^{71}\mathrm{Ge/day}~{},\\\ r_{\mathrm{Out-
meas.}}&=59.8\pm 3.2~{}\mathrm{atoms}^{71}\mathrm{Ge/day}~{}.\end{split}$ (25)

These are 3% and 6% increase in rates, respectively. The statistical
uncertainty with 68% confidence increases by 19% in the two volumes. The
quadratic combination of all systematic uncertainties is -2.7/+2.9%. The
values are summarized in Table 17.

The measured production rates in the K and L peaks, including both statistical
and systematic uncertainties combined in quadrature are

$\begin{split}r_{\mathrm{In-meas.}}&=57.7\pm
3.2(\mathrm{stat.})^{+1.7}_{-1.6}(\mathrm{syst.})\\\
&=57.7^{+3.6}_{-3.6}~{}\mathrm{atoms}^{71}\mathrm{Ge/day}~{},\\\
r_{\mathrm{Out-meas.}}&=59.8\pm
3.2(\mathrm{stat.})^{+1.7}_{-1.6}(\mathrm{syst.})\\\
&=59.8^{+3.6}_{-3.6}~{}\mathrm{atoms}^{71}\mathrm{Ge/day}~{}.\end{split}$ (26)

The ratio of measured to predicted production rates are

$\begin{split}R&{}_{\mathrm{In}}=\frac{57.7^{+3.6}_{-3.6}}{69.4^{+2.5}_{-2.0}}=0.83^{+0.06}_{-0.06}~{},\\\
R&{}_{\mathrm{Out}}=\frac{59.8^{+3.6}_{-3.6}}{72.6^{+2.6}_{-2.1}}=0.82^{+0.06}_{-0.06}~{}.\end{split}$
(27)

These are 2.9$\sigma$ and 3.1$\sigma$ less than unity. The ratio of the outer
to the inner result is $0.99\pm 0.08$ and there is no difference in the
capture rates in the two zones. In Fig. 14, the allowed region for the BEST
result and for all gallium anomaly experiments are illustrated.

##  Appendix C 51Cr Branching Ratio Uncertainty

The calorimetric heat measurement relies on the branching ratio of the 320 keV
51Cr emission to normalize to activity. If the branching value is in error, so
would be the source strength.

Table 18: A summary of the branching ratio of the 320 keV emission from 51Cr. Branching ratio | Reference | Method
---|---|---
0.1030(19) | [70] | Ge(Li)
0.0990(8) | [71] | NaI
0.1008(11) | [71] | HPGe
0.099(1) | [72] | HPGe (Beta-gamma coincidence)
0.0987(3) | [73] | Si(Li) with fixed activity

Table 18 summarizes various measurements of the branching ratio to the excited
state. The branching ratios are claimed to be known to a precision $\sim$0.1%,
and vary within themselves by 4%. This is much smaller than the observed
gallium anomaly and is not enough to explain our results.

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