PHYSICAL REVIEW D 94, 052010 (2016)

Solar neutrino measurements in Super-Kamiokande-IV
K. Abe,1,32 Y. Haga,1 Y. Hayato,1,32 M. Ikeda,1 K. Iyogi,1 J. Kameda,1,32 Y. Kishimoto,1,32 Ll. Marti,1 M. Miura,1,32
S. Moriyama,1,32 M. Nakahata,1,32 T. Nakajima,1 S. Nakayama,1,32 A. Orii,1 H. Sekiya,1,32 M. Shiozawa,1,32 Y. Sonoda,1A. Takeda,1,32 H. Tanaka,1 Y. Takenaga,1 S. Tasaka,1 T. Tomura,1 K. Ueno,1 T. Yokozawa,1 R. Akutsu,2 T. Irvine,2 H. Kaji,2
T. Kajita,2,32 I. Kametani,2 K. Kaneyuki,2,32,* K. P. Lee,2 Y. Nishimura,2 T. McLachlan,2 K. Okumura,2,32 E. Richard,2
L. Labarga,3 P. Fernandez,3 F. d. M. Blaszczyk,4 J. Gustafson,4 C. Kachulis,4 E. Kearns,4,32 J. L. Raaf,4 J. L. Stone,4,32
L. R. Sulak,4 S. Berkman,5 S. Tobayama,5 M. Goldhaber,6,* K. Bays,7 G. Carminati,7 N. J. Griskevich,7 W. R. Kropp,7
S. Mine,7 A. Renshaw,7 M. B. Smy,7,32 H. W. Sobel,7,32 V. Takhistov,7 P. Weatherly,7 K. S. Ganezer,8 B. L. Hartfiel,8 J. Hill,8W. E. Keig,8 N. Hong,9 J. Y. Kim,9 I. T. Lim,9 R. G. Park,9 T. Akiri,10 J. B. Albert,10 A. Himmel,10 Z. Li,10 E. O’Sullivan,10
K. Scholberg,10,32 C. W. Walter,10,32 T. Wongjirad,10 T. Ishizuka,11 T. Nakamura,12 J. S. Jang,13 K. Choi,14 J. G. Learned,14
S. Matsuno,14 S. N. Smith,14 M. Friend,15 T. Hasegawa,15 T. Ishida,15 T. Ishii,15 T. Kobayashi,15 T. Nakadaira,15
K. Nakamura,15,32 K. Nishikawa,15 Y. Oyama,15 K. Sakashita,15 T. Sekiguchi,15 T. Tsukamoto,15 Y. Nakano,16
A. T. Suzuki,16 Y. Takeuchi,16,32 T. Yano,16 S. V. Cao,17 T. Hayashino,17 T. Hiraki,17 S. Hirota,17 K. Huang,17 K. Ieki,17
M. Jiang,17 T. Kikawa,17 A. Minamino,17 A. Murakami,17 T. Nakaya,17,32 N. D. Patel,17 K. Suzuki,17 S. Takahashi,17
R. A. Wendell,17,32 Y. Fukuda,18 Y. Itow,19,20 G. Mitsuka,19 F. Muto,19 T. Suzuki,19 P. Mijakowski,21 K. Frankiewicz,21
J. Hignight,22 J. Imber,22 C. K. Jung,22 X. Li,22 J. L. Palomino,22 G. Santucci,22 I. Taylor,22 C. Vilela,22 M. J. Wilking,22
C. Yanagisawa,22,† D. Fukuda,23 H. Ishino,23 T. Kayano,23 A. Kibayashi,23 Y. Koshio,23,32 T. Mori,23 M. Sakuda,23J. Takeuchi,23 R. Yamaguchi,23 Y. Kuno,24 R. Tacik,25,34 S. B. Kim,26 H. Okazawa,27 Y. Choi,28 K. Ito,29 K. Nishijima,29
M. Koshiba,30 Y. Totsuka,30,* Y. Suda,31 M. Yokoyama,31,32 C. Bronner,32 R. G. Calland,32 M. Hartz,32 K. Martens,32
Y. Obayashi,32 Y. Suzuki,32 M. R. Vagins,32,7 C. M. Nantais,33 J. F. Martin,33 P. de Perio,33 H. A. Tanaka,33 A. Konaka,34
S. Chen,35 H. Sui,35 L. Wan,35 Z. Yang,35 H. Zhang,35 Y. Zhang,35 K. Connolly,36 M. Dziomba,36 and R. J. Wilkes36(Super-Kamiokande Collaboration)
1

Kamioka Observatory, Institute for Cosmic Ray Research, University of Tokyo,
Kamioka, Gifu 506-1205, Japan
2
Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research,
University of Tokyo, Kashiwa, Chiba 277-8582, Japan3
Department of Theoretical Physics, University Autonoma Madrid, 28049 Madrid, Spain
4
Department of Physics, Boston University, Boston, Massachusetts 02215, USA
5
Department of Physics and Astronomy, University of British Columbia,
Vancouver, British Columbia V6T1Z4, Canada
6
Physics Department, Brookhaven National Laboratory, Upton, New York 11973, USA
7
Department of Physics and Astronomy, University of California,
Irvine, Irvine, California 92697-4575, USA
8
Department of Physics, California State University, Dominguez Hills, Carson, California 90747, USA
9
Department of Physics, Chonnam National University, Kwangju 500-757, Korea
10
Department of Physics, Duke University, Durham North Carolina 27708, USA
11
Junior College, Fukuoka Institute of Technology, Fukuoka, Fukuoka 811-0295, Japan
12
Department of Physics, Gifu University, Gifu, Gifu 501-1193, Japan
13
GIST College, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea
14
Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822, USA
15High Energy Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan
16
Department of Physics, Kobe University, Kobe, Hyogo 657-8501, Japan17
Department of Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan18
Department of Physics, Miyagi University of Education, Sendai, Miyagi 980-0845, Japan
19
Institute for Space-Earth Enviromental Research, Nagoya University, Nagoya, Aichi 464-8602, Japan
20
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,
Nagoya University, Nagoya, Aichi 464-8602, Japan
21
National Centre For Nuclear Research, 00-681 Warsaw, Poland
22
Department of Physics and Astronomy, State University of New York at Stony Brook,
New York 11794-3800, USA
23
Department of Physics, Okayama University, Okayama, Okayama 700-8530, Japan
24
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
25
Department of Physics, University of Regina, 3737 Wascana Parkway, Regina,
Saskatchewan S4SOA2, Canada

2470-0010=2016=94(5)=052010(32)052010-1© 2016 American Physical Society

K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)26

Department of Physics, Seoul National University, Seoul 151-742, Korea
Department of Informatics in Social Welfare, Shizuoka University of Welfare,
Yaizu, Shizuoka, 425-8611, Japan
28
Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
29
Department of Physics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
30
The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
31
Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
32
Kavli Institute for the Physics and Mathematics of the Universe (WPI), The University of Tokyo Institutes
for Advanced Study, University of Tokyo, Kashiwa, Chiba 277-8583, Japan
33
Department of Physics, University of Toronto, 60 St., Toronto, Ontario M5S1A7, Canada
34
TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia V6T2A3, Canada
35
Department of Engineering Physics, Tsinghua University, Beijing 100084, China
36
Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA(Received 23 June 2016; published 20 September 2016)27

Upgraded electronics, improved water system dynamics, better calibration and analysis techniques
allowed Super-Kamiokande-IV to clearly observe very low-energy 8 B solar neutrino interactions, withrecoil electron kinetic energies as low as ∼3.5 MeV. Super-Kamiokande-IV data-taking began in
September of 2008; this paper includes data until February 2014, a total livetime of 1664 days. The
62measured solar neutrino flux is ð2.308  0.020ðstatÞþ0.039−0.040 ðsystÞÞ × 10 =ðcm secÞ assuming no oscillations. The observed recoil electron energy spectrum is consistent with no distortions due to neutrino
oscillations. An extended maximum likelihood fit to the amplitude of the expected solar zenith angle
variation of the neutrino-electron elastic scattering rate in SK-IV results in a day/night asymmetry of
ð−3.6  1.6ðstatÞ  0.6ðsystÞÞ%. The SK-IV solar neutrino data determine the solar mixing angle assin2 θ12 ¼ 0.327þ0.026−0.031 , all SK solar data (SK-I, SK-II, SK III and SK-IV) measures this angle to beþ1.5
2−5 eV2 .
sin2 θ12 ¼ 0.334þ0.027−0.023 , the determined mass-squared splitting is Δm21 ¼ 4.8−0.8 × 10
DOI: 10.1103/PhysRevD.94.052010

I. INTRODUCTION
Solar neutrino flux measurements from SuperKamiokande (SK) [1] and the Sudbury Neutrino
Observatory (SNO) [2] have provided clear evidence forsolar neutrino flavor conversion in which electron flavor
neutrinos convert to either muon or tau flavor neutrinos. Thisflavor conversion is well described by flavor oscillations of
three neutrinos. In particular, the extracted oscillation parameters agree with nuclear reactor antineutrino measurements[3]. However, while oscillations of reactor antineutrinos at thesolar frequency were observed, there is still no clear evidence
that the solar neutrino flavor conversion is indeed due toneutrino oscillations and not caused by another mechanism.
Currently there are two types of testable signatures uniqueto neutrino oscillations, the first being the observationand precision test of the Mikheyev–Smirnov–Wolfenstein(MSW) resonance curve [4], the characteristic energy
dependence of the flavor conversion (assuming oscillation
parameters extracted from solar neutrino and reactor antineutrino measurements): higher energy solar neutrinos(higher energy 8 B and hep neutrinos) undergo adiabatic
resonant conversion within the Sun (present data imply a*

Deceased.
Also at BMCC/CUNY, Science Department, New York,New York, USA.
†survival probability of about 30%), while the flavor changesof the lower energy solar neutrinos (pp, 7 Be, pp, CNO andlower energy 8 B neutrinos) arise only from vacuum oscillations. These averaged vacuum oscillations lead to an
average survival probability which—for sufficiently small
1–3 mixing—must exceed 50% (present data imply about60%). The transition from the matter-dominated oscillations
within the Sun to the vacuum-dominated oscillations should
occur near three MeV. This makes 8 B neutrinos the best
choice when looking for a transition point within the energy
spectrum. An observed deviation from the expected behavior
in the transition region would imply new physics, e.g.
nonstandard interactions [5] or mass-varying neutrinos [6].
A second signature unique to oscillations arises from theeffect of the terrestrial matter density on solar neutrino
oscillations. This effect is tested directly by comparing solar
neutrinos that pass long distances through the Earth atnighttime to those which do not pass through the Earthduring the daytime. Those neutrinos which pass through the
Earth will generally have an enhanced electron neutrino
content, leading to an increase in the nighttime electron elastic
scattering rate (or any charged-current interaction rate), andhence a negative “day/night asymmetry” ðrD − rN Þ=rave ,where rD (rN ) is the daytime (nighttime) rate and rave ¼
1
2 ðrD þ rN Þ is the average rate. This day/night asymmetry
depends on the solar mass squared splitting and therefore

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PHYSICAL REVIEW D 94, 052010 (2016)

constitutes a measurement of this oscillation parameter. It isalso sensitive to new physics. SK is sensitive to 8 B and hep
solar neutrinos in the energy range around 4 to 18.7 MeVand
precisely measures the neutrino interaction time. It is therefore a good detector to search for both solar neutrino
oscillation signatures.
SK [7] is a large, cylindrical, water Cherenkov detector
containing 50,000 tons of ultrapure water. It is located1,000 m beneath the peak of Mount Ikenoyama, in Kamioka
Town, Japan. The SK detector is optically separated into a
32.5 kton cylindrical inner detector (ID) surrounded by a
∼2.5 meter water shield, ∼2 m of which is the active vetoouter detector (OD). The structure dividing the detector
regions contains an array of photomultiplier tubes (PMTs).
SK started data-taking in April of 1996, with 11,146 ID and1,885 OD PMTs, and was then shut down for maintenance inJune of 2001. This period is called SK-I [1]. While refilling
the tank with water in November of 2001, a PMT implosion
caused a chain reaction which destroyed 60% of the PMTs.
The surviving and new PMTs were redistributed andcovered with fiber-reinforced plastic (FRP) and acrylic
cases, in order to avoid another accidental chain reaction.
Data-taking restarted with 5,182 ID and 1,885 OD PMTs in
December of 2002, and the period until October of 2005 iscalled SK-II [8]. In October of 2006, newly manufactured
PMTs replaced those which had been destroyed, and with
11,129 ID and 1,885 OD PMTs data-taking resumed as the
SK-III phase [9]. The fourth phase of SK (SK-IV) began in
September of 2008, with new front-end electronics (QTC
Based Electronics with Ethernet, QBEE [10]) for both the ID
and OD, new data acquisition system, and continues to thisday. This paper will include data taken up until the beginningof February 2014.
Improvements in the front-end electronics, the water
circulation system, calibration techniques and the analysis
methods have allowed the SK-IV solar neutrino measurements to be made with a lower energy threshold and smaller
systematic uncertainties, compared to SK-I, II and III. The
hardware and software improvements are summarized inSec. II, while the SK-IV data set, data reduction, and its
systematic uncertainty estimations on the total flux aredetailed in Sec. III. The simulation of solar neutrino events
in SK is described also in Sec. III. Unfortunately, thesimulation code for the SK-III period used in [9] wasinaccurate, which affected the input recoil electron spectrum.
The details (and the correction applied) as well as a reanalysisof the SK-III data are briefly described in Sec. III andAppendix A.
In Sec. IV, the energy spectrum results of SK-IVas well as
all SK phases combined are discussed. Section V presents
the SK-IV day/night asymmetry analysis. Finally, Sec. VI
contains an oscillation analysis of SK-IV data by themselves
and in combination with other SK phases, and also a global
analysis which combines the SK results with other relevant
experiments.

In previous SK solar neutrino publications [1,8,9]“energy” meant total recoil electron energy, while in thispaper we subtract the electron mass me ¼ 511 keV toobtain kinetic energy. The kinetic energy threshold of the
SK-IV data analysis is thus 3.49 MeV, corresponding to thetotal energy of 4.00 MeV.
II. DETECTOR PERFORMANCE
A. Electronics, data acquisition system
To ensure stable observation and to improve the sensitivity of the detector, new front-end electronics called
QBEEs were installed, allowing for the development of
a new online data acquisition system. The essential
components on the QBEEs used for the analog signal
processing and digitization are the QTC (high-speed
charge-to-time converter) ASICs [10], which achieve very
high speed signal processing and allow the integration of
the charge and recording of the time of every PMT signal.
These PMT signal times and charge integrals are sent toonline computers, where a software trigger searches fortiming coincidences within 200 ns to pick out events in asimilar fashion as the hardware “hitsum trigger” did in SK-I
through III [1,8,9]. The energy threshold of this coincidence trigger is determined by the number of coincident
PMT signals that are required: a smaller coincidence level
will be more sensitive to lower energy events, but will result
in larger event rates. The definitions of the different trigger
types and the corresponding typical event rates are summarized in Table I. Since all PMT signals are digitized andrecorded, there is no deadtime of the detector from a large
trigger rate, so the efficiency of triggering on HE events
does not limit the maximum possible rate of SLE triggers;
only the processing capability of the online computers
limits this maximum rate. The software trigger system uses
flexible event time periods (1.3 μ sec for SLE, 40 μ sec for
LE and HE). The trigger efficiencies for the thresholds are∼84% (∼99%) between 3.49 and 3.99 MeV (3.99 and4.49 MeV) and 100% above 4.49 MeV.
B. Water system
To keep the long light attenuation length of the SK water
stable, the water is continuously purified with a flow rate of
60 ton=hour. Purified water supplied to the bottom of thedetector replaces water drained from its top. A higher
temperature of the supply water than the detector temperature
TABLE I. Normal data-taking trigger types along with thethreshold of hits and average trigger rates.
Trigger type
Super low energy (SLE)Low energy (LE)High energy (HE)052010-3

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PHYSICAL REVIEW D 94, 052010 (2016)250

C. Event reconstruction
The methods used for the vertex, direction, and energy
reconstructions are the same as those used for SK-III [9].
The Cartesian coordinate system for the SK detector isshown in Fig. 1.
1. Vertex

200
150
100
50
0

5

7.5
10
12.5
True Electron Energy (MeV)

15

FIG. 2. Vertex resolution for SK-I, II, III, and IV shown by thedotted, dashed-dotted, dashed, and solid lines, respectively. The
SK-III vertex resolution improvement over SK-I comes fromusing an improved vertex reconstruction while the slightly
improved timing resolution and better agreement between dataand simulated events are responsible for the further improvementin SK-IV.

event directions. During the SK-III phase an energy
dependence was included in the likelihood and the angularresolution was improved by about 10% (10 MeV electrons)compared to SK-I. The angular resolution in SK-IV issimilar to that in SK-III.
3. Energy
The energy reconstruction is based on the number of PMT
hits within a 50 ns time window, after the photon travel timefrom the vertex is subtracted. This number is then corrected
for water transparency, dark noise, late arrival light (due toscattering and reflection), multiphoton hits, etc., producing aneffective number of hits N eff (see [9]). Simulations ofmonoenergetic electrons are used to produce a function
relating N eff to the recoil electron energy (MeV).

10 cm

10

z (m)

The vertex reconstruction is a maximum likelihood fit to
the arrival times of the Cherenkov light at the PMTs [8].
Figure 2 shows the vertex resolution for each SK phase. The
large improvement in SK-III compared to SK-I is the result ofusing an advanced vertex reconstruction program, while theimproved timing resolution and slightly better agreement of
the timing residuals between data and Monte Carlo (MC)simulated events are responsible for the additional improvement of SK-IV. We observed a bias in the reconstructed vertex
called the vertex shift. This vertex shift is measured with agamma-ray source at several positions within the SK detector:
neutrons from spontaneous fission of 252 Cf are thermalized inwater and then captured on nickel in a spherical vessel [7,11].
The nickel then emits 9 MeV gammas (Ni calibration source).
Figure 3 shows the shift of the reconstructed vertex of these Ni
gammas in SK-IV from their true position (assumed to be thesource position). The SK-IV vertex shift is improved compared with SK-I, II, and III [7–9].

Vertex Resolution (cm)results in convection throughout the detector volume. This
convection transports radioactive radon gas, which is produced by radioactive decays from the U/Th chain near theedge of the detector into the central region of the detector.
Radioactivity coming from the decay products of radon gas(most commonly 214 Bi beta decays) mimics the lowest energy
solar neutrino events. In January of 2010, a new automatedtemperature control system was installed, allowing for controlof the supply water temperature at the 0.01 degree level. By
controlling the water flow rate and the supply water temperature with such high precision, convection within the tank is
kept to a minimum and the background level in the central
region has since become significantly lower.

2. Direction
A maximum likelihood fit comparing the Cherenkov
ring pattern of data to MC simulations is used to reconstruct

0

-10

0

5

10

15

r (m)

FIG. 1.

Definition of the SK detector coordinate system.

FIG. 3. Vertex shift of the Ni calibration events in SK-IV.
The start of the arrow is at the true Ni-Cf source position and the
direction indicates the averaged vertex shift at that position. The
length of the arrow indicates the magnitude of the vertex shift. To
make the vertex shifts easier to see this length is scaled up by afactor of 20.

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PHYSICAL REVIEW D 94, 052010 (2016)

140
130
120
110

38
37.5
37
36.5
36

2010

2012
Year

2014

FIG. 4. Top: Time variation of the water transparency asmeasured by decay electrons. Bottom: Time variation of themean reconstructed energy of μ decay electrons before (after)
water-transparency correction in triangles (circles). Before
the correction, a water transparency of 90 m is assumed, thenthe mean value of the distribution is adjusted to that of the
distribution after the correction. After the correction the meanenergy is stable within 0.5% (dashed lines).

The water transparency parameter used in the energy
reconstruction is measured using decay electrons from raymuons. This method of obtaining the water transparency is thesame as for SK-I, II, and III [1,8,9]: exploiting the azimuthal
symmetry of the Cherenkov cone, we determine the lightintensity as a function of light travel distance and fit it with an
exponential light attenuation function. The top panel of Fig. 4
shows the time variation of the measured water transparency,
while the bottom panel shows the reconstructed mean energyof μ decay electrons in triangles (circles) before (after) watertransparency corrections have been applied. The stability of
the water transparency corrected energy reconstruction iswithin 0.5% (dashed lines).
4. Multiple scattering goodness (MSG)
Even at the low energies of the recoil electrons from 8 B
solar neutrino-electron scattering, the PMT hit pattern fromthe Cherenkov cone reflects the amount of multiple Coulomb
scattering recoil electrons experience. Very low-energy
electrons will incur such scattering more than higher energy
electrons and thus have a more isotropic PMT hit pattern.
Radioactive background events, such as 214 Bi beta decays,
generally have less energy than 8 B recoil electrons.
Radioactive background events with γ emission will be moreisotropic still. The “goodness” of a directional fit characterizes this hit pattern anisotropy: it is constructed by firstprojecting 42° cones from the vertex position, centeredaround each PMT that was hit within a 20 ns time window(after time of flight subtraction). Pairs of such cones are thenused to define “event direction candidates,” which are vectors
along the intersection lines of the two cones. Only cone pairs
which intersect twice are used to define event direction
candidates. Figure 5 shows a schematic view of how the event
direction candidates are found. The gray points represent hit

FIG. 5. Schematic view of the event direction candidates used
to calculate the multiple scattering goodness. The gray points
represent PMT hits and the dotted circles surrounding them are
the projections of the 42° cones centered around each hit. The
black crosses give the intersection points of the cones. The
vectors from the event vertex position to these intersection points
are taken as event direction candidates. The black dot shows theevent best fit direction and the black solid circle is the projectionof its Cherenkov cone onto the inner detector wall. Theintersections will cluster around the event direction.

PMTs, which will roughly be found around the Cherenkov“ring,” the projection of the cone onto the inner detector wallshown by the gray circle. As seen in the figure, for pairs of
PMTs with positions located near the Cherenkov ring, one of
the intersection lines shown by the black crosses will fallclose to the best fit direction vector shown as the black pointon the inner detector wall which this vector passes through.
Clusters of these event direction candidates are then found by
associating other event direction candidates which are within
50° of a “central event direction” seeded by the candidates
themselves. Once an event direction candidate has been
associated to a cluster, it then will not seed another cluster.
The event direction candidate vectors of a cluster are added
together to adjust the central event direction. Several iterations of this adjustment with subsequent cluster reassignment will center the clusters and maximize the magnitude of
the vector sum. The vector sum with the largest magnitude is
kept as the “goodness direction.” The multiple scattering
goodness (MSG) is then defined by the ratio of thismagnitude and the number of event direction candidates
within the 20 ns time window. Electrons undergoing more
multiple Coulomb scattering will have a lower MSG value
than those undergoing less. The filled squares (error bars) andsolid (dotted) lines of Fig. 6 compare the LINAC data and
MC MSG distributions for 4.38 MeV (8.16 MeV) electrons.
As expected, higher energy electrons have a larger mean
MSG since they undergo less multiple Coulomb scattering.
D. Energy calibration
The absolute energy scale is determined by an electronlinear accelerator (LINAC) [12]. The LINAC calibration

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1.5

0.04

1
(MC-Data)/Data (%)

0.03

0.02

0.01

0.5
0
-0.5
(x,y)=(−12.4,−0.7)m(x,y)=(−3.9,−0.7)m(x,y)=(0.4,−12.0)m

-1

0

0

0.2
0.4
0.6
0.8
Multiple Scattering Goodness

1

-1.5
-15

FIG. 6. MSG for LINAC data (points) and MC (histogram),
normalized by the number of events. The solid (dotted) lines andpoints on that correspond to 4.38 MeV (8.16 MeV) electrons.

system injects single monoenergetic electrons into SK in
the downward direction. The energy of the momentumselected electrons is precisely measured by a germanium(Ge) detector using a thin titanium window similar to thatused under the water. To determine the energy scale, 6.28
and 12.93 MeV electron data are compared to simulated
events. Figure 7 shows the z dependence of this comparison. We cross check the energy scale obtained from the
LINAC energy with 16 N β=γ decays, which originate fromthe (n,p) reaction of 16 O with neutrons produced by adeuterium-tritium (DT) fusion neutron generator [13]. The
10.5 MeV endpoint 16 N decays of the DT calibration are
isotropic, with 66% of the decays emitting a 6 MeV γ inconjunction with an electron. DT-produced 16 N data aretaken at a much larger number of positions in SK than
LINAC data. Figure 8 compares the reconstructed energyof 16 N simulated events with data, as a function of the z
position of the production. The observed dependence on z
is probably due to an imperfect model of the z dependence

-10

-5

0
z (m)

(x,y)=(−0.4,12.0)m(x,y)=(0.4,−0.7)m(x,y)=(11.0,−0.7)m

5

10

15

FIG. 8. Difference of the mean reconstructed energy between
data and simulated events, at each position, coming from the
SK-IV DT calibration.

of the optical parameters (see subsection II E). Figure 9shows the directional dependence of the energy scale, withrespect to the detector zenith angle. The two bins betweencos θzSK ¼ 0.6 and 1 are affected by increased shadowing
from the DT generator. Conservatively, we fit the entire
data with a linear combination of a constant and an
exponential function to estimate the systematic uncertainty
on the day/night asymmetry due to the directional dependence of the bias of the reconstructed energy.
The systematic uncertainty of the energy scale due toposition (direction) dependence is estimated to be 0.44%(0.1%). The effect of the water transparency variation
during LINAC calibration is estimated to be 0.2%, while
the uncertainty of the LINAC electron beam energy (asmeasured by the Ge detector), is estimated to be 0.21%.
The total systematic uncertainty of the absolute energy
0.45
0.4

(x,y)=(-12.4,-0.7)m, E=12.93 MeV (x,y)=(-3.9,-0.7)m, E=12.93 MeV

0.35(x,y)=(-3.9,-0.7)m, E=6.28 MeV(MC-Data)/Data (%)(x,y)=(-12.4,-0.7)m, E=6.28 MeV(MC-Data)/Data (%)

1

0

0.3
0.25
0.2
0.15
0.1
0.05
0
-1 -0.8 -0.6 -0.4 -0.2

-1

-10

0
z (m)

0 0.2 0.4 0.6 0.8
cosθzSK

1

10

FIG. 7. LINAC calibration z position dependence of theabsolute energy scale of SK-IV.

FIG. 9. Difference of the mean energy between data andsimulated events as a function of the zenith angle in the
SK-IV detector for DT calibration. After subtracting the absoluteoffset, the uncertainty is estimated to be 0.1%.

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scale thus becomes 0.54%, calculated by adding all thecontributions in quadrature, and is summarized in Table II.
These uncertainties are similar to those in SK-III (0.53%).
The detector’s energy resolution is determined using thesame method as described in [9]. Monoenergetic electrons
are simulated and used to determine the relationshipbetween the effective number of hits in the detector andthe electron energy in MeV. Using the width of Gaussianfits to the energy distributions resulting from these simulated electrons, the energy dependence of the energy
resolution is well described by the functionpﬃﬃﬃﬃ
σðEÞ ¼ −0.0839 þ 0.349 E þ 0.0397E;

ð2:1Þ

in units of MeV, where E is electron total energy. This iscomparable to the SK-III
pﬃﬃﬃﬃ energy resolution, given asσðEÞ ¼ −0.123 þ 0.376 E þ 0.0349E in [9].
E. Light propagation in water
1. Water parameters
The water transparency in the MC simulation is determined using absorption and scattering coefficients as afunction of wavelength (full details of this and other moregeneral detector calibrations can be found in [11]). These
coefficients are independently measured by a nitrogen laser
and laser diodes at five different wavelengths: 337 nm,
375 nm, 405 nm, 445 nm, and 473 nm. Based on thesemeasurements, the dominant contribution to the variation of
the water transparency is a variation in the absorption
length. The absorption coefficient is time and position
dependent, as explained below. This SK-IV solar neutrino
analysis only varies the absorption, and uses a single set oftime independent scattering coefficients, as measured by
the laser diodes [11].

the decay electron water transparency parameter as afunction of time.
In order to connect the absorption time dependence in the
MC to the water transparency parameter measured bydecay electrons we generate monoenergetic electron samples throughout the detector for a wide range of absorption
coefficients with nine different energies between 4 and50 MeV. Each MC sample is assigned a particular decay
electron water transparency parameter that minimizes thedifference between input energy and average reconstructedenergy. As expected, the relationship between water transparency and MC absorption coefficient does not significantly depend on the generated energy. The same procedure
establishes the relationship between the (corrected) numberof PMT hits and energy. Figure 10 shows the obtained
relationship between absorption coefficient and watertransparency parameter. For convenience we measure theabsorption coefficient relative to the coefficient at the timeof the LINAC calibration data-taking, which defines theenergy scale (see [11]). We employ a linear interpolation
between the data points. The mean energy of these decay
electrons is used to evaluate the systematic uncertainty ofthe time dependence of the energy scale (see bottom panelof Fig. 4). After correction for the time variation of theabsorption coefficient, the apparent time dependence ofthe μ decay electron mean energy becomes smaller than
0.5%.
3. Position dependence
As already explained, the water in the SK detector iscontinuously recirculated through the SK water purification
system. Water is drained from the detector top, purified,and reinjected at the bottom. Due to careful temperature
control of the injected water, the convection inside the SK
tank is suppressed everywhere but at the bottom part of thetank below z ¼ −11 m. Figure 11 shows the typical water

2. Time dependence

1.8

To track the absorption time dependence, we measure thelight attenuation of Cherenkov light from decay electrons(from cosmic-ray muons stopping throughout the SK innerdetector volume). This measurement uses the azimuthal
symmetry of the emitted Cherenkov cone to compare
different light propagation path lengths within the same
event and assumes a simple exponential attenuation. Thiseffective attenuation length is one of the energy
reconstruction parameters. The top panel of Fig. 4 shows
TABLE II.

Relative Absorption Coeff.

1.6
1.4
1.2
1
0.8
0.6
0.4

Systematic uncertainty of the energy scale.

Position dependence
Direction dependence
Water transparency
LINAC energy

0.44%
0.10%
0.20%
0.21%

Total

0.54%

0.2

11000

12000

13000
WT (cm)

14000

15000

FIG. 10. Change in the absorption coefficient, relative to thecoefficient when the absolute energy scale calibration was done, asa function of the μ decay electron measured water transparency.

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PHYSICAL REVIEW D 94, 052010 (2016)

includes data taken from October 6, 2008 until February 1,
2014. The total livetime is 1664 days. The entire data
period was taken with a new very low energy threshold of
34 hits within 200 ns (cf. Table I). To reduce the required
data storage capacity, obvious backgrounds are removed
using faster and less-stringent implementations of theanalysis cuts on fiducial volume, energy, ambient eventsand external events, before the data is permanently stored.
By applying these precuts, the data load was reduced to∼1% of its original size.

13.25

o

Temperature ( C)

13.3

13.2

13.15

13.1

A. Event selection
-10

0
z (m)

10

FIG. 11. Typical z dependence of the water temperature in SK
detector. Below −11 m the temperature is constant due toconvection, and so the absorption coefficient is assumed to beconstant below this point.

temperature as a function of z in the SK detector. The
temperature is uniform below z ¼ −11 m, where convection is occurring and increases steadily above that. We
assume that absorption is strongly correlated with theamount of convection and model the position dependenceof the absorption length as constant below −11 m andlinearly changing above −11 m:
αabs ðλ; z; tÞαðλ; tÞð1 þ βðtÞ · zÞ; for z ≥ −11 m¼αðλ; tÞð1 − βðtÞ · 11Þ; for z ≤ −11 m;ð2:2Þ

where β parametrizes the z-dependence of the absorption.
The β parameter is determined by studying the distributionof hit PMTs of Ni calibration data (see Sec. II C) [11] in the“top,” “bottom,” and “barrel” regions of the detector (seeFig. 1). After other detector asymmetries like quantum
efficiency variations of the PMTs are taken into account,the hit rate of the top region in the detector is 3 ∼ 5% lower
than that of the bottom region. β is then fit using the hit
asymmetry of Ni calibration events. Since the Ni calibration hit pattern varies with time, both α and β depend ontime. The Xe flash lamp scintillator ball calibration system[11] tracks the β time dependence: a Xe flash lamp powers
a scintillator ball located near the middle of the detector.
The time dependence of β is also monitored by Ni
calibration data. The introduction of β into the MC
simulation has helped to reduce the systematic uncertainty
on the energy scale, as it addresses a significant contribution to its directional dependence. This is important for thesolar neutrino day/night asymmetry analysis.
III. DATA ANALYSIS
After installation of the new front-end electronics, SK-IV
physics data-taking started on October 6, 2008. This paper

Most of the cuts used are the same as those used in
SK-III [9], but some of the cut values and the energy
regions in which they are applied are changed to optimize
the significance: if S (BG) is the number of signal (backpﬃﬃﬃﬃﬃﬃﬃ
ground) events, we define the significance as S= BG.
Also, as was the case in SK-III, below 4.99 MeV thefiducial volume is reduced since backgrounds appear
localized at the bottom of the detector and at large radii.
1. Ambient background reduction
As in [1,8,9], several cuts remove low-energy radioactive
backgrounds. These backgrounds originate mostly from the
PMT enclosures, the PMT glass, and the detector wall
structure. While the true vertices lie outside the fiducial
volume, some radioactive background events are misreconstructed inside the fiducial volume. The quality of theevent reconstruction is tested by variables describing its
goodness. The first variable is a timing goodness gt testing
the “narrowness” of the PMT hit timing residuals, which isdefined in [8] [Sec. III B, Eq. (3.1)]. The second is a hit
pattern goodness gp testing the azimuthal symmetry of theCherenkov cone (gp ¼ 0 is perfectly symmetric, gp ¼ 1 iscompletely asymmetric). Good single electron events musthave g2t − g2p greater than 0.2. Events below 6.99 MeV(4.99 MeV) must have g2t − g2p greater than 0.25 (0.29). The
same cut was applied for SK-III.
We also check the consistency of the observed light patternwith a single 42° Cherenkov cone as in [1] [Sec. VII C,
Eq. (7.4)]. This cut will remove events with multiple
Cherenkov cones, e.g., from beta decays to an excited nuclear
state with subsequent gamma emission. The hit pattern is
assigned a likelihood based on the direction fit likelihood
function. Figure 12 shows the likelihood and cut criteria inthree different energy ranges. Further details are found in [14].
A small hit cluster cut targets radioactive background
events in the PMT enclosures or glass, which coincide withan upward fluctuation of the PMT dark noise. Only eventswith a reconstructed r2 bigger than 155 m2 (120 m2 ), areconstructed z smaller than −7.5 m (−3 m), or a reconstructed z bigger than 13 m, for reconstructed energies in
4.49–4.99 MeV (3.49–4.49 MeV), are subject to this cut.
To characterize small hit clusters, we select PMT hits with

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0.9

300
250

>11.5MeV

MC
4000

0.8

200

3500

0.7

150

3000

eff

0.6
20ns

/N

100

N

50
0
2500

7.49-11.5MeV

1500
1000

0.2

1500

500

0.1
0.9

1000

Data

0.8

500

-2.4

2000
0.4
0.3

2000

0.7

0
25000
20000

0.6

20ns

/N

eff

5.99-7.49MeV

N

4000
3500
3000
2500
2000
1500
1000
500

2500
0.5

15000
0.5
0.4

10000

0.3
5000
0.2

-2.3

-2.2

-2.1

-2

-1.9

-1.8

-1.7

-1.6

-1.5

0.1

-1.4

L

0

200

400

600

800 1000 1200 1400

0radius [cm]pattern

FIG. 12. Left: Hit pattern likelihood distributions in three different energy ranges for data (black error bars) and MC (red histogram).
The cut point is shown by the blue dashed line. Right: Removal of small hit clusters. The top panel shows the MC cluster size vs thecluster radius, the bottom panel is the data. Events below the dashed black line are removed.

times coincident within 20 ns (after time-of-flight subtraction, see Sec. II C 3), and then find the smallest spherearound any of the selected PMTs that encloses at least 20%of all selected PMTs. This radius is multiplied by the ratioof PMT hits coincident within 20 ns (without time-of-flightcorrection) divided by N eff (see Sec. II C 3). Solar neutrinos
near the edge of the fiducial volume have a bigger radius ×
hit ratio (see also Sec. III C in [9], Fig. 17 and 18) than theradioactive background. As in SK-III, we remove eventswith radius × hit ratio less than 75 cm as shown in Fig. 12.
Finally, we remove spurious events due to various calibration sources (mostly radioactive decays), if they are below
TABLE III. Locations used by the calibration source cut. The
sources are described in detail in [11].
Sourcex (cm)y (cm)z (cm)

Xenon flasher
LED
TQ Diffuser Ball
DAQ Rate Test Source
Water Temp. Sensors 1
Water Temp. Sensors 2

353.5
35.5
−176.8
−35.3
−35.3
70.7

−70.7
−350.0
−70.7
353.5
1200
−777.7

0.0
150.0
100.0
100.0
−2000
−2000

4.99 MeV. A reconstructed position closer than 2 m to thesource, or closer than 1 m to the source or water temperature
sensor cable (all cables run along the z axis from the top down
to the source position) means the event is removed. The loss inthe fiducial volume is about 0.48 kton. Table III lists thevarious calibration sources which are considered.
2. External event cut
To remove radioactive background coming from the
PMTs or the detector wall structure, we calculate thedistance to the PMT-bearing surface from the reconstructedvertex looking back along the reconstructed event direction.
Radioactive backgrounds tend to appear “incoming,” so weremove events where this distance is small. Solar neutrino
candidates above 7.49 MeV (above 4.99 MeV and below
7.49 MeV) must have a distance of at least 4 m (6.5 m). In
the energy region below 4.99 MeV we distinguish between
the “top” (cylinder top lid), “barrel” (cylinder side walls)
and “bottom” (cylinder bottom lid) surfaces, shown in
Fig. 1. Candidates which come from the “top” (“bottom”)
must have a distance of at least 10 m (13 m), while “barrel”
event candidate distances must exceed 12 m. SK-III applied
the same cuts.

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3. Spallation cut
Some cosmic-ray μ’s produce radioactive elements bybreaking up an oxygen nucleus [15]. A spallation event
occurs when these radioactive nuclei eventually decay andemit β’s and/or γ’s. A spallation likelihood function is madefrom the distance of closest approach between the preceding μ track(s) and a solar neutrino candidate, their timedifference, and the charge deposited by the preceding μðsÞ.
By using the likelihood function spallation-like events are
rejected, see [1,16] for details.
When lower energy cosmic-ray μ− ’s are captured by 16 O
nuclei in the detector, 16 N can be produced which decays
with gamma-rays and/or electrons with a half-life of
7.13 sec. In order to reject these events, the correlation
between stopping μ’s in the detector and the remaining
candidate events are checked. The cut criteria for 16 N eventsis as follows; (1) reconstructed vertex is within 250 cm to thestopping point of the μ, (2) the time difference is between
100 μ sec and 30 sec.
To measure their impact on the signal efficiency, thespallation and 16 N cuts are applied to events that cannot becorrelated with cosmic-ray muons (e.g. candidates preceding muons instead of muons preceding candidates). This“random sample” then measures the accidental coincidences rate between the muons and subsequent candidate
events. The spallation (16 N) cut reduces signal efficiency by
about 20% (0.53%).
4. Fiducial volume cut
Events which occur near the wall of the detector(reconstructed within 2 m from the ID edge) are rejected.
The volume of this fiducial volume is 22.5 kton. Below
4.99 MeV this cut is tightened. Figure 13 shows the r2
ð¼x2 þ y2 Þ vs z data vertex distribution for 3.49 to
3.99 MeV, after the above cuts. Each bin shows the rate(events/day/bin), with blue showing a lower rate and red a

150
× jz − 4.25j4 ≤ 150;
ð3:1Þ11.754
where the coordinates are given in meters. This function waschosen in order to approximately follow the contours ofconstant event rate. For the energy range of 4.49 to 4.99 MeV,
events which have r2 > 180 m2 or z < −7.5 m are cut.
r2 þ

5. Other cuts
Short runs (<5 minutes), runs with hardware and/or
software problems, and calibration runs are not used for thisanalysis. Cosmic-ray μ events are removed by rejecting
events with more than 400 hit PMTs, which corresponds toabout 60 MeV for electron type events.
6. Summary
Figure 14 shows the energy spectrum after each reduction step and Fig. 15 shows the reduction efficiency of thecorresponding steps. The final sample of SK-IV data isshown by the filled squares and for comparison the SK-III
final sample is superimposed (dashed lines). Above
5.99 MeV, the efficiency for solar neutrinos in the final
sample is almost the same as in SK-III, while for 4.99 to
5.99 MeV, the SK-IV efficiency is better than SK-III. The
reason for the improvement is the removal of a fiducial
104
103

0.25

5

0.2

0

0.15

-5

0.1

-10

0.05

-15
0

50

100

150

200

events/day/bin

10

Events/day/kt/0.5MeV

0.3

15

z[m]

higher rate. We expect solar neutrino events to be uniformly
distributed throughout the detector volume, and the regionswith high event rates are likely dominated by background.
To increase the significance in the final data sample for thisenergy region (3.49 to 4.49 MeV), we have reduced thefiducial volume to the region shown by the black line in the
figure and described by

102
10
1
10-1
10-2
10-3

5

7.5
10
12.5
Recoil Electron Kinetic Energy (MeV)0r2[m2]

FIG. 13. Vertex distribution for 3.49 to 3.99 MeV data.
Radioactive background leads to a large event rate at the bottom
and large radii. The black line indicates the reduced fiducial
volume in this energy region.

FIG. 14. Energy spectrum after each reduction step in the22.5 kton fiducial volume. The open circles (filled inverted
triangles) correspond to the reduction step after the spallation(ambient) cut. The stars give the spectrum after the external event
cut, and the final SK-IV sample after the tight fiducial volume cut
is given by the filled squares. The dashed line shows the final
sample of SK-III.

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100

cm /MeV)

1

75

2
-46

dσ/dTe (x10Signal Efficiency

0.75

0.5

0.25

50

25

00
0

5

2.5

5

7.5

10

Recoil Electron Kinetic Energy(MeV)

7.5
10
12.5
Recoil Electron Kinetic Energy (MeV)

FIG. 15. Signal efficiency after each reduction step. The open
stars are the trigger efficiency in the 22.5 kton fiducial volume,the open circles (filled inverted triangles) correspond to the
reduction step after the spallation (ambient) cut. The filledsquares give the final reduction efficiency, with the step fromthe filled circles (after the external event cut) to the filled squares
indicates the reduction in fiducial volume at low energy. The
dashed line shows the efficiency of SK-III.

volume cut based on the “second vertex fit” [1,9] andmaking a looser ambient event cut. The reduced fiducial
volume and a tighter ambient event cut for 3.49 to 4.99 MeV
results in a lower efficiency than SK-III, but in exchange thebackground level has been reduced by ∼40%.

FIG. 16. Differential cross section of ðνe ; eÞ (solid) and ðνμ;τ ; eÞ(dashed) elastic scattering for the case of 10 MeV incident
neutrino energy.

where the ratio dσ νe =dEe and dσ νμ;τ =dEe depends on therecoil electron energy Ee. Figure 16 shows the differential
cross section of ðνe ; e− Þ (solid) and ðνμ;τ ; e− Þ (dashed)elastic scattering, for the case of 10 MeV incident neutrinoenergy. This recoil electron energy dependence of the crosssection was accidentally omitted in the SK-III flux calculation in [9]. Therefore, wrong recoil electron kinematics
were generated for the SK-III analysis, primarily affecting
the lowest energy. We reanalyzed SK-III with the correctenergy dependence (leaving everything else unchanged),
the results of which can be found in Appendix A.

B. Simulation of solar neutrinos

C. Total flux

There are several steps in simulating solar neutrino
events at SK: generate the solar neutrino fluxes and crosssections, determine the recoil electron kinematics, track theCherenkov light in water and simulate the response of the
PMTs and electronics. We used the 8 B solar neutrino
spectrum calculated by Winter et al. [17] and the hep solar
neutrino spectrum. The systematic uncertainties from theseflux calculations are incorporated in the energy-correlated
systematic uncertainty of the recoil electron spectrum. The
simulated event times are chosen according to the livetime
distribution of SK-IV so that the solar zenith angle
distribution of the solar neutrinos is reflected correctly
across the simulated events. The recoil electron energy
spectrum is calculated by integrating the differential crosssection between zero and T max . T max is the maximum
kinetic energy of the recoiling electron, which is limited by
the incident neutrino energy.
Because νe ’s scatter via both W  and Z0 exchange, whileνμ;τ ’s interact only in the neutral-current channel, theðνe ; e− Þ cross section is approximately six times larger
than ðνμ;τ ; e− Þ. For the total and differential cross sections of
those interactions, we adopted the calculation from [18], inwhich the radiative corrections are taken into account and−In the case of ðν; e Þ interactions of solar neutrinos in SK,
the incident neutrino and recoil electron directions arehighly correlated. Figure 17 shows the cos θsun distribution
for events in the energy range 3.49 to 19.5 MeV, as well as
the definition of cos θsun . In order to obtain the number ofsolar neutrino interactions, an extended maximum likelihood fit is used. This method is also used in the SK-I [1], II[8], and III [9] analyses. The likelihood function is defined as
L ¼ e−ðPiBi þSÞ

ni
N bin YYðBi · bij þ S · Y i · sij Þ;ð3:2Þ

i¼1 j¼1

where N bin is the number of energy bins. The flux analysis of
SK-IV has N bin ¼ 23 energy bins; 20 bins of 0.5 MeV width
between 3.49 and 13.5 MeV, two energy bins of 1 MeV
between 13.5 MeV and 15.5 MeV, and one bin between
15.5 MeVand 19.5 MeV. ni is the number of observed eventsin the ith energy bin. S and Bi , the free parameters of thislikelihood function, are the number of solar neutrino
interactions in all bins and the number of background eventsin the ith energy bin, respectively. Y i is the fraction of
signal events in the ith energy bin, calculated from solar
neutrino simulated events. The background weights

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PHYSICAL REVIEW D 94, 052010 (2016)

Events/day/kton/bin

0.3

Vertex shift systematic uncertainty [%]K. ABE et al.
zθsunrrec θz
0.2

rν
Sun

0.1

0-1

-0.5

0

0.5

1

cosθsunFIG. 17. Solar angle distribution for 3.49 to 19.5 MeV. θsun is
the angle between the incoming neutrino direction rν and thereconstructed recoil electron direction rrec . θz is the solar zenith
angle. Black points are data while the histogram is the best fit to
the data. The dark (light) shaded region is the solar neutrinosignal (background) component of this fit.
sun
bij ¼ βi ðcos θsunij Þ and the signal weights sij ¼ σðcos θij ;Eij Þ are calculated from the expected shapes of the background and solar neutrino signal, respectively (probability
density functions). The background shapes βi are based on
the zenith and azimuthal angular distributions of real data,
while the signal shapes σ are obtained from the solar
neutrino simulated events. The values of S and Bi are
obtained by maximizing the likelihood. The histogram ofFig. 17 is the best fit to the data, the dark (light) shaded
region is the solar neutrino signal (background) componentof that best fit. The systematic uncertainty for this method ofsignal extraction is estimated to be 0.7%.

1. Vertex shift systematic uncertainty
The systematic uncertainty resulting from the fiducial
volume cut comes from event vertex shifts. To calculate theeffect on the elastic scattering rate, the reconstructed vertex
positions of solar neutrino MC events are artificially shifted
following the arrows in Fig. 3, and the number of events
passing the fiducial volume cut with and without theartificial shift are compared. Figure 18 shows the energy
dependence of the systematic uncertainty coming from theshifting of the vertices. The increase below 4.99 MeV
comes from the reduced fiducial volume (smaller surface tovolume ratio), not from an energy dependence of the vertex
shift. The systematic uncertainty on the total rate is 0.2%.
2. Trigger efficiency systematic uncertainty
The trigger efficiency depends on the vertex position,water transparency, number of hit PMTs, and response of
the front-end electronics. The systematic uncertainty

0.5
0.4
0.3
0.2
0.1
0

4

6
8
10
12
14
16
18
Recoil electon kinetic energy [MeV]

FIG. 18. Vertex shift systematic uncertainty on the flux. The
increase below 4.99 MeV comes from the tight fiducial volume
cut. (see text).

from the trigger efficiency is estimated by comparing
Ni-calibration data (see Sec. II C) with MC simulation. For 3.49–3.99 MeV and 3.99–4.49 MeV, the difference between data and MC is −3.43  0.37% and−0.86  0.31%, respectively [14]. Above 4.49 MeV thetrigger efficiency is 100% and its uncertainty is negligible.
The resulting total flux systematic uncertainty due to thetrigger efficiency is 0.1%.
3. Angular resolution systematic uncertainty
The angular resolution of electrons is defined as theangle which includes 68% of events in the distribution of
the angular difference between their reconstructed direction
and their true direction. The MC prediction of the angularresolution is checked and the systematic uncertainty is
estimated by comparing the difference in the reconstructed
and true directions of LINAC data and LINAC (see [12])simulated events. This difference is shown in Table IV forvarious energies. To estimate the systematic uncertainty on
the total flux, the signal shapes sangþ
and sang−
are varied byij
ij
shifting the reconstructed directions of the simulated solar
neutrino events by the uncertainty in the angular resolution.
These new signal shapes are used when extracting the total
TABLE IV. Angular resolution difference between LINAC dataand simulated LINAC events for each SK phase. The energy
refers to the electron’s in-tank kinetic energy.
Energy (MeV)

SK-I(%)

SK-II(%)

SK-III(%)

SK-IV(%)

4.0
4.4
5.3
6.3
8.2
10.3
12.9
15.6
18.2


−1.64
−1.38
2.32
2.33
1.52
1.07
0.88





5.93
7.10

6.50




0.74


0.40

−0.27
0.39


0.64
0.68

0.02
0.06

0.22

0.31

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flux, and the resulting 0.1% change in the extracted
flux is taken as the systematic uncertainty from angularresolution.

TABLE VI.

SK measured solar neutrino flux by phase.
Flux (×106 =ðcm2 secÞ)2.380  0.024þ0.084−0.076SK-I

4. Result
The systematic uncertainty on the total flux (between 3.49
and 19.5 MeV) is summarized in Table V. The energy scale
dominates the total systematic uncertainty which is calculated as the quadratic sum of all components, and found to be
1.7%. This is the smallest systematic uncertainty of allphases of SK. In particular, the systematic uncertainties thatare energy-correlated (arising from the energy scale andresolution uncertainty) are smallest: while SK-IV’s livetimeis the same for all energy bins, previous phases have less
livetime below 5.99 MeV recoil electron kinetic energy. For
example, SK-III data below 5.99 MeV has only about half
the livetime as the full SK-III phase. The improved livetime
below 5.99 MeV, a higher efficiency in that energy region,and the additional data below 4.49 MeVall lessen the impact
of energy scale and resolution uncertainties on the flux
determination compared to previous phases. Other contributions to the reduction come from the removal of thefiducial volume cut based on an alternate vertex fit, andbetter control of vertex shift, trigger efficiency and angularresolution systematic effects. The number of solar neutrino
events (3.49–19.5 MeV) extracted from Fig. 17 is31; 891þ283−281 ðstatÞ  543ðsystÞ. This number corresponds
to a 8 B solar neutrino flux of
TABLE V. Summary of the systematic uncertainty on the total
rate for each SK phase. The details are also explained in [9,14].
Threshold (MeV)

SK-I
4.49

SK-II
6.49

SK-III
3.99

SK-IV
3.49

Trigger efficiency
Angular resolution
Reconstruction goodness
Hit pattern
Small hit cluster
External event cut
Vertex shift
Second vertex fit
Background shape
Multiple scattering goodness
Livetime
Spallation cut
Signal extraction
Cross section

0.4%
1.2%
þ1.9
−1.3 %
0.8%

0.5%
1.3%
0.5%
0.1%

0.1%
0.2%
0.7%
0.5%

0.5%
3.0%
3.0%


1.0%
1.1%
1.0%
0.4%
0.4%
0.1%
0.4%
0.7%
0.5%

0.5%
0.7%
0.4%
0.3%
0.5%
0.3%
0.5%
0.5%
0.1%
0.4%
0.1%
0.2%
0.7%
0.5%

0.1%
0.1%
0.1%
0.5%
þ0.5
−0.4 %
0.1%
0.2%

0.1%
0.4%
0.1%
0.2%
0.7%
0.5%

Subtotal

2.8%

4.8%

1.6%

1.2%

Energy scale
Energy resolution
8 B spectrum

1.6%
0.3%
þ1.1
−1.0 %

þ4.2−3.9 %

1.2%
0.2%
þ0.3
−0.4 %

þ1.1−1.2 %
þ0.3−0.2 %
þ0.4−0.3 %

Total

þ3.5−3.2 %

þ6.7−6.4 %

2.2%

1.7%

0.3%
1.9%

SK-II

2.41  0.05þ0.16−0.15

SK-III2.404  0.039  0.053

SK-IV

2.308  0.020þ0.039−0.040Combined

2.345  0.014  0.036

Φ8 B ðSK − IVÞ
62¼ ð2.308  0.020ðstatÞþ0.039−0.040 ðsystÞÞ × 10 =ðcm secÞ;assuming a pure νe flavor content.
As seen in Table VI, the SK-IV measured flux agrees
with that of previous phases within systematic uncertainty.
It can then be combined with the previous three SK flux
measurements to give the SK measured flux as
Φ8 B ðSKÞ
¼ ð2.345  0.014ðstatÞ  0.036ðsystÞÞ
× 106 =ðcm2 secÞ:IV. ENERGY SPECTRUM
Present values of Δm221 and sin2 θ12 imply that solar
neutrino flavor oscillations above about three MeV are
dominated by the solar MSW [4] resonance, while lowenergy solar neutrino flavor changes are mostly due tovacuum oscillations. Since the MSW effect rests solely on
standard weak interactions, it is rather interesting tocompare the expected resonance curve with data as any
deviation would imply new (weak interaction) physics.
Unfortunately multiple Coulomb scattering prevents the
kinematic reconstruction of the neutrino energy in neutrinoelectron elastic scattering interactions. However, the energyof the recoiling electron still provides a lower limit to theneutrino’s energy. Thus, the neutrino spectrum is inferredstatistically from the recoil electron spectrum. Moreover,the differential cross section of νμ;τ ’s is not just a factor of
about six smaller than the one for νe ’s, but also has a softer
energy dependence. In this way, the observed recoil
electron spectrum shape depends both on the flavor
composition and the energy dependence of the compositionof the solar neutrinos (see Sec. III B in particular Fig. 16).
Thus, even a flat composition of 33% νe and 67% νμ;τwould still distort the recoil electron spectrum compared to
one with 100% νe . The energy dependence of the day/nighteffect and rare hep neutrino interactions (with a higher
endpoint than 8 B ν’s) also distort the spectrum.
Since the transition between MSW resonance andvacuum oscillations lies around 3 MeV, the lowest energy

052010-13

K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)

þ551063þ124−122 ðstatÞ−54 ðsystÞ events:

0.4
0.2
MSG < 0.35

0.35 < MSG < 0.45

00.3

A. SK-IV spectrum results

MSG > 0.45

3.99 MeV < Ekin< 4.49 MeV

0.2
0.1
MSG < 0.35

0.35 < MSG < 0.45

00.3

MSG > 0.45

4.49 MeV < Ekin< 4.99 MeV

0.2
0.1
0

As outlined in Sec. III C [in particular Eq. (3.2)], the solar
neutrino signal of SK-IV is extracted by an extended
maximum likelihood fit. While the 8 B flux analysis uses
all 23 energy bins at once (and constrains the energy
spectrum to the one expected from unoscillated simulation
via the Y i factors), here we extract the solar neutrino energy
spectrum by fitting one recoil electron energy bin i at a time,with Y i ¼ 1. Below 7.49 MeV, each energy bin is split into
three subsamples according to the MSG of the events, withboundaries set at MSG ¼ 0.35 and 0.45. These threesubsamples are then fit simultaneously to a single signal
and three independent background components. The signal
fraction Y ig in each MSG bin g is determined by solar
neutrino simulated events in the same manner as the Y ifactors in the 8 B flux analysis. Similar to the 8 B flux analysis,
the signal and background shapes depend on the MSG bin g:
the signal shapes σ g are calculated from solar neutrino
simulated events and the background shapes βig are takenfrom data. Figure 20 shows the measured angular3.49 MeV < Ekin< 3.99 MeV

0.6

Events/day/kton/bin

solar neutrinos show the largest deviation from the resonance electron survival probability. Here, we report for the
first time, a clear solar neutrino signal with high statistics inthe energy range 3.49–3.99 MeV observed over the entire
data-taking period of SK-IV. Figure 19 shows the solar
angle distribution for this energy bin, with a distinct peak(above the background) coming from solar neutrinos. The
number of solar neutrino interactions (measured in thisenergy range from fits to the distributions of Fig. 20
discussed below) is

MSG < 0.35

0.35 < MSG < 0.45

MSG > 0.45

6.99 MeV < Ekin< 7.49 MeV

0.4
0.2
MSG < 0.35

0-1 -0.5

0.35 < MSG < 0.45

0 0.5

1-1 -0.5

0 0.5

cosθsun

MSG > 0.45

1-1 -0.5

0 0.5

1FIG. 20. Solar angle distribution for the electron energy
ranges 3.49–3.99 MeV, 3.99–4.49 MeV, 4.49–4.99 MeV and
6.99–7.49 MeV (from top to bottom), for each MSG bin (left toright). The style definitions are same as Fig. 17.

distributions (as well as the fits) for the energy ranges
3.49–3.99 MeV, 3.99–4.49 MeV, 4.49–4.99 MeV, and
6.99–7.49 MeV (from top to bottom), for each MSG bin(left to right). As expected in the lowest energy bins, where
the dominant part of the background is due to very lowenergy β=γ decays, the background component is largest in
the lowest MSG subsample. Also as expected, the solar
neutrino elastic scattering peak sharpens as MSG is increased.

0.15
0.55

Data/MC (Unoscillated)

Events/day/kton/bin

0.5

0.1

0.05

0.45
0.4
0.35
0.3
0.25
0.2

0
-1

-0.5

0

0.5

5

10
15
Recoil Electron Kinetic Energy [MeV]1cosθsunFIG. 19. Solar angle distribution for events with electronenergies between 3.49 and 3.99 MeV. The style definitions aresame as Fig. 17.

FIG. 21. SK-IV energy spectrum using MSG sub-samples
below 7.49 MeV, shown as the ratio of the measured rate to
the simulated unoscillated rate. The horizontal dashed line gives
the SK-IV total average (0.440). Error bars shown are statisticalplus energy-uncorrelated systematic uncertainties.

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SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

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1.3

8 B and hep neutrino fluxes within uncertainties. The 8 B

1.1

flux is constrained to ð5.25  0.20Þ × 106 =ðcm2 secÞ andthe hep flux to ð8  16Þ × 103 =ðcm2 secÞ (motivated by
SNO’s measurement [19] and limit [20]). The χ 2 isdescribed in detail in Sec. VI.

1

B. Systematic uncertainties on the energy spectrum

Data/MC

1.2

0.9
0.3

0.4

0.5

0.6

Multiple Scattering Goodness

FIG. 22. MSG scaling functions applied to simulated events toestimate the systematic uncertainty on the energy spectrum. The
dotted, dashed and solid lines correspond to 16.1, 8.67, and4.89 MeV LINAC data over simulated events.

Using this method for recoil electron energy bins below
7.49 MeV gives ∼10% improvement in the statisticaluncertainty on the number of extracted signal events (theadditional systematic uncertainty is small compared to thestatistical gain). Figure 21 shows the resulting SK-IV
energy spectrum, where below 7.49 MeV MSG has beenused and above 7.49 MeV the standard signal extraction
method without MSG is used. Table XVI in Appendix B
gives the measured and expected rate in each energy bin,as well as that measured for the day and night times
separately, along with the 1σ statistical deviations. We
reanalyzed the SK-III spectrum below 7.49 MeV with thesame method, the same MSG bins and the same energy
bins as SK-IV, down to 3.99 MeV. We also refit the entire
SK-II (which has poorer resolution) spectrum using thesame three MSG sub-samples. The gains in precision are
similar to SK-IV. The SK-II and III spectra are given inSec. IV C.
To analyze the spectrum, we simultaneously fit the SK-I,
II, III and IV spectra to their predictions, while varying the

Since we simultaneously fit multiple samples defined by
the multiple Coulomb scattering goodness in the lowest
recoil electron energy region, a systematic shift in thisgoodness of the data compared to solar 8 B (or hep) neutrino
simulated events would affect the measured event rate inthat energy region. To estimate the systematic effect ofusing MSG sub-samples, MSG distributions of LINAC
data and simulated LINAC events were compared, as seenin Fig. 6. The simulated solar neutrino MSG distributions
are adjusted using the observed ratio of the LINAC data andsimulated events at the nearest LINAC energy. Thischanges the solar signal shapes σ g and the ratios of
expected signal events Y ig for MSG bin g. The cos θsun
distributions are then refit, using the new angular distributions and signal ratios and the change in the extracted
number of signal events is taken as the systematic uncertainty. The scaling functions for three LINAC energies canbe seen in Fig. 22.
The change for each energy bin and all other energyuncorrelated systematic uncertainties of the SK-IV recoil
electron energy spectrum are summarized in Table VII. The
total energy-uncorrelated systematic uncertainty in thistable is calculated as the sum in quadrature of each of
the components. Since we assume no correlations between
the energy bins in the SK-IV spectrum analysis, thecombined uncertainty is added in quadrature to the statistical error of that energy bin.
The 8 B neutrino spectrum uncertainty (a shift of∼  100 keV), the SK-IV energy scale uncertainty(0.54%) and the SK-IV energy resolution uncertainty(1.0% for < 4.89 MeV, 0.6% for > 6.81 MeV) [14], will

TABLE VII. Energy-uncorrelated systematic uncertainties on the spectrum shape. The systematic error of the (unlisted) small hit
cluster cut (only applied below 4.99 MeV) is negligible.
Energy (MeV)
Trigger efficiency
Reconstruction goodness
Hit pattern
External event cut
Vertex shift
Background shape
Signal extraction
Cross section
MSG
Total

3.49–3.99 3.99–4.49 4.49–4.99 4.99–5.49 5.49–5.99 5.99–6.49 6.49–6.99 6.99–7.49 7.49–19.5
þ3.6−3.3 %

0.6%

0.1%
0.4%
2.9%
2.1%
0.2%
0.4%

0.8%
0.7%

0.1%
0.4%
1.0%
2.1%
0.2%
0.4%

þ5.1
−4.9 %

2.6%




0.1%
0.2%
0.8%
2.1%
0.2%
0.3%


0.4%

0.1%
0.2%
0.2%
0.7%
0.2%
0.3%


0.2%

0.1%
0.2%
0.1%
0.7%
0.2%
0.3%


0.1%
0.6%
0.1%
0.2%
0.1%
0.7%
0.2%
1.7%


0.1%
0.6%
0.1%
0.2%
0.1%
0.7%
0.2%
1.7%


0.1%
0.6%
0.1%
0.2%
0.1%
0.7%
0.2%
1.7%


0.1%
0.4%
0.1%
0.2%
0.1%
0.7%
0.2%


þ2.4
−2.3 %

0.9%

0.9%

2.0%

þ2.0
−1.9 %

1.9%

þ0.9−0.8 %

þ0.6−0.5 %

052010-15

K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)

shift all energy bins in a correlated manner. The size andcorrelation of these uncertainties are calculated from theneutrino spectrum, the differential cross section, the energy
resolution function, and the size of the systematic shifts. Wevary each of these three parameters (8 B neutrino spectrum
shift, energy scale, and energy resolution) individually.
Figure 23 shows the result of this calculation. When weanalyze the spectrum, we apply these shifts to the spectral
predictions. When the SK-IV spectrum is combined with
the SK-I, II, and III spectra, the 8 B neutrino spectrum shift
is common to all four phases, while each phase varies its
energy scale and resolution individually (without correlation between the phases).

0.44
0.42

Pee

0.4

0.36
0.34
0.32

C. SK-I/II/III/IV combined spectrum analysis

0.3

In order to discuss the energy dependence of thesolar neutrino flavor composition in a general way,
SNO [19] has parametrized the electron survival probability Pee using a quadratic function centered at
10 MeV:



2
Eν
Eν
Pee ðEν Þ ¼ c0 þ c1
− 10 þ c2− 10 ;
MeV
MeV
ð4:1Þ
where c0 , c1 , and c2 are polynomial parameters.
As seen in Fig. 24, this parametrization does notdescribe well the MSW resonance based on the oscillation parameters of either best fit. This is also true foralternative solutions such as nonstandard interactions [5]and mass-varying neutrinos [6]. However, it is simple,
and the SNO collaboration found that it introduces no10

Systematic uncertainty [%]

8
6
4
2
0
-2
-4
-6
-8
-10

4

6

8

10

12

14

16

0.38

18

Recoil electron kinetic energy [MeV]

FIG. 23. Energy-correlated systematic uncertainties. The dotdashed, solid, and dashed distributions correspond to the systematic uncertainties of the 8 B spectrum shape, energy resolution,
and absolute energy scale, respectively.

4

6

8

10

12

14

Eν in MeV

FIG. 24. νe survival probability Pee based on the oscillation
parameters fit to SK (thick solid green) and all solar neutrino and
KamLAND data (thick solid blue). The solid yellow (cyan) line isthe best exponential approximation to the thick solid green (blue)line. The dashed black (dotted green) line is the best quadratic(cubic) approximation to the thick solid green line and the dashedred (dotted pink) line the best quadratic (cubic) approximation to
the thick solid blue.

bias when determining oscillations parameters. In addition to this quadratic function we have explored two
different alternatives to parametrize the survival probability in order to study any limitations the quadratic
function might have: an exponential fit and a cubic
extension of the quadratic fit. The exponential fit isparametrized asEνe 
Pee ðEν Þ ¼ e0 þ 1 ee2 ðMeV−10Þ − 1 :ð4:2Þ
e2This particular functional form allows direct comparison of e0 and e1 to the quadratic coefficients c0 and c1 ,if c1 and e1 are small. The parameter e2 controls the“steepness” of the exponential fall or rise. Both exponential and cubic parametrizations describe the MSW
resonance curve reasonably well as shown in Fig. 24.
This is true for both the SK-only and the solar þ
KamLAND best-fit oscillation parameters discussed in
the oscillation section below. Table VIII lists the
exponential and cubic coefficients that best describe
those two MSW resonance curves. The definition of
the spectrum χ 2 and the best-fit values are given inSec. VI.
To ease the comparison between SK spectral data and
SNO’s results, we also performed a quadratic fit to SK
data. Table IX gives the best quadratic coefficients forboth the SK-only and the solar þ KamLAND case.

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TABLE VIII. Best approximations to the MSW resonances
using exponential and polynomial parametrizations of Pee .
sin2 θ12 ¼ 0.304
Δm221 ¼ 7.41 × 10−5sin2 θ12 ¼ 0.314
Δm221 ¼ 4.90 × 10−50.3205−0.0062−0.270770.69
2.31

0.3106−0.0026−0.3549
68.99
0.61

Exponential e0
Exponential e1
Exponential e2
Exponential χ 2
Exponential Δχ 2
Polynomial c0
Polynomial c1
Polynomial c2
Polynomial c3
Polynomial χ 2
Polynomial Δχ 2

0.3194−0.0071
þ0.0012
0
70.79
2.46

0.3204−0.0059
þ0.0009−0.0001
70.71
7.07

0.3095−0.0033
þ0.0008
0
68.87
0.54

0.3105−0.0021
þ0.0005−0.0001
69.06
5.43

For each set of parameters, the expected rate in eachenergy bin is adjusted according to the average day/night
enhancement expected from sin2 θ12 ¼ 0.304 andΔm2 ¼ 4.90 × 10−5 eV2 . Figure 25 shows the SK spectral
data. They are expressed as the ratio of the observed
elastic scattering rates of each SK phase over MC
expectations, assuming no oscillations (pure electron
flavor composition), a 8 B flux of 5.25 × 106 =ðcm2 secÞand a hep flux of 8 × 103 =ðcm2 secÞ. Table XVII in
Appendix B lists the data shown in Fig. 25, with the given
errors including statistical uncertainties as well as energyuncorrelated systematic uncertainties.
Table XV in Appendix B gives the SK exponential andpolynomial best-fit coefficients and their correlations. We
compare the best χ 2 of the full MSW calculation to that of
the best exponential, cubic, and quadratic function fits, aswell as a simple energy-independent suppression of theelastic scattering rate in SK. In the case of the flat (energyindependent) suppression, 0.4268 was chosen as the ratioof observed elastic scattering over expectation assuming
no neutrino oscillations. The value 0.4268 corresponds toa constant Pee ¼ 0.317 if the cross section ratio was

TABLE IX.
FitParam.

Spectrum fit χ 2 comparison.

MSW (solþKamLAND)
2

sin θ12 , sin θ13 , Δm22120.304, 0.02,
7.50×10−5 eV2

χ2
SK-I
SK-II
SK-IIISK-IVdσ νμ =dσ νe ¼ 0.16 independent of energy. In reality, thatratio becomes larger at lower energy, leading to a small
low-energy “upturn” even for a constant Pee ¼ 0.317. Theenergy dependence of the day/night effect (which iscorrected for in the polynomial and exponential fits) leads
to a small “downturn.” In case of this flat suppression we
fit with and without the day/night correction. Tables IX
and X compare the various χ 2 , while Table VIII gives theχ 2 from the best exponential (quadratic, cubic) approximations of the MSW resonance curve as well as thedifference in χ 2 from the exponential (quadratic, cubic)
best fit. The exponential and quadratic fits are consistent
with a flat suppression as well as the MSW resonance“upturn.” In either case an “upturn” fits slightly better (byabout 1.0σ), but the coefficients describing the MSW
resonance are actually slightly disfavored by 1.5σ (exponential) and 1.6σ (quadratic), for the best-fit Δm221 from
KamLAND data, and by 0.8σ (exponential) and 0.7σ
(quadratic) for the best-fit Δm221 from solar neutrino data.
The cubic fit disfavors the flat suppression by 2.3σ; asseen in Fig. 27 the fit prefers an inflection point in the
spectrum occurring near 8 MeV, a shape which cannot be
accommodated by the other two parametrizations. From
Table IX the SK-II and SK-IV minimum χ 2 s of the cubic
fit are similar to the quadratic and exponential fit, however
the SK-I (SK-III) data favor the cubic fit by about 1.7σ
(1.2σ). The reason for that preference is mostly due to data
above ∼13 MeV (see Fig. 25). We checked these data but
found no reason to exclude them. However, conservatively, we disregard the cubic best fit in our conclusions.
Therefore, we find no significant spectral “upturn” (ordownturn) at low energy, but our data is consistent withthe “upturn” predicted by the MSW resonance curve(disfavoring the one based on solar þ KamLAND bestfit parameters by about 1.5σ). Figure 25 shows thepredictions for the best MSW fits, the best exponential/
quadratic and the best cubic fit. Figure 26 statisticallycombines the different SK phases ignoring differences inenergy resolutions and systematic uncertainties. It isΦ8 B =
cm2 sec

Φhep =
cm2 sec2

MSW (solar)

Exponential

Quadratic

Cubic

2

e0 e1 , e2c0 , c1 , c2

0.334, −0.001, −0.12

0.33, 0, 0.001

c0 , c1 , c2 , c3
0.312, −0.031,
0.0095, 0.0044

sin θ12 , sin θ13 , Δm221
0.304, 0.02,
4.84×10−5 eV2χ2

Φ8 B =
cm2 sec

Φhep =
cm2 secχ2

Φ8 B =
cm2 sec

Φhep =
cm2 secχ2

Φ8 B =
cm2 sec

Φhep =
cm2 secχ2

Φ8 B =
cm2 sec

Φhep =
cm2 sec

19.71 5.26×106 39.4×103 19.12 5.47×106 41.0×103 18.82 5.22×106 41.4×103 18.94 5.24×106 36.8×103 16.14 5.25×106 5.1×103
5.39 5.33×106 55.1×103 5.35 5.53×106 56.8×103 5.31 5.27×106 56.9×103 5.38 5.30×106 51.5×103 5.15 5.34×106 11.9×103
29.06 5.34×106 15.7×103 28.41 5.55×106 14.7×103 28.07 5.29×106 13.8×103 28.02 5.31×106 10.9×103 26.59 5.30×106 −3.6×103
14.43 5.22×106 12.2×103 14.00 5.44×106 11.4×103 14.29 5.20×106 10.8×103 14.15 5.22×106 8.2×103 14.07 5.22×106 −4.2×103

Combined 71.04 5.28×106 14.1×103 69.03 5.49×106 13.4×103 68.38 5.25×106 13.1×103 68.33 5.26×106 11.9×103 63.63 5.25×106 −0.7×103

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K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)
SK II Spectrum

0.7

0.7

0.6

0.6

Data/MC (unoscillated)

Data/MC (unoscillated)

SK I Spectrum

0.5

0.4

0.3

0.5

0.4

0.3

0.2

6

8

10

12

E

kin

14

16

0.2

18

10

8

in MeV

16

E

in MeV

18

SK IV Spectrum

0.7

0.7

0.6

0.6

Data/MC (unoscillated)

Data/MC (unoscillated)

14

kin

SK III Spectrum

0.5

0.4

0.3

0.2

12

0.5

0.4

0.3

4

6

8

10

12

E

in MeV

kin

14

16

0.2

18

4

6

8

10

12

E

in MeV

kin

14

16

18

FIG. 25. SK-I, II, III and IV recoil electron spectra divided by the nonoscillated expectation. The green (blue) line represents the best
fit to SK data using the oscillation parameters from the fit to all solar (solar þ KamLAND) data. The orange (black) line is the best fit to
SK data of a general exponential or quadratic (cubic) Pee survival probability. Error bars on the data points give the statistical plus
systematic energy-uncorrelated uncertainties while the shaded purple, red, and green histograms give the energy-correlated systematic
uncertainties arising from energy scale, energy resolution, and neutrino energy spectrum shift.

included only as an illustration and should not be fit topredictions.
Section B of the appendix discusses the measured
coefficients, their uncertainties, and their correlations of
all three parametrizations of Pee . It also compares thequadratic coefficients obtained from SK data with thosefrom SNO data, and the coefficients of the SK-SNO

combined fit. Figure 27 compares the allowed survival
probability Pee based on the exponential fit with thatbased on the cubic and quadratic fits. Between about 5.5
and 12.5 MeV, the different parametrizations agree while
outside this energy region parametrization-dependent
extrapolation effects become significant. While thestrength of the SK data constraints on Pee is comparable

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SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

PHYSICAL REVIEW D 94, 052010 (2016)

Spectrum fit χ 2 comparison for the “flat suppression” of 0.4268 of the expected rate assuming no

TABLE X.
neutrino oscillation.
Fit

χ2

With D/N correction
Φ8 B =cm2 sec
Φhep =cm2 sec

χ2

SK-I
SK-II
SK-III
SK-IV

18.92
5.30
27.94
15.50

5.38 × 106
5.43 × 106
5.45 × 106
5.37 × 106

41.4 × 103
56.3 × 103
12.0 × 103
9.4 × 103

18.81
5.27
27.98
14.99

5.47 × 106
5.52 × 106
5.55 × 106
5.46 × 106

42.6 × 103
58.4 × 103
13.1 × 103
10.2 × 103

Combined

69.30

5.41 × 106

12.3 × 103

68.75

5.50 × 106

12.7 × 103

0.6
0.5
0.4

Pday
ee

to that of SNO data, its low energy constraints are tighter
and its high energy constraints weaker. The reason for thisis the absence of a nuclear threshold in elastic electronneutrino scattering, and the direct correlation of neutrinoenergy and electron energy in neutrino-deuteron charged
current interactions. SK data prefers a slight “upturn”,
SNO data prefer a “downturn.” The combined fit favors
an upturn more strongly than SK data by themselves since
SK data prefer a higher average Pee than SNO data, andthe tighter SK constraints force the combined fit to thishigher average probability at low energy, while the tighter
SNO constrains force the combined fit to the lower SNO
value at high energy. Figures 28 and 29 (combined fit)display the 1σ allowed bands of Pee ðEν Þ. Figure 30
superimposes the same combined band (on a logarithmic
scale) on the SSM [21] solar neutrino spectrum. Also
shown are the pp and CNO neutrino flux constraints fromall solar data [22] and the 7 Be, the pep and the 8 B flux
measurement of the Borexino experiment [23]. The SK
and SNO combined allowed band (and the other solar

0.3
0.2
0.1
0.0

6

7

8

9 10 11 12
Eν in MeV

67

8 9 10 11 12 13
Eν in MeV

FIG. 27. Allowed survival probability 1σ band from SK-IV data(left) and all SK data (right). The red (blue) area is based on an
exponential (quadratic) fit and the green band is based on a cubic
fit. The 8 B flux is constrained to the measurement from SNO. The
absolute value of the 8 B flux does not affect the shape constraint
much, just the average value. Also shown are predictions basedon the oscillation parameters of a fit to all solar data (green) and a
fit to all solar þ KamLAND data (blue).

SK I/II/III/IV LMA Spectrum

0.6
0.58

0.6

0.56

0.5

0.54
0.52

0.4

0.5

Pday
ee

Data/MC (unoscillated)

Without D/N correction
Φ8 B =cm2 sec
Φhep =cm2 sec

0.48

0.3

0.46

0.2

0.44
0.42
0.4

0.1
4

6

8

10
12
E in MeV

14

16

18

0.0

kin

FIG. 26. SK-I þ II þ III þ IV recoil electron spectrum compared to the no-oscillation expectation. The green (blue) shape is
the MSW expectation using the SK (solar þ KamLAND) best-fit
oscillation parameters. The orange (black) line is the best fit to SK
data with a general exponential/quadratic (cubic) Pee survival
probability.

3

4

5

6

7

8

9

10

11

12

13

14

15

Eν in MeV

FIG. 28. Allowed survival probability 1σ band from SK (solid
green) and SNO (dotted blue) data. Also shown are predictions
based on the oscillation parameters of a fit to all solar data (green)
and a fit to all solar þ KamLAND data (blue).

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PHYSICAL REVIEW D 94, 052010 (2016)

0.6
0.5

Pday
ee

0.4
0.3
0.2
0.1
0.0

3

4

5

6

7

8

9

10

11

12

13

14

15

Eν in MeV

FIG. 29. Allowed survival probability 1σ band from the
combined data of SK and SNO (red). Also shown are predictions
based on the oscillation parameters of a fit to all solar data (green)
and a fit to all solar þ KamLAND data (blue). The pastel coloredbands are the separate SK (green) and SNO (blue) fits.

7

pp

Be

pep

CNO

8

B

Hep

0.7

109
108
107
106

0.5

105
104

0.4

103

0.3

102

0.2

2

0.6

Pee

1010

Flux in /keVcm s

0.8

10
1

0.1

10
0
10

-1

10
1

-1
-2

10

ν Energy in MeV

FIG. 30. Predicted solar neutrino spectra [21]. Overlaid are
expected MSW survival probabilities, green is that expectedassuming oscillation parameters from the solar best fit and blue
from the solar þ KamLAND best fit. The 1σ band of Pee from the
combined data of SK and SNO is shown in red. Also shown are
Pee measurements of the 7 Be (green point), the pep (light green
point) and the 8 B flux (red point) by Borexino [23], as well as pp(blue point) and CNO values (gold point) extracted from otherexperiments [22].

data) are in good agreement with the MSW curves (basedon different parameters: blue ¼ solar þ KamLAND best
fit, green ¼ solar best fit).
V. DAY/NIGHT ASYMMETRY
The matter density of the Earth affects solar neutrino
oscillations while the Sun is below the horizon. Thisso called “day/night effect” will lead to an enhancementof the νe flavor content during the nighttime for mostoscillation parameters. The most straightforward test of thiseffect uses the solar zenith angle θz (defined in Fig. 17) at
the time of each event to separately measure the solar
neutrino flux during the day ΦD (defined as cos θz ≤ 0)
and the night ΦN (defined as cos θz > 0). The day/night
asymmetry ADN ¼ ðΦD − ΦN Þ= 12 ðΦD þ ΦN ) defines aconvenient measure of the size of the effect; it is sensitive
to Δm221 .
A more sophisticated method to test the day/night effectis given in [1,24]. For a given set of oscillation parameters,
the interaction rate as a function of the solar zenith angle ispredicted. Only the shape of the calculated solar zenith
angle variation is used; the amplitude is scaled by anarbitrary parameter. The extended maximum likelihood fit
to extract the solar neutrino signal (see Sec. III C) is
expanded to allow time-varying signals. The likelihood
is then evaluated as a function of the average signal rates,
the background rates and a scaling parameter, termed the“day/night amplitude”. The equivalent day/night asymmetry is calculated by multiplying the fit scaling parameterwith the expected day/night asymmetry. In this manner the
day/night asymmetry is measured more precisely statistically and is less vulnerable to some key systematiceffects.
Because the amplitude fit depends on the assumed
shape of the day/night variation (given for each energy
bin in [24] and [1]), it necessarily depends on theoscillation parameters, although with very little dependence expected on the mixing angles (in or near thelarge mixing angle solution and for θ13 values consistent
with reactor neutrino measurements [25]). The fit isrun for parameters covering the MSW region of oscillation parameters (10−9 eV2 ≤ Δm221 ≤ 10−3 eV2 and10−4 ≤ sin2 θ12 < 1), and values of sin2 θ13 between 0.015
and 0.035.
A. Systematic uncertainty on the solar neutrino
amplitude fit day/night flux asymmetry
1. Energy scale
True day (night) solar neutrino events will mostly be
coming from the downward (upward) direction, and so
the directional dependence of the SK light yield orenergy scale will affect the observed interaction rate asa function of solar zenith angle and energy. To quantify
the directional dependence of the energy scale, theenergy of the DT-produced 16 N calibration data and its
simulation are compared as a function of the reconstructed detector zenith angle (Fig. 9). The fit from Fig. 9
is used to shift the energy of the 8 B MC events, while
taking energy-bin correlations into account, and theunbinned amplitude fit was rerun. The resulting 0.05%
change in the equivalent day/night asymmetry is taken as

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SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

PHYSICAL REVIEW D 94, 052010 (2016)

the systematic uncertainty coming from the directional
dependence of the energy scale. The large reduction
compared to SK-I (0.8%) comes from the use of a depthdependent water transparency parameter, introduced
at the beginning of SK-III. The further reduction from
SK-III (0.2%) to SK-IV comes from an increase in
DT calibration statistics and the improved timing agreement between data and MC, a result of the electronics
upgrade.
2. Energy resolution
Throughout the different phases of SK, the energy
resolution function relating the true and reconstructed
recoil electron energies was found by two slightly different
methods. During the SK-I and SK-IV phases, 8 B simulated
events were used to set up a “transfer matrix” relatingreconstructed to true recoil electron energy (and reconstructed recoil electron energy to neutrino energy.) Thismethod, by construction, considers the effect of all analysis
cuts on energy resolution. For the SK-II and III phases,
dedicated monoenergetic simulated events were produced
to parametrize the energy resolution with a Gaussianfunction, modeling only some analysis cuts. The two
methods produce slightly different results; in particular,
the predicted day/night asymmetries differ by 0.05%. To
estimate the systematic uncertainty on the day/night asymmetry coming from the energy resolution function, theamplitude fit was performed using both methods, with theresulting 0.05% difference taken as the systematic
uncertainty.
3. Background shape
Although there is only one background component fit in
the day/night asymmetry fit (any time dependence of the
background should be much slower than the day/nightvariation), different cos θsun background shapes must be
used for different solar zenith angle bins. We use one for the
day and six for the night (in accordance with Table XII).
The systematically different shapes come from the detector’s directional bias when reconstructing background
events (directions alongside and perpendicular to thedetector symmetry axis are preferred). The backgroundis first fit as functions of the detector zenith and azimuthal
angles. These fits also yield a covariance matrix V for the fit
parameters. The parameters of each of the zenith andazimuthal fits are varied by the one sigma statisticaldeviation, one at a time, giving a new background shapefor each solar zenith angle bin. Because the background
distributions are calculated as projections of the detector
zenith and azimuthal angles on the solar direction, theshape deviations as a function of solar zenith angle are fully
correlated and must be varied simultaneously. The day/night amplitude fit is then rerun for each set of new
background shapes. The difference in the central value istaken as the error of the day/night asymmetry due to thatparticular zenith or azimuthal fit parameter. These errors arethen propagated to a total systematic uncertainty using thecovariance matrix V of the fit to the detector zenith andazimuth angles. The total uncertainty on the day/night
asymmetry coming from the background shapes is 0.6%,
and is the largest contribution to the total.
4. Event selection
Most of the analysis cuts affect the day and night solar
neutrino interaction rates equally, so their effect on thesystematic uncertainty on the day/night asymmetry can beneglected. However, the vertex shift and angular resolutiondifference between data and simulated events can cause abias in the external event cut efficiency when used inconjunction with the tight fiducial volume cut. To estimate
the size of the effect, we artificially shift the reconstructedvertex and direction and estimate the fraction of events
which are rejected by the cuts during daytime and during
nighttime. The associated estimated systematic uncertainty
is 0.1%.
5. Earth model
Different models of Earth’s density profile can change
the signal rate zenith profiles, thus leading to changes in themeasured day/night asymmetry value. For this reason it isessential to model the Earth as precisely as possible, mostfrequently done using the Preliminary Reference EarthModel (PREM) [26] and an Earth which is assumed to be
spherical. A spherical description of Earth using theequatorial radius leads to a ∼0.2% change in the day/nighteffect from a spherical description using an average radius.
To better represent the Earth we have modeled an ellipsoidal Earth, using the equatorial and polar radii as thesemimajor and semiminor axes of an ellipse. The ellipse isthen rotated around its minor axis to produce an ellipsoid
and the spherical PREM model density boundaries are
mapped accordingly.
Due to SK’s location on Earth and using the above
procedure of modeling an ellipsoidal Earth, the event rate is
no longer rotationally symmetric about the detector azimuthal angle and the day/night zenith amplitude fit must take
into account the change in the expected signal rate as theazimuthal angle is varied. This was done by varying theazimuthal angle and the zenith angle when tracing neutrinos through the Earth, and then using the detector
livetime to average over the azimuthal angle. The resulting
expected solar zenith angle dependent signal rates werethen used in the day/night amplitude fit and the resultscompared to the results when assuming a spherical Earthwith an average radius. The 0.01% change in the day/night
asymmetry is taken as the systematic uncertainty comingfrom the Earth shape.
As a final step in estimating the systematic uncertainty
coming from the model of the Earth, the PREM model was
replaced with the more recent PREM500 model [27], which

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K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)

6. Summary of the systematic uncertainty
The total estimated systematic uncertainty on the measured day/night asymmetry is calculated by adding thecomponents in quadrature, the result of which can be seenin Table XI. The large reduction in systematics from SK-I
[1] to SK-IV comes from the introduction of a z-dependent
absorption into the simulation and a better method of
estimating the systematic uncertainty using DT data. The
directional dependence of the energy scale is now better
understood, bringing the total systematic uncertainty
to 0.6%.
B. SK day/night asymmetry results
The SK-IV livetime during the day (night) is 797 days
(866 days). The solar neutrino flux between 4.49 and19.5 MeV assuming no oscillations is measured as ΦD ¼
6
2
ð2.250þ0.030−0.029 ðstatÞ  0.038ðsysÞÞ × 10 =ðcm secÞ during
the day and ΦN ¼ ð2.364  0.029ðstatÞ  0.040ðsysÞÞ ×
106 =ðcm2 secÞ during the night. Figure 31 shows thesolar zenith angle variation of the ratio of the measured
rate to the unoscillated simulated rate (assuming 5.25 ×
106 =ðcm2 secÞ for the 8 B flux) in the seven energy regions
shown in Table XII. Overlaid is the expected zenith
variation for best-fit oscillation parameters coming froma fit to all solar neutrino data (solar þ KamLAND data) in
red (blue). Table XII lists the data used in Fig. 31, the errors
are statistical uncertainties only. Figure 32 shows the dataover simulated rate ratio between 4.49 and 19.5 MeV(assuming no oscillations) as a function of cos θz , divided
into five day and six night bins (corresponding to themantle 1–5 and core definitions of Table XII). By comparing the separately measured day and night fluxes, themeasured day/night asymmetry for SK-IV is found to
be ADN ¼ ð−4.9  1.8ðstatÞ  1.4ðsystÞÞ%.

TABLE XI. SK-IV amplitude fit day/night asymmetry systematic uncertainties. The total is found by adding the contributionsin quadrature.
Energy scale
Energy resolution
Background shape
Event selection
Earth model

0.05%
0.05%
0.6%
0.1%
0.01%

Total

0.6%

4.49-4.99 MeV

4.99-5.99 MeV

5.99-7.49 MeV

0.7

13.0-15.5 MeV

0.6
Data/MC (unoscillated)

gives an updated and more detailed description of thedensity profile of Earth. This resulted in a 0.01% shift in themeasured day/night asymmetry. When added in quadrature
to the uncertainty coming from the Earth shape, 0.014%
gives the total estimated uncertainty coming from the
Earth model.

0.7

0.5

0.6

0.4
0.5

0.6

7.49-8.99 MeV

8.99-11.0 MeV

11.0-13.0 MeV

0.5

0.4

0.4

0.3

-0.5

0

0.5 -0.5

0

0.5 -0.5

cos θz00.5

1 -0.50

0.5

cos θz1

FIG. 31. SK-IV data/MC (unoscillated) rate dependence on thesolar zenith angle, for various energy regions (zenith angle andenergy bins defined in Table XII, panels are ordered by energy
with the upper, left panel being the lowest). The unoscillated
rate assumes a 8 B (hep) flux of 5.25 × 106 =ðcm2 secÞ
ð8 × 103 =ðcm2 secÞÞ. Overlaid green (blue) lines are predictions
when using the solar neutrino data (solar neutrino data þ
KamLAND) best-fit oscillation parameters and the assumed
neutrino fluxes fit to best describe the data. The error bars shown
are statistical uncertainties only.

The SK-IV day/night asymmetry resulting from theday/night amplitude fit method, for an energy range of
4.49–19.5 MeV and oscillations parameters preferred by
SK (Δm221 ¼4.84×10−5 eV2 , sin2 θ12 ¼0.311 and sin2 θ13 ¼
0.020), is found to be
Afit;SK-IV¼ ð−3.6  1.6ðstatÞ  0.6ðsystÞÞ%:DN
The expected day/night asymmetry for the above set ofoscillation parameters is −3.3%. For the case of a global fit
to solar neutrino data and KamLAND [3], the mass squared
splitting changes to Δm221 ¼ 7.50 × 10−5 eV2 , and the
expected day/night asymmetry goes to −1.7%. However,the day/night amplitude fit measured SK-IV day/night
asymmetry is only slightly reduced toAfit;SK-IV¼ ð−3.3  1.5ðstatÞ  0.6ðsystÞÞ%:DN
Within the Δm221 range of the large mixing angle (LMA)
region, all measured values of the day/night asymmetry
coming from the day/night amplitude fit are within 0.3%
of −3.3%. If the above measurement is combined with theprevious three phases of SK, the SK combined measured
day/night asymmetry is
Afit;SK
DN ¼ ð−3.3  1.0ðstatÞ  0.5ðsystÞÞ%:Previously, we published Afit;SK
DN ¼ ð−3.2  1.1ðstatÞ 
0.5ðsystÞÞ% in [28] which was the first significant

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SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

PHYSICAL REVIEW D 94, 052010 (2016)

TABLE XII. The observed zenith angle dependence of event rates (events/year/kton) in each energy region, at 1 AU. The errors are
statistical uncertainties only. The reduction efficiencies are corrected and the expected event rates are for a flux of 5.25 × 106 =ðcm2 secÞ.
Energy(MeV)
4.49–4.99
4.99–5.99
5.99–7.49
7.49–8.99
8.99–11.0
11.0–13.0
13.0–15.5

DAY
MANTLE1cos θz ¼ −1 ∼ 0
0 ∼ 0.16
79.4þ5.1−5.0

124.2þ3.8−3.7
139.5þ3.3−3.2
93.5þ2.7−2.7
52.0þ1.8−1.8
15.5þ0.9−0.9
3.83þ0.46−0.40

75.5þ13.4−12.2

116.8þ9.5−9.0
134.2þ8.6−8.2
89.3þ7.1−6.6
55.7þ5.1−4.7
17.4þ2.6−2.2
5.69þ1.54−1.18

MANTLE2
0.16 ∼ 0.33

Observed rate
MANTLE3
0.33 ∼ 0.50

MANTLE4
0.50 ∼ 0.67

MANTLE5
0.67 ∼ 0.84

CORE
0.84 ∼ 1

74.5þ12.1−11.1

91.6þ10.9−10.2

80.3þ10.6−9.9

85.1þ11.1−10.3

86.9þ11.4−10.6

127.0þ8.9−8.5
133.3þ8.1−7.7
90.5þ6.7−6.3
57.8þ4.7−4.4
17.3þ2.5−2.1
2.53þ1.07
−0.73

þ8.0
123.9−7.6
þ7.5
155.7−7.2
þ5.9
88.6−5.6
þ4.0
47.7−3.7
þ2.0
15.3−1.8
2.49þ0.91−0.65

indication that matter effects influence neutrino oscillations. The slightly larger significance (2.9σ) here is due to asomewhat larger data set.
VI. OSCILLATION ANALYSIS
SK measures elastic scattering of solar neutrinos withelectrons, the rate of which depends on the flavor content of
the solar neutrino flux, so it is sensitive to neutrino flavor
oscillations. To constrain the parameters governing theseoscillations, we analyze the integrated scattering rate, therecoil electron spectrum (which statistically implies theData/MC (Unoscillated)

0.5

0.48

0.46

0.44

0.4

All
Day
Night

0.42

-1

0

1

cosθz

FIG. 32. SK-IV solar zenith angle dependence of the solar
neutrino data/MC (unoscillated) interaction rate ratio (4.49–
19.5 MeV). The day data are subdivided into five bins, while
the night data are divided into six bins. Solar neutrinos in the lastnight bin pass through the Earth’s outer core. Overlaid green(blue) lines are predictions when using the solar neutrino data(solar neutrino data þ KamLAND) best-fit oscillation parameters
and the assumed neutrino fluxes fit to best describe the data. The
error bars shown are statistical uncertainties only.

126.7þ7.7−7.4
148.5þ7.1−6.9
94.0þ5.8−5.6
54.4þ4.0−3.7
14.9þ2.0−1.7
4.19þ1.03−0.80

133.9þ8.4−8.1
136.1þ7.5−7.2
88.1þ6.2−5.9
56.4þ4.4−4.1
15.2þ2.2−1.9
þ1.15
3.84−0.86

þ8.5
112.3−8.1

þ8.3
153.0−7.9
þ7.2
102.2−6.8
þ5.1
65.5−4.8
þ2.5
17.7−2.2
4.48þ1.33−1.01

Unoscillated rate
8
B
hep
167.8

0.323

283.6

0.611

321.4

0.799

196.6

0.647

122.2

0.619

36.0

0.365

7.45

0.204

energy-dependence of the electron-flavor survival probability), and the time of the interactions which defines theneutrino path through the earth during night time, andtherefore controls the earth matter effects on solar neutrino
oscillations. An expansion of the likelihood used in theextended maximum likelihood fit to extract the solar
neutrino signal (see Sec. III C) could make full use of
all information (timing, spectral information and rate), but
is CPU time intensive. Instead, we separate the log(likelihood) into a time-variation (day/night variation) portionlog LDN and a spectral portion: logL¼logLDN þlogLspec
where Lspec , the likelihood for assuming no time variation,is replaced by − 12 χ 2spec. This χ 2spec fits the calculated elastic
scattering rate in energy bin e of a particular SK phase p to
the measurement dpe  σ pe . The calculated event rate rpe isthe sum of the expected elastic scattering rate bpe from 8 B
neutrinos scaled by the parameter β and hep from hep
neutrinos scaled by the parameter η: rpe ¼ βbpe þ ηhpe . The
calculation includes neutrino flavor oscillations of threeflavors; they depend on the mixing angles θ12 , θ13 andthe mass squared difference Δm221 . rpe is then multipliedby the spectral distortion factor f pe ðτ; ϵp ; ρp Þ which
describes the effect of a systematic shift of the 8 B neutrino
spectrum scaled by the constrained nuisance parameter τ, adeviation in the SK energy scale in phase p described by
the constrained nuisance parameter ϵp, and a systematicchange in the SK energy resolution based on a third
constrained nuisance parameter ρp. If N p is the numberof energy bins of phase p, we minimize

χ 2p ðβ; ηÞ ¼

Np  pXde − f pe rpe ðsin2 θ12 ; sin2 θ13 ; Δm2 Þ 2
e¼1

σ pe

21

over all systematic nuisance parameters and the flux scaling
parameters:

052010-23

K. ABE et al.

χ 2spec;1 ¼

PHYSICAL REVIEW D 94, 052010 (2016)Min ðχ 2p;dat þ τ2 þ ϵ2p þ ρ2p þ ΦÞ;τ;ϵp ;ρp ;β;η

ð6:1Þ

η−η0 20 2
where Φ ¼ ðβ−β
σ β Þ þ ð σ η Þ constrains the flux parameters

to prior uncertainties: β is constrained to result in a 8 B fluxof ð5.25  0.20Þ × 106 =ðcm2 secÞ (motivated by the
SNO NC measurement of the total 8 B neutrino flux[19]), η is only slightly constrained to correspond to a
hep flux of ð8  16Þ × 103 =ðcm2 secÞ. The nuisanceparameters τ, ϵp , and ρp are constrained to 0  1 (i.e. they
are defined as standard Gaussian variables) by the “penalty
ρp −0 22 ϵp −0 2
terms” ðτ−0
1 Þ , ð 1 Þ , and ð 1 Þ . We rewrite Eq. (6.1) as aquadratic form with the 2 × 2 curvature matrix Cp and thebest-fit flux parameters βpmin and ηpmin asχ 2spec;αp ¼ χ 2p;min þ αp ðβ − βpmin ; η − ηpmin Þ
β − βpmin;· Cp ·
η − ηpmin

ð6:2Þ

for αp ¼ 1. The parameter αp ≠ 1 is introduced to scale thepﬃﬃﬃﬃﬃ
a posteriori constraints on the flux parameters by 1= αp
without affecting the χ 2 minimum in order to take into
account additional systematic uncertainties of the total
rate. These uncertainties are not covered by σ pe or f pe.
Table V (subtotal) lists these additional uncertainties
integrated over all energies. To incorporate them we choose
σ 2p;stat. To best compare this three-flavor analysis
2p;stat þσ p;syst

αp ¼ σ 2

to two-flavor analyses performed for previous phases, we
also perform an analysis with an a priori constraint on θ13

χ 2spec ¼

1010Δm =(3.2
2
21

+2.8
-0.2

-5

2

sin2(Θ13)=0.0219±0.0014

sin2(Θ12)=0.334+0.027-0.023) 10 eV

-4

2

-5

10

2

10

10

Δm in eV2
2

Δm in eV

10

-6
-7

10

-8

10

-9

10

p¼1

8 3σ
6
4 2σ
2

2
+0.026-0.031sin (Θ12)=0.327

10

p¼1


4
X
2
2
ðϵp þ ρp Þ þ Φ :

1σ

2

10

ν;ϵp ;ρp ;β;η

χ 2p;αp þ τ2 þ

Each SK phase is represented by a separate day/night
likelihood ratio, where the flux and nuisance parameters
are taken from the combined fit. Figure 33 shows the SK
combined allowed areas based on rate, spectrum, and

1σ

10

Min

X
4

ð6:3Þ

Δχ

Δχ

2

8 3σ
6
4 2σ
2

coming from reactor neutrino experiments [25]. Unlike thetwo-flavor analyses, θ13 is not fixed to zero, but constrained2
13 −0.0219 2
Þ . The time-variation
to a nonzero value by ðsin θ0.0014
likelihood log LDN ¼ log Lwith − log Lwithout is simply thedifference between the likelihoods with and without thepredicted day/night variation assuming the best-fit flux andnuisance parameters from the spectrum χ 2 minimization.
As the uncertainties in each spectral bin are closely
approximated by Gaussian uncertainties, the total χ 2 isthen given by χ 2spec − 2 log LDN. Figure 33 shows allowed
regions of oscillation parameters from SK-IV data withthe external constraint from reactor neutrino data on θ13
at the 1, 2, 3, 4, and 5σ confidence level. SK-IV deter2
mines sin2 θ12 to be 0.327þ0.026−0.031 , as well as Δm21 to be
þ2.8
−5
2
ð3.2−0.2 Þ × 10 eV . A secondary region appears at about
the 3σ level at Δm221 ≈ 8 × 10−8 eV2 . Small mixing is only
very marginally allowed at about the 5σ confidence level.
We combined the SK-IV constraints with those ofprevious SK phases, as well as other solar neutrino
experiments [19,22]. For the combined SK fit, the spectrum
and rate χ 2 is

1σ

-4

10

-3

10

-2

10

-1
2

tan (θ)

1

10

2σ

3σ

2 4 6 8
2

Δχ

10

-5
2
Δm221=(4.8+1.5-0.8) 10 eV

sin2(Θ13)=0.0219±0.0014

-4
-5
-6
-7
-8
-9

10

1σ

-4

10

-3

10

-2

10

-1
2

tan (θ)

1

10

2σ

3σ

2 4 6 8Δχ 2

FIG. 33. Contours of Δm221 vs tan2 θ12 from the SK-IV (left panel) and SK-I/II/III/IV (right panel) spectral þ day=night data with a 8 B
flux constraint of 5.25  0.20 × 106 =ðcm2 secÞ at the 1, 2, 3, 4, and 5σ confidence levels. The filled regions give the 3σ confidence level2
13 −0.0219 2
results. θ13 is constrained by ðsin θ0.0014
Þ.

052010-24

PHYSICAL REVIEW D 94, 052010 (2016)
8
6
4
2

2

8 3σ
6
4 2σ
2

Δχ

Δχ

2

SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

1σsin2(Θ12)=0.310±0.014

-4

2

-5
-6

10
10

2

2

10

-7
-8
-9

10

-4

10

-3

10

-2

-1

10
1
2
tan (θ)

1σ

10

1σ
2

-5

Δm in eV102σsin2(Θ12)=0.316+0.034-0.0262

10sin2(Θ13)=0.0219±0.0014

Δm in 10 eV

10

-5
2
Δm221=(4.8+1.3-0.6) 10 eV

3σ

2σ

3σ

2 4 6 8Δχ 2(Θ12)=0.308±0.014
18 sinsin2(Θ12)=0.307+0.013-0.012
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0.1
0.2

-5
2
Δm221=(7.54+0.19
sin2(Θ13)=0.0219±0.0014-0.18) 10 eV-52Δm221=(4.85+1.33-0.59) 10 eV
2
-5
2
+0.19
Δm21=(7.49 -0.18) 10 eV

1σ

0.3

0.4

0.5

2σ

3σ

2 4 6 8Δχ2

sin (θ12)

2

FIG. 34. Left: comparison of the oscillation parameter determination of the SK and SNO combined analysis (red) to the oscillation
constraints of SNO by itself (blue). Right: allowed contours of Δm221 vs sin2 θ12 from solar neutrino data (green), KamLAND data
(blue), and the combined result (red). For comparison, the almost identical result of the SK þ SNO combined fit is shown by the dasheddotted lines. The filled regions give the 3σ confidence level results, the other contours shown are at the 1 and 2σ confidence level (for the2
13 −0.0219 2
Þ.
solar analyses, 4 and 5σ confidence level contours are also displayed). θ13 is constrained by ðsin θ0.0014

Δχ

2

The combined allowed contours based on SK, SNO[19] and other solar neutrino experiments’ [22] data,
KamLAND’s constraints and the combination of the twoare shown in Figs. 34 and 35. SK and SNO dominate
the combined fit to all solar neutrino data. This can beseen from the two almost identical sets of green
contours in Fig. 34. The low energy measurement of
the 7 Be day/night asymmetry [29] does not change the

8 3σ
6
4 2σ
2
0.2 1σ2sin (Θ12)=0.326+0.044-0.037sin2(Θ12)=0.310+0.022-0.017sin2(Θ12)=0.309+0.014-0.0132sin (Θ13)

day/night variation. SK selects large mixing (0.5>sin2 θ12 >0.2) over small mixing by more than five standard
deviations and very strongly (3.6σ) favors the Δm221 of
the LMA solution (below 2 × 10−4 eV2 and above
2 × 10−5 eV2 ) over any other oscillation parameters. SK
2determines sin2 θ12 to be 0.334þ0.027−0.023 , as well as Δm21 to
þ1.5
−5
2
be ð4.8−0.8 Þ × 10 eV .
Figure 34 compares the SK þ SNO combined constraints to those based on SNO data alone [19]. While
SNO’s measurement of the mixing angle is more precise2(sin2 θ12 ¼ 0.299þ0.023−0.020 ) than SK’s, its Δm21 constraints are
þ1.9
−5
2
poorer [ð5.6−1.4 Þ × 10 eV ]. Also, SNO very slightly
favors the low mass (Low) solution (region near 10−7 eV2 )and allows small mixing at the 3.6σ level. The combined
analysis of SK and SNO is particularly powerful: as SNO
and SK both measure 8 B neutrinos in a very similar energy
range but in a different way and with different systematiceffects, the combined analysis profits from correlations andis better than a mere addition of χ 2 ’s. The SK þ SNO
combined analysis measures sin2 θ12 ¼ 0.310  0.014 and−5Δm221 ¼ ð4.8þ1.3
eV2 . Oscillation parameter val−0.6 Þ × 10
ues outside the LMA are very strongly excluded: the solar
mixing angle lies within 0.12 ≤ sin2 θ12 ≤ 0.45 at about the7.5σ C.L., Δm221 < 1.33 × 10−5 eV2 (which includes the“small mixing angle” and “low mass” regions) is ruledout at the 5.5σ C.L., and Δm221 >1.9×10−4 eV2 is excluded
at 7.5σ C.L. The hep flux constraint used by SNO isð7.9  1.2Þ × 103 =ðcm2 secÞ from the solar standard model[21]. The SK and SNO combined analysis also uses thistighter constraint.

sin2(Θ13)=0.010+0.033-0.034sin2(Θ13)=0.027+0.024-0.027sin2(Θ13)=0.028±0.015

0.1

0.0
0.0

1σ

0.1

0.2

0.3
2

sin (Θ12)

0.4

2σ

3σ

0.5 2 4 6 8
2Δχ

FIG. 35. Allowed contours of sin2 θ13 vs sin2 θ12 from solar
neutrino data (green) at 1, 2, 3, 4, and 5σ and KamLAND
measurements (blue) at the 1, 2, and 3σ confidence levels. Also
shown is the combined result in red. The yellow band is the θ13
measurement from reactor neutrino data [25].

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K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)

constraints in the LMA region, but independently
excludes the Low solution. In the right panel of thisfigure, some tension between the solar neutrino andreactor antineutrino measurements of the solar Δm221 isevident, stemming from the SK day/night measurement.
Even though the expected day/night amplitude agrees
within ∼1.1σ with the fitted amplitude for any Δm221, ineither the KamLAND or the SK range, the SK dataslightly favor the shape of the day/night variation
predicted by values of Δm221 that are smaller than
KamLAND’s. Figure 35 shows the results of the θ13
unconstrained fit. Solar neutrinos by themselves weakly
favor a nonzero θ13 by about one standard deviation
because for low energy solar neutrinos the survival
probability (e.g 7 Be) is about ð1 − 12 sin2 ð2θ12 ÞÞcos4 ðθ13 Þ
while the MSW effect causes a high energy (8 B) solar
neutrino survival probability of sin2 ðθ12 Þ cos4 ðθ13 Þ. This
results in a correlation of sin2 ðθ12 Þ and sin2 ðθ13 Þ forhigh energy neutrinos and an anticorrelation for low
energy neutrinos. KamLAND reactor neutrino data has
the same anticorrelation as the low energy solar neutrinos because in both cases matter effects play a minor
role. Therefore the significance of nonzero θ13 increases
in the solar þ KamLAND data combined fit to ∼2σ,favoring sin2 θ13 ¼ 0.028  0.015.

to be sin2 θ12 ¼ 0.334þ0.027−0.023 , the determined squared split−5 eV2 . A θ constrained fit toting is Δm221 ¼ 4.8þ1.5
13−0.8 × 10
all solar neutrino data and KamLAND yields sin2 θ12 ¼
þ0.19
2
−5
0.307þ0.013
eV2 . When
−0.012 and Δm21 ¼ ð7.49−0.18 Þ × 10
this constraint is removed, solar neutrino experiments
and KamLAND measure sin2 θ13 ¼ 0.028  0.015, a value
in good agreement with reactor neutrino measurements.
ACKNOWLEDGMENTS
The authors gratefully acknowledge the cooperation of
the Kamioka Mining and Smelting Company. Super-K has
been built and operated from funds provided by theJapanese Ministry of Education, Culture, Sports, Scienceand Technology, the U.S. Department of Energy, and the
U.S. National Science Foundation. This work was partially
supported by the Research Foundation of Korea (BK21 and
KNRC), the Korean Ministry of Science and Technology,
the National Research Foundation of Korea (NRF20110024009), the National Science Foundation ofChina (Grant No. 11235006), the European Union
H2020 RISE-GA641540-SKPLUS, and the National
Science Centre, Poland (2015/17/N/ST2/04064, 2015/18/
E/ST200758).
APPENDIX A: REVISED SK-III RESULTS

The fourth phase of SK measured the solar 8 B neutrinoelectron elastic scattering-rate with the highest precisionyet. SK-IV measured a solar neutrino flux of ð2.308 
6
20.020ðstatÞþ0.039
assuming no−0.040 ðsystÞÞ × 10 =ðcm secÞoscillations. When combined with the results from theprevious three phases, the SK combined flux is ð2.345 
0.014ðstatÞ  0.036ðsystÞÞ × 106 =ðcm2 secÞ. A quadratic
fit of the electron-flavor survival probability as a functionof energy to all SK data, as well as a combined fit with SNO
solar neutrino data, very slightly favor the presence ofspectral distortions, but are still consistent with an energyindependent electron neutrino flavor content. The SK-IV
solar neutrino elastic scattering day/night rate asymmetry
is measured as ð−3.6  1.6ðstatÞ  0.6ðsystÞÞ%. Combining this with other SK phases, the SK solar zenith
angle variation data gives the first significant indication
for matter-enhanced neutrino oscillation: the significancecompared to no day/night asymmetry is 2.9σ. This leads
SK to having the world’s most precise measurement−5of Δm221 ¼ ð4.8þ1.5
eV2 , using neutrinos rather−0.8 Þ × 10
than antineutrinos. There is a slight tension of 1.5σ
between this value and KamLAND’s measurement using
reactor antineutrinos. The tension increases to 1.6σ,
if other solar neutrino data are included. The SK-IV
solar neutrino data determine the solar mixing angle assin2 θ12 ¼ 0.327þ0.026−0.031 , all SK solar data measures this angle

Since the publication of the previous report [9], two
mistakes were found. One is in how energy-dependent
systematic errors are calculated and the other is related to
the flux calculation in SK-III. The estimates of the energycorrelated uncertainties in the main text of that report are
based on the Monte Carlo (MC) simulated 8 B solar
neutrino events. It is found that this evaluation method
was not accurate enough. The statistical error of the MC
×103
True spectrum

160

Spectrum for the previous SK-III

140
Arbitrary unit

VII. CONCLUSION

120
100
80
60
40
20
0

0

2

4
6
8
10
12
Total electron energy [MeV]14

FIG. 36. Energy spectrum shapes of recoil electrons from 8 B
solar neutrinos for SK-III. The blue dotted and red solid lines
show the true theoretical calculation and incorrect spectrum usedin the SK-III analysis in the previous report [9].

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SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

PHYSICAL REVIEW D 94, 052010 (2016)

simulation distorted the shapes of the energy-correlated
uncertainties systematically.
The energy dependence of the differential interaction
cross section between neutrinos and electrons was accidentally eliminated only for the SK-III flux calculation in
the main text. Figure 36 shows the energy distributions ofrecoil electrons from 8 B solar neutrinos. The blue dotted
histogram shows the true energy spectrum shape from atheoretical calculation considering the detector resolutions.
The red solid plot shows the energy spectrum shape used in
the SK-III analysis in the previous report. The expected
total flux was normalized correctly, but the expected 8 B
energy spectrum shape was improperly distorted in theanalysis.
These mistakes have been fixed in this paper. In thisappendix, the revised SK-III solar neutrino results aredescribed. The latest oscillation results, including bothrevised SK-III data and SK-IV data, are reported in themain text of this report.
1. Systematic uncertainties
The energy-correlated systematic uncertainties are
obtained by counting the number of events in the solar
neutrino MC simulation with artificially shifted energy
scale, energy resolution and 8 B solar neutrino energy
spectrum. In the SK-III analysis in the previous report,
this estimation was done with the generated solar
neutrino MC events. However, in the high energy region,
not enough MC events were generated to accurately
estimate the small systematic errors. In the current
analysis, this estimation is performed with a theoretical
calculation considering the detector resolutions, thus
eliminating the statistical effects introduced by the small
MC statistics.

TABLE XIII Revised summary of the systematic uncertainty ofthe total flux in Etotal ¼ 5.0–20.0 MeV in SK-III. This is arevision of Table IV in the previous paper [9].
SourceTotal Flux

Energy scale
Energy resolution
8
B spectrum
Trigger efficiency
Angular resolution
Vertex shift
Event quality cuts
- Reconstruction Goodness- Hit pattern- Second vertex
Spallation cut
Gamma-ray cut
Cluster hit cut
Background shape
Signal extraction
Livetime
Cross section

1.4%
0.2%
0.4%
0.5%
0.67%
0.54%
0.4%
0.25%
0.45%
0.2%
0.25%
0.5%
0.1%
0.7%
0.1%
0.5%

Total

2.2%

The revised results of the energy-correlated systematic
uncertainties are shown in Fig. 37. In this update, theuncertainty from 8 B spectrum shape was improved.
The systematic uncertainties on total flux in SK-III
are also revised. The revised uncertainties are summarized in Table XIII. The 8 B spectrum error wasunderestimated in the analysis in the main text of[9]. The revised systematic uncertainty on the total fluxin Etotal ¼ 5.0–20.0 MeV in SK-III is estimated tobe 2.2%.

Energy correlated uncertainty

2. 8 B solar neutrino flux results
The observed number of solar neutrino events is
also updated. In this analysis, the extracted number of8
B solar neutrinos with the ES reaction in Etotal ¼
5.0–20.0 MeV for a live time of 548 days of SK-III data8
was 8148þ133−131 ðstatÞ  176ðsysÞ. The corresponding B flux
is obtained to be:

0.1

02.404  0.039ðstatÞ  0.053ðsysÞ × 106 cm−2 sec−1 :

-0.1
5

10
15
Total electron energy (MeV)20

FIG. 37 Revised energy-correlated systematic uncertainties
in SK-III. The solid, dotted, and dashed lines show the uncertainties of the 8 B spectrum, the energy scale, and the energy
resolution, respectively. This is a revision of Fig. 25 in theprevious paper [9].

Fixing the cross section problem, a 3.4% increase wasobserved.
The observed and expected fluxes are reestimated in eachenergy region. Table XIV shows the revised event rate ineach energy region. Figure 38 shows the revised observed
energy spectrum divided by the 5.79×106 cm−2 sec−1 flux
value without oscillations.

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K. ABE et al.

PHYSICAL REVIEW D 94, 052010 (2016)

TABLE XIV Revised observed energy spectra expressed in units of event/kton/year in SK-III in each recoil
electron total energy region. The errors in the observed rates are statistical only. The expected rates neglecting
oscillations are for a flux value of 5.79 × 106 cm−2 sec−1 . θz is the angle between the z-axis of the detector and thevector from the Sun to the detector. This is a revision of Table VI in the previous paper [9].
Energy(MeV)

ALL−1 ≤ cos θz ≤ 1

Observed rateDAY
−1 ≤ cos θz ≤ 0

NIGHT
0 < cos θz ≤ 1

82.3þ10.3−9.9
66.4þ6.4−6.1
62.9þ4.9−4.7
54.8þ2.7−2.6
53.8þ2.5−2.4
40.4þ2.2−2.1
36.4þ1.9−1.8
30.5þ1.7−1.6
22.4þ1.4−1.3
19.1þ1.2−1.2
14.3þ1.0−1.0
13.7þ1.0−0.9
9.41þ0.79−0.73
5.63þ0.64−0.57
4.91þ0.57−0.50
3.03þ0.44−0.38
1.92þ0.35−0.29
1.32þ0.29−0.23
2.15þ0.36−0.30
0.832þ0.234−0.175
þ0.1300.112−0.064

93.4þ15.7−14.9
73.7þ9.8−9.3
55.3þ7.0−6.5
50.8þ3.8−3.7
55.6þ3.6−3.5
39.6þ3.1−3.0
37.2þ2.7−2.6
28.4þ2.3−2.2
19.8þ1.9−1.8
17.7þ1.7−1.6
15.0þ1.5−1.4
14.7þ1.4−1.3
9.36þ1.17−1.03
5.24þ0.90−0.76
4.08þ0.79−0.66
2.67þ0.61−0.49
1.59þ0.47−0.35
1.13þ0.39−0.27
2.00þ0.51−0.40
0.381þ0.289−0.1580.244þ0.238−0.11772.6þ13.7−13.0
59.9þ8.4−7.9
70.4þ7.1−6.7
58.7þ3.8−3.7
52.1þ3.5−3.3
41.1þ3.1−2.9
35.7þ2.6−2.5
32.6þ2.4−2.2
24.9þ2.1−1.9
20.3þ1.8−1.7
13.6þ1.4−1.3
12.9þ1.3−1.2
þ1.11
9.44−0.98
þ0.94
6.04−0.81
þ0.85
5.69−0.73
3.38þ0.65
−0.53
þ0.55
2.25−0.43
1.48þ0.47−0.35
2.31þ0.53−0.42
þ0.3851.208−0.275
þ0.1230.000−0.401

5.0–5.5
5.5–6.0
6.0–6.5
6.5–7.0
7.0–7.5
7.5–8.0
8.0–8.5
8.5–9.0
9.0–9.5
9.5–10.0
10.0–10.5
10.5–11.0
11.0–11.5
11.5–12.0
12.0–12.5
12.5–13.0
13.0–13.5
13.5–14.0
14.0–15.0
15.0–16.0
16.0–20.0

APPENDIX B: PARAMETRIZED SURVIVALPROBABILITY FIT
We fit the SK spectral data to the exponential,
quadratic, and cubic survival probability in the same

Data/SSM(BP2004)

0.8

0.6

0.4

0.2

0

5

10

15

20

Total electron energy (MeV)

FIG. 38 Revised ratio of observed and expected energy spectra
in SK-III. The dashed line represents the revised SK-III average.
This is a revision of Fig. 27 in the previous paper [9].

Expected rate
hep

8B

189.7
172.2
155.2
134.3
117.1
101.2
85.8
71.7
58.5
47.1
37.0
28.5
21.45
15.76
11.21
7.79
5.22
3.39
3.49
1.227
0.513

0.334
0.321
0.310
0.289
0.271
0.257
0.240
0.223
0.205
0.186
0.169
0.151
0.134
0.118
0.102
0.088
0.074
0.062
0.092
0.059
0.068

manner as we fit them to the MSW prediction. Figure 39
shows the resulting allowed areas of the exponential
coefficients e1 and e2 . The “baseline” (average Pee ) e0 is
profiled; the e0 constraint results from the comparison of
the electron elastic scattering rate in SK and the SNO
neutral-current interaction rate on deuterium. The contours deviate from a multivariate Gaussian. As there is no
significant deviation from an undistorted spectrum, thedata impose no constraint on e2 , the “steepness” of the
exponential. Table XVII uses the best quadratic form
approximation of the χ 2 of the fit as a function of theparameters to extract the values, uncertainties and correlations. Figure 40 shows the allowed shape parameters(c1 and c2 ) and the allowed slope (c1 ) vs the baseline(c0 ) of the quadratic fit. The SK-IV contours show somedeviations from a multivariate Gaussian at 3σ, while
the SK combined result is consistent with it. Overlaid inblue are the constraints from the SNO measurements. The
corresponding coefficients of Table XVII differ slightly
from those in [19] which fits both the survival probability
to a quadratic function and the energy dependent
day/night asymmetry to a linear function. Here, we

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PHYSICAL REVIEW D 94, 052010 (2016)
8 3σ
6
4 2σ
2

2

8 3σ
6
4 2σ
2
0.3 1σ
0.2
0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
-0.7
-0.8
-0.9

Δχ

Δχ

2

SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

1σ

e2

c2

0.01

0

-0.01

1σ

-0.04

0.0

2σ

3σ

0.04 2 4 6 8

1σ

-0.04

0.00

Δχ2

e1

2σ

3σ

0.04 2 4 6 8

Δχ2

c1

FIG. 39 Allowed areas of the shape parameters (e1 and e2 on left, c1 and c2 on the right) of an exponential (left) and quadratic (right) fit
to the survival probability Pee of SK-IV (solid lines) and all SK data (dashed lines) at the 1, 2 (filled region) and 3σ confidence levels.
The oscillation parameter set corresponding to the SK (or all solar neutrino) data best-fit is indicated by the white star. The solar þ
KamLAND best-fit (black star) is also shown.

assume the energy dependence of the day/night effectcalculated from standard earth matter effects. The resulting reduction in the degree of freedom leads to somewhat
tighter constraints as well as a slight shift in the best fit
value. The precision of the SK constraint is similar to thatbased on SNO data, and also statistically consistent.

Since SK’s correlation between c1 and c2 is opposite tothat of SNO’s, a combined fit is rather powerful inconstraining the shape. The c1 − c2 correlation is slightly
smaller. The addition of SK data to SNO data not only
significantly increases the precision of the c0 determination, but the uncertainties on the shape are reduced.

TABLE XV SK exponential and polynomial best-fit coefficients and their correlations. Also given are SNO’s quadratic fit coefficients(slightly different than the published value since the day/night asymmetry is not fit) as well as SK and SNO combined measured
quadratic fit coefficients and their respective correlations.
Data set
SK-IV
SK

e0

e1

0.326  0.024 −0.0029  0.0073
0.336  0.023 −0.0014  0.0051

e2

e0 -e1 corr.

no constraint
no constraint

þ0.202
þ0.077

quadratic function
SK-IV
c0
c1
c2
c3
SK

cubic function

c1
c2
c0
c1
c2
c3
c0
0.324  0.025 −0.0030  0.0097 0.0012  0.0040 0.313  0.028 −0.018  0.021 0.0059  0.0074 0.0021  0.0028
1−0.125−0.412−0.1251
þ0.683−0.412
þ0.683011
þ0.388−0.602−0.488

þ0.3881−0.580−0.892−0.602−0.5801
þ0.8390.334  0.023 −0.0003  0.0065 0.0008  0.0029 0.313  0.024 −0.031  0.016 0.0097  0.0051 0.0044  0.0020
1þ0.258−0.449−0.327

þ0.2581−0.599−0.916−0.449−0.5991
þ0.814c0 -c1 corr.

c0 -c2 corr.

c1 -c2 corr.

0.315  0.017 −0.0007  0.0059 −0.0011  0.0033−0.301−0.391−0.312SK þ SNO 0.311  0.015 −0.0034  0.0036 þ0.0004  0.0018−0.453−0.407

þ0.301

c0
c1
c2
c3

SNO

−0.488−0.892
þ0.8391

1−0.131−0.345−0.1311
þ0.649−0.345
þ0.6491052010-29−0.327−0.916
þ0.814
1

PHYSICAL REVIEW D 94, 052010 (2016)
8 3σ
6
4 2σ
2

2

8 3σ
6
4 2σ
2

Δχ

Δχ

2

K. ABE et al.

1σ

1σ

0.03
0.01

0.02

c1

c2

0.01
0

0
-0.01

-0.01

-0.02
-0.03
1σ

-0.04

0.00

c1

2σ

3σ

1σ

0.04 2 4 6 8

Δχ

0.3

2

0.4

2σ

3σ

2 4 6 82Δχc0

FIG. 40. Left: Allowed areas of the shape parameters (c1 and c2 ) of a quadratic fit to the survival probability Pee of SK (solid green)
and SNO (dashed blue) data at the 1, 2 (filled region) and 3σ confidence levels. Right: Allowed areas of the slope (c1 ) and baseline (c0 )of a quadratic fit to the survival probability Pee of SK (solid green) and SNO (dashed blue) data at the 1, 2 (filled region) and 3σ
confidence levels. Also shown is a combined fit (dotted red). The oscillation parameter set corresponding to the SK (all solar neutrino)
data best-fit is indicated by the dark green (light blue) star. The solar þ KamLAND best-fit (dark blue) is also shown.

APPENDIX C: TABLESTABLE XVI The observed event rates in each energy bin (events/year/kton), at 1 AU. The errors are statistical errors only. The
reduction efficiencies are corrected and the expected event rates are for a flux of 5.25 × 106 =ðcm2 secÞ.
Energy(MeV)
3.49–3.99
3.99–4.49
4.49–4.99
4.99–5.49
5.49–5.99
5.99–6.49
6.49–6.99
6.99–7.49
7.49–7.99
7.99–8.49
8.49–8.99
8.99–9.49
9.49–9.99
9.99–10.5
10.5–11.0
11.0–11.5
11.5–12.0
12.0–12.5
12.5–13.0
13.0–13.5
13.5–14.5
14.5–15.5
15.5–19.5

ALL−1 ≤ cos θz ≤ 1

Observed rateDAY
−1 ≤ cos θz ≤ 0

NIGHT
0 < cos θz ≤ 1

92.2þ10.8−10.6
76.7þ5.2−5.1
82.1þ3.4−3.3
69.3þ2.1−2.1
59.6þ1.6−1.6
54.2þ1.4−1.4
47.8þ1.3−1.3
40.6þ1.2−1.1
35.7þ1.0−1.0
29.1þ0.9−0.9
24.0þ0.8−0.8
18.5þ0.7−0.7
14.5þ0.6−0.6
10.7þ0.5−0.5
8.43þ0.43−0.41
6.60þ0.37−0.35
4.40þ0.30−0.28
3.04þ0.25−0.23
2.14þ0.20−0.18
1.47þ0.17−0.15
1.59þ0.17−0.15
0.469þ0.102−0.0820.186þ0.072−0.05196.0þ16.8−16.3
64.6þ7.9−7.6
79.4þ5.1−5.0
65.9þ3.1−3.0
58.3þ2.3−2.2
51.0þ2.1−2.0
45.7þ1.9−1.8
41.8þ1.7−1.7
35.0þ1.5−1.5
28.6þ1.3−1.3
24.1þ1.2−1.1
17.9þ1.0−0.9
14.5þ0.9−0.8
10.2þ0.7−0.7
7.73þ0.61−0.56
6.60þ0.54−0.49
3.83þ0.41−0.37
3.04þ0.35−0.31
2.41þ0.31−0.27
1.48þ0.25−0.21
1.54þ0.25−0.21
0.486þ0.151−0.1120.150þ0.108−0.06581.5þ14.0−13.6
85.2þ6.9−6.7
84.6þ4.6−4.5
72.5þ3.0−2.9
60.5þ2.2−2.2
56.9þ2.0−2.0
49.9þ1.9−1.8
39.5þ1.6−1.6
36.1þ1.5−1.4
28.9þ1.3−1.2
23.7þ1.1−1.1
19.2þ1.0−0.9
14.4þ0.8−0.8
11.1þ0.7−0.7
9.23þ0.64−0.60
6.72þ0.53−0.49
4.89þ0.44−0.40
3.06þ0.36−0.32
1.93þ0.29−0.25
1.47þ0.25−0.21
1.63þ0.25−0.22
0.493þ0.161−0.1210.203þ0.113−0.071052010-30

8B

Expected rate

196.8
182.8
167.8
153.3
137.8
121.9
106.8
92.1
78.0
65.2
53.4
42.9
33.8
26.0
19.55
14.34
10.24
7.10
4.80
3.11
3.18
1.117
0.464

hep
0.346
0.335
0.323
0.312
0.298
0.282
0.266
0.250
0.232
0.214
0.197
0.179
0.162
0.144
0.128
0.112
0.097
0.083
0.070
0.059
0.088
0.056
0.064

SOLAR NEUTRINO MEASUREMENTS IN SUPER- …

PHYSICAL REVIEW D 94, 052010 (2016)

TABLE XVII Elastic scattering rate ratios and energy-uncorrelated uncertainties (statistical plus systematic) for
each SK phase.
Energy (MeV)

SK-I

SK-II

SK-III

SK-IV

3.49–3.99−−−

3.99–4.49−−

4.49–4.99−

4.99–5.49

0.453þ0.043−0.0420.430þ0.023−0.0220.448þ0.100−0.096

þ0.0600.468−0.059

5.49–5.99

0.449  0.018−

5.99–6.49

0.444  0.015−

6.49–6.99

0.461þ0.016−0.0150.439þ0.050−0.048

6.99–7.49

0.476  0.016

7.49–7.99
8.49–8.99

0.457þ0.017−0.016
0.431þ0.017−0.0160.454þ0.018−0.017

8.99–9.49

0.464  0.019

9.49–9.99

0.456þ0.021−0.0209.99–10.5

0.409  0.021

10.5–11.0

0.472þ0.025−0.0240.439þ0.028−0.0260.460þ0.033−0.0310.465þ0.039−0.036
0.461þ0.048−0.0430.582þ0.064−0.0570.475þ0.059−0.0520.724þ0.120−0.1020.575þ0.173−0.130

7.99–8.49

11.0–11.5
11.5–12.0
12.0–12.5
12.5–13.0
13.0–13.5
13.5–14.5
14.5–15.5
15.5–19.5−

0.448þ0.043−0.041
0.461þ0.037−0.0360.473þ0.036−0.0350.463þ0.036−0.0340.499þ0.038−0.0370.474þ0.038−0.036
0.481þ0.041−0.0390.452þ0.043−0.0400.469þ0.046−0.0430.482þ0.052−0.0480.419þ0.054−0.0490.462þ0.063−0.0570.444þ0.070−0.0620.430þ0.066−0.0590.563þ0.100−0.0870.648þ0.123−0.103[1] J. Hosaka et al., Phys. Rev. D 73, 112001 (2006).
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0.472þ0.058−0.0560.420þ0.039−0.0370.457þ0.035−0.0340.433þ0.023−0.0220.504þ0.025−0.0240.424þ0.024−0.0230.467þ0.024−0.0230.469þ0.026−0.0250.420þ0.026−0.0250.444þ0.029−0.0270.423þ0.031−0.0290.529þ0.037−0.0350.481þ0.041−0.0370.391þ0.044−0.0400.479þ0.055−0.0490.425þ0.061−0.0530.400þ0.073−0.061
0.422þ0.093−0.074
0.663þ0.110−0.0930.713þ0.201−0.1500.212þ0.248−0.1220.419  0.030
0.488  0.023
0.451  0.014
0.432  0.012
0.444  0.015
0.447  0.015
0.440  0.015
0.455  0.014
þ0.0150.439−0.014
þ0.0160.445−0.015

0.430  0.016
þ0.0180.426−0.017
þ0.0190.408−0.018
þ0.0230.432−0.021
þ0.026
0.461−0.025
þ0.0290.423−0.027
þ0.0350.425−0.032
þ0.0430.445−0.039
þ0.0550.465−0.049
þ0.0540.485−0.048
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052010-32