PHYSICAL REVIEW D 98, 052006 (2018)

Measurement of the tau neutrino cross section in atmospheric neutrino
oscillations with Super-Kamiokande
Z. Li,9 K. Abe,1,37 C. Bronner,1 Y. Hayato,1,37 M. Ikeda,1 K. Iyogi,1 J. Kameda,1,37 Y. Kato,1 Y. Kishimoto,1,37 Ll. Marti,1
M. Miura,1,37 S. Moriyama,1,37 M. Nakahata,1,37 Y. Nakajima,1,37 Y. Nakano,1 S. Nakayama,1,37 A. Orii,1 G. Pronost,1
H. Sekiya,1,37 M. Shiozawa,1,37 Y. Sonoda,1 A. Takeda,1,37 A. Takenaka,1 H. Tanaka,1 S. Tasaka,1 T. Tomura,1,37 R. Akutsu,2T. Kajita,2,37 Y. Nishimura,2 K. Okumura,2,37 K. M. Tsui,2 P. Fernandez,3 L. Labarga,3 F. d. M. Blaszczyk,4 J. Gustafson,4C. Kachulis,4 E. Kearns,4,37 J. L. Raaf,4 J. L. Stone,4,37 L. R. Sulak,4 S. Berkman,5 S. Tobayama,5 M. Elnimr,6 W. R. Kropp,6
S. Locke,6 S. Mine,6 P. Weatherly,6 M. B. Smy,6,37 H. W. Sobel,6,37 V. Takhistov,6,* K. S. Ganezer,7 J. Hill,7 J. Y. Kim,8
I. T. Lim,8 R. G. Park,8 A. Himmel,9 E. O’Sullivan,9 K. Scholberg,9,37 C. W. Walter,9,37 T. Ishizuka,10 T. Nakamura,11J. S. Jang,12 K. Choi,13 J. G. Learned,13 S. Matsuno,13 S. N. Smith,13 J. Amey,14 R. P. Litchfield,14 W. Y. Ma,14 Y. Uchida,14
M. O. Wascko,14 S. Cao,15 M. Friend,15 T. Hasegawa,15 T. Ishida,15 T. Ishii,15 T. Kobayashi,15 T. Nakadaira,15
K. Nakamura,15,37 Y. Oyama,15 K. Sakashita,15 T. Sekiguchi,15 T. Tsukamoto,15 KE. Abe,16 M. Hasegawa,16 A. T. Suzuki,16
Y. Takeuchi,16,37 T. Yano,16 T. Hayashino,17 T. Hiraki,17 S. Hirota,17 K. Huang,17 M. Jiang,17 M. Mori,17 KE. Nakamura,17
T. Nakaya,17,37 N. D. Patel,17 R. A. Wendell,17,37 L. H. V. Anthony,18 N. McCauley,18 A. Pritchard,18 Y. Fukuda,19
Y. Itow,20,21 M. Murase,20 F. Muto,20 P. Mijakowski,22 K. Frankiewicz,22 C. K. Jung,23 X. Li,23 J. L. Palomino,23
G. Santucci,23 C. Vilela,23 M. J. Wilking,23 C. Yanagisawa,23,† G. Yang,23 S. Ito,24 D. Fukuda,24 H. Ishino,24 A. Kibayashi,24
Y. Koshio,24,37 H. Nagata,24 M. Sakuda,24 C. Xu,24 Y. Kuno,25 D. Wark,26,32 F. Di Lodovico,27 B. Richards,27
S. M. Sedgwick,27 R. Tacik,28,41 S. B. Kim,29 A. Cole,30 L. Thompson,30 H. Okazawa,31 Y. Choi,33 K. Ito,34 K. Nishijima,34
M. Koshiba,35 Y. Suda,36 M. Yokoyama,36,37 R. G. Calland,37 M. Hartz,37 K. Martens,37 M. Murdoch,37 B. Quilain,37
C. Simpson,37,26 Y. Suzuki,37 M. R. Vagins,37,6 D. Hamabe,38 M. Kuze,38 Y. Okajima,38 T. Yoshida,38 M. Ishitsuka,39
J. F. Martin,40 C. M. Nantais,40 H. A. Tanaka,40 T. Towstego,40 A. Konaka,41 S. Chen,42 L. Wan,42 Y. Zhang,42
A. Minamino,43 and R. J. Wilkes44(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
Department of Physics and Astronomy, University of California, Irvine,
Irvine, California 92697-4575, USA
7
Department of Physics, California State University, Dominguez Hills, Carson, California 90747, USA
8
Department of Physics, Chonnam National University, Kwangju 500-757, Korea
9
Department of Physics, Duke University, Durham, North Carolina 27708, USA
10
Junior College, Fukuoka Institute of Technology, Fukuoka, Fukuoka 811-0295, Japan
11
Department of Physics, Gifu University, Gifu 501-1193, Japan
12
GIST College, Gwangju Institute of Science and Technology, Gwangju 500-712, Korea
13
Department of Physics and Astronomy, University of Hawaii, Honolulu, Hawaii 96822, USA
14
Department of Physics, Imperial College London, London SW7 2AZ, United Kingdom
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, University of Liverpool, Liverpool L69 7ZE, United Kingdom
19
Department of Physics, Miyagi University of Education, Sendai, Miyagi 980-0845, Japan
20
Institute for Space-Earth Environmental Research, Nagoya University, Nagoya, Aichi 464-8602, Japan
21
Kobayashi-Maskawa Institute for the Origin of Particles and the Universe, Nagoya University,
Nagoya, Aichi 464-8602, Japan
22
National Centre For Nuclear Research, 00-681 Warsaw, Poland
23
Department of Physics and Astronomy, State University of New York,
Stony Brook, New York 11794-3800, USA

2470-0010=2018=98(5)=052006(16)052006-1

Published by the American Physical Society

Z. LI et al.

PHYS. REV. D 98, 052006 (2018)24

Department of Physics, Okayama University, Okayama, Okayama 700-8530, Japan
25
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
26
Department of Physics, Oxford University, Oxford OX1 3PU, United Kingdom
27
School of Physics and Astronomy, Queen Mary University of London, London E1 4NS, United Kingdom
28
Department of Physics, University of Regina, 3737 Wascana Parkway,
Regina, Saskatchewan S4SOA2, Canada
29
Department of Physics, Seoul National University, Seoul 151-742, Korea
30
Department of Physics and Astronomy, University of Sheffield, Sheffield S10 2TN, United Kingdom
31
Department of Informatics in Social Welfare, Shizuoka University of Welfare,
Yaizu, Shizuoka 425-8611, Japan
32
STFC, Rutherford Appleton Laboratory, Harwell Oxford, and Daresbury Laboratory,
Warrington OX11 0QX, United Kingdom
33
Department of Physics, Sungkyunkwan University, Suwon 440-746, Korea
34
Department of Physics, Tokai University, Hiratsuka, Kanagawa 259-1292, Japan
35
The University of Tokyo, Bunkyo, Tokyo 113-0033, Japan36
Department of Physics, University of Tokyo, Bunkyo, Tokyo 113-0033, Japan
37
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
38
Department of Physics, Tokyo Institute of Technology, Meguro, Tokyo 152-8551, Japan
39
Department of Physics, Faculty of Science and Technology, Tokyo University of Science,
Noda, Chiba 278-8510, Japan40
Department of Physics, University of Toronto, Ontario M5S 1A7, Canada
41
TRIUMF, 4004 Wesbrook Mall, Vancouver, British Columbia, V6T2A3 Canada
42
Department of Engineering Physics, Tsinghua University, Beijing 100084, China
43
Faculty of Engineering, Yokohama National University, Yokohama 240-8501, Japan
44
Department of Physics, University of Washington, Seattle, Washington 98195-1560, USA(Received 27 November 2017; published 13 September 2018)
Using 5326 days of atmospheric neutrino data, a search for atmospheric tau neutrino appearance has
been performed in the Super-Kamiokande experiment. Super-Kamiokande measures the tau normalization
to be 1.47  0.32 under the assumption of normal neutrino hierarchy, relative to the expectation of unity
with neutrino oscillation. The result excludes the hypothesis of no-tau appearance with a significance levelof 4.6σ. The inclusive charged-current tau neutrino cross section averaged by the tau neutrino flux at SuperKamiokande is measured to be ð0.94  0.20Þ × 10−38 cm2 . The measurement is consistent with theStandard Model prediction, agreeing to within 1.5σ.
DOI: 10.1103/PhysRevD.98.052006

I. INTRODUCTION
In the three-flavor neutrino framework, the three neutrino
flavor states (νe , νμ , ντ ) are superpositions of three neutrinomass states (ν1 , ν2 , ν3 ). The oscillation parameters in theframework have been measured in atmospheric neutrino
experiments [1–3], solar neutrino experiments [4–6], reactor
neutrino experiments [7–10], and long-baseline neutrino
experiments [11–13]. Atmospheric neutrino observations*

Also at Department of Physics and Astronomy, UCLA,
California 90095-1547, USA.
†
Also at BMCC/CUNY, Science Department, New York,New York, USA.
Published by the American Physical Society under the terms of
the Creative Commons Attribution 4.0 International license.
Further distribution of this work must maintain attribution to
the author(s) and the published article’s title, journal citation,
and DOI. Funded by SCOAP3.

are characterized by a large deficit of muon events. The
deficit is largely explained by the quantum mechanicalmixing of the propagating mass states such that weakinteraction at the detector is comprised of a mixture of muonand tau flavors, whereas most of the tau neutrino flux has
energy below the tau lepton production threshold. The
objective of this paper is to observe those tau neutrino
interactions that are above that threshold. A direct detectionof tau neutrinos from neutrino oscillation is important
for an unambiguous confirmation of three-flavor neutrino
oscillations.
However, the detection of tau neutrino appearance ischallenging. Charged-current neutrino interactions are
required to determine the flavor in neutrino detection.
Charged-current tau lepton appearance has an energy
threshold of 3.5 GeV, and the charged-current tau neutrino
cross section is greatly suppressed at low energies due tothe large mass of the tau lepton relative to the electron or

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MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION …
muon. The DONUT experiment first directly observed thetau neutrino by measuring charged-current interactions
using a high-energy neutrino beam that contained tau
neutrinos [14]. Long-baseline experiments tuned for maximum oscillation have the bulk of their neutrinos below
this energy. In addition, the tau lepton has an extremely
short lifetime, making a direct detection very difficult.
Nevertheless, the long-baseline neutrino experiment
OPERA measured tau neutrino appearance in a highenergy muon neutrino beam by observing five ντ eventswith a background expectation of 0.25 events [15].
Atmospheric neutrinos are mostly electron or muon
neutrinos at production [16]. Tau neutrino appearance is
expected in the atmospheric neutrinos from neutrino
oscillations. In three-flavor neutrino oscillation in thevacuum, the probability of ντ appearance can be approximately expressed asLPνμ →ντ ≃ cos2 θ13 sin2 ð2θ23 Þsin2 1.27Δm232
;
ð1aÞ
E


2
2
2
2 L
Pνe →ντ ≃ sin ð2θ13 Þ cos ðθ23 Þ sin 1.27Δm32;
ð1bÞ
E
where Δm232 ≡ m23 − m22 is the mass splitting in eV2 , θij is a
mixing angle in the Pontecorvo-Maki-Nakagawa- Sakata(PMNS) matrix, L is neutrino path length in km, and E isneutrino energy in GeV. Atmospheric neutrinos have energies
spanning many orders of magnitude from 10 MeV to morethan 1 TeV; the high energy component of the atmospheric
neutrinos have enough energy for charged-current tau neutrino
interactions. Super-Kamiokande is expected to detect roughly
one charged-current tau neutrino interaction per kiloton ofwater per year. Super-Kamiokande previously published ameasurement of atmospheric tau neutrino appearance consistent with three-flavor neutrino oscillation with data collected in SK-I through SK-III [17]. This analysis has been
updated with data collected in SK-IV between 2008 and 2016,
and the simulation and reconstruction have been improved.
Using the measured charged-current tau neutrino events,
Super-Kamiokande also measures the charged-current tau
neutrino cross section.
This paper proceeds as follows: Section II describes
some basic features of the Super-Kamiokande experiment(Super-K, SK). Section III describes Monte Carlo simulations of both the charge-current tau neutrino signal andthe atmospheric neutrino background. Section IV describes
standard data selection and reconstruction algorithms usedin the analysis. Section V describes a neural network
algorithm developed to select the tau signal. Section VI
describes a search for atmospheric tau neutrino appearance,
and a measurement of charge-current tau neutrino crosssection. Section VII presents our results and conclusion.
II. THE SUPER-KAMIOKANDE DETECTOR
Super-Kamiokande is a 50 kiloton cylindrical waterCherenkov detector located in the Kamioka mine under

PHYS. REV. D 98, 052006 (2018)about 1 km rock overburden (2.7 km water equivalent) at
the Ikenoyama mountain in Japan [18,19]. The detector isarranged into two optically separated regions: the innerdetector (ID) and the outer detector (OD). The ID is
instrumented with 11 129 20-inch inward-facing photomultiplier tubes (PMTs) and the OD is instrumented with
1,885 8-inch outward-facing PMTs. The PMTs collectCherenkov light produced in the ultrapure water in thedetector. A fiducial volume of the ID is defined as thecylindrical volume 2 meters inward from the ID wall, andhas a mass of 22.5 kilotons [18].
Super-K has been in operation since 1996, and has had
four data-taking periods. The first period, called SK-I,
began in April 1996, with 11 146 PMTs covering 40% of
the ID surface. The SK-I period continued until July 2001,
totaling 1489.2 live-days. An accident in November 2001
destroyed half of the ID PMTs. The remaining 5,182 PMTs
were rearranged uniformly on the ID surface, covering 19%of the surface. The data-taking period with this decreased
photocoverage between December 2002 and October 2005
is called SK-II. This period lasted 798.6 live-days. A fullreconstruction of the detector restored the photocoverage to
40% in 2006. The third data-taking period, called SK-III,lasted between July 2006 and September 2008, comprising518.1 live-days. The detector was upgraded with improved
electronics in the summer of 2008 [20]. The period after the
upgrade is referred to as SK-IV. In this paper, SK-IV data
are used up to March 2016, totaling 2519.9 live-days. The
complete SK-I through SK-IV data set comprises a total
exposure of 5,326 live-days.
III. SIMULATION
In order to predict the rate of both tau signal andatmospheric neutrino background, a full Monte Carlo(MC) simulation is used to model both the neutrino interactions and the detector response of Super-K. Since the four
Super-K periods have different detector configurations,separate sets of MC for both tau signal and atmospheric
neutrino background are generated for each period.
Atmospheric neutrinos are produced from the decays ofcharged mesons and muons in the cosmic-ray induced
atmospheric showers, and are mostly νμ and νe at production. The intrinsic tau neutrinos in the atmospheric neutrino
flux are negligible for this analysis [21]. Three-dimensional
neutrino fluxes of νμ and νe are modeled from thecalculation of Honda et al. [16]. The calculation predicts
the fluxes of electron and muon neutrinos as a function ofneutrino direction and neutrino energy at the Super-K site.
A. Neutrino fluxes
Although atmospheric neutrinos consist of νμ and νe at
production, ντ are expected to appear due to neutrino
oscillations. The probabilities of ντ appearance from neutrino
oscillations of νμ or νe in vacuum are shown in Eq. (1).

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However since neutrinos coming from below travel throughthe Earth, the oscillation probabilities are altered by thematter effect. Therefore, a custom code [22] is used to
calculate the oscillation probabilities, which takes into
account the effect of neutrino types, path lengths, neutrinoenergies and the matter effect. The oscillation parameters
used are Δm232 ¼ 2.1 × 10−3 eV2 , Δm221 ¼ 7.6 × 10−5 eV2 ,sin2 2θ23 ¼ 1.0, sin2 2θ13 ¼ 0.099, δCP ¼ 0 [23]. A methodfrom [24] is used to account for the matter effect in thecalculation of oscillation probabilities based on the matter
density structure of the Earth in Ref. [25]. Figure 1 illustrates
the tau neutrino appearance probabilities from muon neutrino
or electron neutrino oscillations in three-flavor neutrino
oscillation under the assumption of the normal hierarchy.
For a neutrino with a given energy and path length, a muon
neutrino has a larger probability than a electron neutrino to be
detected as a tau neutrino. Following the oscillation calculation, we can predict the atmospheric tau neutrino flux at
Super-K. Figure 2 shows the expected atmospheric tau
neutrino fluxes from neutrino oscillations at Super-K.
B. Neutrino interactions
The NEUT code [26] is used to model the neutrino nucleon
interactions including quasielastic scattering, single meson
production, coherent pion production, and deep-inelastic
scattering (DIS). In the simulation of atmospheric neutrino
background, all νμ and νe interactions are included. Allflavors of neutrinos interact with neutral-current (NC)

Neutrino Flux (GeV-1/m2/s)

103
102

ντ flux

10ντ flux

1
10-1
10-2
10-3
10-4
-5

10

20

40

60

Neutrino Energy (GeV)

Neutrino Flux (/m2/s)

10FIG. 1. Probabilities of tau neutrino appearance from the muon
neutrino (top) or the electron neutrino (bottom) as a function ofneutrino energy and zenith angle, Δm232 ¼ 2.1 × 10−3 eV2 ,Δm221 ¼ 7.6 × 10−5 eV2 , sin2 2θ23 ¼ 1.0, sin2 2θ13 ¼ 0.099,
δCP ¼ 0 and assuming the normal hierarchy. The cosine of thezenith angle equal to 1 corresponds to the downward-goingdirection of neutrinos, and the cosine of the zenith angle equal to−1 corresponds to the upward-going direction of neutrinos.

ντ flux
ντ flux

1

10-1

10-2

10-3
-1

-0.5

0

0.5

1

Cosine of Zenith Angle

FIG. 2. Fluxes of atmospheric ντ (red) and ν̄τ (blue) fromneutrino oscillations as a function of neutrino energy (top) andcosine of zenith angle (bottom) at Super-Kamiokande basedon Honda flux calculation [16] and tau neutrino appearance
probabilities.

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PHYS. REV. D 98, 052006 (2018)

2

10

1

pτsinθ (GeV)

Cross Section (10-38cm2)

2.2

10-1
NEUT CC ν τ cross section

1.6

Eν =10 GeV

P=1

1.4
1.2
1
0.8

0.4
NEUT CC ν τ cross section

0.2
0

-3

0

ντ+N→τ-+X

0.6

10-2

10

1.8

20

40

60

0

2

4

6

8

10

12

pτcosθ (GeV)

Neutrino Energy (GeV)
2.2

FIG. 3. Total charged-current cross section as a function ofneutrino energy for ντ (red) and ν̄τ (blue) from 3.5 to 70 GeV.
pτsinθ (GeV)

interactions, and hence are unaffected by oscillations.
Atmospheric neutrino neutral current events are simulated
based on the total neutrino flux. The simulation of tau signal
contains only charged-current (CC) ντ interactions whosecross sections are calculated following the same models as
those used for νμ and νe . The relatively large mass of the tau
lepton produced in the interactions greatly suppresses the
cross section of charged-current tau neutrino interactions at
low energies and results in an energy threshold of 3.5 GeV.
Figure 3 shows the total cross section of charged-current
interactions for ντ and ν̄τ in the simulations. Tau leptons
produced in the CC tau neutrino interactions are polarized,
and the polarization affects the distributions of its decay
particles. Therefore, a polarization model from Ref. [27] isalso included in the simulation. Figure 4 shows the polarization of τ− =τþ in the simulation for interactions of neutrinos
with energy of 10 GeV.
This analysis selects events at relatively high neutrinoenergies, at which the CC interactions contain a highpercentage of DIS (45%) in the background, with CC ντ
events containing 60% DIS. The GRV98 [28] parton
distribution functions are used in the calculation of the
DIS cross sections. In order to smoothly match the DIS
cross sections with the resonance region, an additionalcorrection developed by Bodek and Yang [29] is also
applied.
The tau lepton has a mean lifetime of 2.9 × 10−13 s and it
decays very quickly after production in the detector.
The decays of tau lepton are divided into leptonic decays
and hadronic decays based on the particles produced. The
kinematics of tau lepton decay is simulated with TAUOLA
version 2.6 [30].
The particles produced in both atmospheric neutrino
background interactions and CC ντ interactions are input toa custom detector simulation based on GEANT3 [31]. The
code simulates the propagation and Cherenkov lightemission of the particles and the Super-K detector [19].

21.8ντ+N→τ++X

1.6

Eν =10 GeV

P=1

1.4
1.2
1
0.8
0.6
0.4
0.2
0

0

2

4

6

8

10

12

pτcosθ (GeV)

FIG. 4. Polarization of τ− (top) and τþ (bottom) produced in theinteractions of neutrinos with energy of 10 GeV in Super-K
simulation. θ is the scattering angle of the tau lepton relative tothe neutrino direction. The τ polarizations are shown by thearrows. The length of the arrows gives the degree of polarization,and the direction of arrows gives that of the τ spin in the τ rest
frame. The size of the 100% polarization (P ¼ 1) arrow is shown
as a reference. The three contours represent regions of 68.3%,
95.4%, and 99.7% of MC events.

IV. REDUCTION AND RECONSTRUCTION
This analysis only uses fully contained (FC) multi-GeV
events in the fiducial volume. Fully contained events aredefined as events which only have activity in the ID, andFC events in the fiducial volume are selected by requiring
the reconstructed event vertex be at least 2 meters awayfrom the ID wall. In addition, the events are required tohave more than 1.3 GeV of visible energy (Evis ), which isdefined as the energy to produce the observed light in theevent if it were produced by a single electron. As shown in
Fig. 5, the Evis cut selects the majority of the tau signal but
rejects the bulk of low-energy atmospheric neutrino background events. The selection efficiencies for this set ofcuts are 86% for the ντ CC signal and 23% for the
background in four Super-K periods. The same selection
is applied to the simulations and the observed data.

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PHYS. REV. D 98, 052006 (2018)
Selected

Events / 5326 Days

104

103

102

10

1

0

10

20

30

Visible Energy (GeV)

FIG. 5. The distribution of Evis in simulations of atmospheric
neutrino background (black histogram) and tau signal (grayshaded histogram) scaled to the live time of SK-I through
SK-IV. The atmospheric neutrino background has a bulk of
events with Evis less than 1.3 GeV, but the majority of the tau
signal has Evis more than 1.3 GeV.

The selected events are passed through a reconstruction
program to determine the event vertex, the number of
Cherenkov rings, the particle type and momentum of each
ring, and the number of Michel electrons. The samereconstruction algorithms are applied to the MCs and theobserved data. Events are assumed to originate from asingle vertex, and the reconstruction uses the distribution ofobserved charge and the PMT timing to find the vertex andthe brightest Cherenkov ring. A Hough transformation
method [32] is used to find additional rings. Each ringcandidate is tested using a likelihood method to remove
fake rings and determine the final number of rings. A
likelihood method based on the ring pattern and ring
opening angle is used to identify each ring as e-like(showering type from e or γ) or μ-like (nonshoweringtype). Michel electrons from stopping muons are tagged bysearching for clusters of hits after the primary event. The
time window for such clusters extends to 20 μs after the
primary event. In the SK-I to SK-III periods, there was an
impedance mismatch in the electronics which caused signal
reflection around 1000 ns after the main event. Therefore,
the time period 800–1200 ns after the main event was
excluded. The improved SK-IV electronics avoids suchsignal mismatch, thus no exclusion is required. As a result,
the tagging efficiency was improved from 80% to 96%
for μþ decays and 63% to 83% for μ− decays between
SK-I-II-III and SK-IV. More details of the reconstruction
can be found in Refs. [33,34].
V. A NEURAL NETWORK ALGORITHM FOR TAU
NEUTRINO SEARCHAs described in Sec. III, tau leptons produced in CC ντinteractions decay quickly to secondary particles. Becauseof the short lifetime of tau lepton, it is not possible todirectly detect them in Super-K. The decay modes of thetau lepton are classified into leptonic and hadronic decay
based on the secondary particles in the decay. The leptonic
decays produce neutrinos and an electron or a muon. These
events look quite similar to the atmospheric CC νe or νμbackground. The hadronic decays of the tau are dominant
and produce one or more pions plus a neutrino. The
existence of extra pions in the hadronic decays of tau
allows the separation of the CC ντ signal from CC νμ , CC νeand NC background. As shown in Fig. 6, CC ντ events
typically produce multiple rings in the detector. Multiplering events are relatively easy to separate from single-ringatmospheric neutrino events. However, the multiring background events, resulting from multipion/DIS atmospheric
neutrino interactions, are difficult to distinguish from thetau signal. Simple selection criteria based on kinematic
variables do not identify CC ντ events efficiently. In orderto statistically identify events with the expected characteristics that differentiate signal and background, a multivariate method is applied in this analysis. Specifically, amultilayer perceptron (MLP) method is used. It is implemented in the ROOT-based TMVA [35] library, and was
also used in our previously published ντ search [17].
A multilayer perceptron is a feed-forward artificial
neural network (NN), which maps between a set of inputs
and a set of outputs. It is typically organized in layers ofinterconnected neurons with one or more hidden layers
between the input and output layer. Neurons in the input
layer receive inputs, then normalize the inputs and forward
them to the neurons in the first hidden layer. Each layer isfully linked to neurons in the next one with weighted
connections. The output of a neuron is scaled by theconnecting weights and fed forward to the neurons in thenext layer. A MLP has the ability to learn through training,
during which the weights in the network are adjusted. Once
trained with representative training data, the MLP can beapplied to new, unseen data.
A MLP is used in this analysis, which has seven inputs,
ten neurons in one hidden layer, and one output. It takes
seven input variables for both the CC ντ signal andatmospheric neutrino background to produce a single
discriminating output variable that separates signal andbackground. To prepare the MLP algorithm for event
identification, three stages are required. They are referred
to as training, testing, and analysis. Separate sets of signal
and background MC are used in each stage. We describe the
MLP that we implemented for this analysis below.
Seven variables are used as inputs to the MLP based on
the expected separation between signal and background inthese variables. The seven variables are(1) The log10 of the total visible energy in MeV (Evis ) of
the event. Due to the energy threshold of CC ντinteractions and the large mass of tau lepton, thesignal events are expected to have higher average

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MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION …

PHYS. REV. D 98, 052006 (2018)visible energy than background. The Evis spectrumof the CC ντ signal peaks around 4 GeV, as shown in
Fig. 7. By contrast, the Evis spectrum of the background falls with increasing Evis .
(2) The particle identification likelihood parameter ofthe ring with maximum energy. Tau leptons decay
quickly to daughter particles after production
through leptonic and hadronic decays. Except for
the leptonic decay to a muon, most decay channels
have at least one showering particle. A showering
particle has a negative value in the definition ofparticle identification likelihood, compared with apositive value for a nonshowering particle. The
particle identification of the most energetic ringfor the signal has a distribution mostly in thenegative region, while the background has a broad
distribution in both negative and positive regions.
(3) The number of decay electron candidates in theevent. Naively, we expect more decay electrons for
signal from pion decays which are produced in
hadronic tau decays. This variable does not dependon ring reconstruction, so it is relatively independentof most other variables.
(4) The maximum distance between the primary interaction point and any decay electron from a pion ormuon decay. Energetic muons can travel a long
distance in water. Therefore, CC νμ background
involving a high energy neutrino is expected to havea large distance between the primary interaction
point and the decay electron from the muon. In
comparison, the pions from hadronic tau decay are
expected to have smaller momentum, resulting in asmaller value of the variable.
(5) The clustered sphericity of the event in the center ofmass system. Sphericity is a measure of how
spherical an event is. A perfectly isotropic event
has sphericity 1, while a perfectly one-directional
event has sphericity 0. We follow the definition from[36], defining the spherical tensor as
αβ

S

FIG. 6. Simulation of a single-ring CC background event(top) with 2.8 GeV visible energy in the ID, a multiring NC
background event (middle) with 2.2 GeV visible energy in
the ID, and a CC ντ event (bottom) with 3.3 GeV visibleenergy in the ID. The tau signal event produces multiple rings,
making it different from the single-ring background event.
The background event with multirings has a similar pattern
to the signal event, and requires more effort to statisticallydistinguish.

052006-7P α βi pi pi
¼ P
;
2
i pi

ð2Þ

where α,β ¼ 1, 2, 3 are three Cartesian momentum
vectors pointing to binned photoelectric charge inthe event. Sphericity is then constructed by finding
the eigenvalues, λ1 > λ2 > λ3 , of the tensor:
3
S ¼ ðλ2 þ λ3 Þ:2ð3ÞThe hadronic decay of the heavy tau lepton is moreisotropic than a typical νμ or νe background. The
spectrum of sphericity is centered near 0.8 for signal,

Z. LI et al.

PHYS. REV. D 98, 052006 (2018)

1000

while the spectrum for background has an almost flat
distribution between 0.1 and 0.8.
(6) The number of possible Cherenkov ring fragments.
In the ring reconstruction, these ring candidates
are formed using a method based on a Houghtransformation to find rings. We expect more ringcandidates for signal because of the multiple charged
particles and pions in hadronic tau decay. Thisvariable is sensitive to even partial ring fragments.
(7) The fraction of the total number of photoelectrons
carried by the most-energetic ring in an event. Thisvariable is calculated from the number of photoelectric charge in each PMT (qi ) and the reconstructed vertex and direction of an event asP
θi <48° qi
rfrac ¼ P
;
ð4Þ
qi

800
600
400
200
0
3

3.5
4
(1) Log of Visible Energy

4.5

3000
2000
1000
0
-300

-200
-100
(2) ID of Maximum Energy Ring

0

2500
2000
1500
1000
500
0
0

2

4
6
8
(3) Number of Decay Electrons

10

4000
3000
2000
1000
0
0

2000
4000
(4) Max Distance to Decay-e

6000

600
400
200
0
0

0.2

0.4
0.6
(5) Clustered Sphericity

5

10
15
(6) Ring Candidates

0.8

1

1000
800
600
400
200
0
0

20

1500
1000
500
0

0.2

0.4
0.6
0.8
(7) Fraction carried in First Ring

1

FIG. 7. The input variables to the neural network for the
downward MC (black histogram) and data (black dots), alongwith tau MC (gray shaded histogram) in SK-I through SK-IV. The
downward MC is normalized to the live time of data, and the tau
signal is normalized for equal statistics.

where θi is the angle between the reconstructed
direction of the first ring and the direction of thereconstructed vertex to a PMT. The variable calculates the ratio of charge within 48° of the direction of
the first ring in the event. The variable is expected tobe smaller for the signal because energy is carried bymultiple particles in the hadronic decay of the tau.
Because the oscillation-induced tau events come frombelow, the downward sample is expected to have no tau
events. Therefore, the data in the downward sample can beused to study the extent to which the atmospheric neutrino
simulation for background events agrees with data.
Figure 7 shows the seven variables for data and MC in
the downward sample, along with the expected tau signal.
The data and background MC have good agreement.
The neural network is trained with a set of signal andbackground MC with the target of output ¼ 1 for signal and
0 for background. The weights in the MLP are adjusted
iteratively during the training such that the difference
between the actual output of the MLP and the target isminimized.
The MC is generated with a Honda flux that calculates
the fluxes of neutrinos at production in the atmosphere [16].
Therefore, the events need to be weighted with oscillation
probabilities before being fed to the MLP. Tau neutrinos
from oscillations are expected to mostly come from below
because the oscillation length of neutrinos in excess of thetau threshold is at least 4,100 km. Used naively, theoscillation weight described in Sec. III will encode this
up-down asymmetry into the signal MC. To avoid training
the neural network to select signal events based on event
direction, the training sample is instead weighted based on
the average probability at its visible energy, as shown in
Fig. 8. In other words, instead of weighting each MC eventwith its oscillation probability, the weight is calculated as
an average of the oscillation probabilities for MC events in

052006-8

MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION …

PHYS. REV. D 98, 052006 (2018)1
Event weight of background events
Event weight of signal events

Background rejection

Event Weight

1.5

1

0.5

0
3

3.5

4

4.5

5

0.8

0.6

0.4

0.2

5.5

Log(E )

0

0.2

vis

FIG. 8. Event weights for atmospheric neutrino background(magenta) and CC ντ signal (black) as a function of log10 ðEvis Þ inthe training sample. The event weight is calculated as an averagevalue of oscillation weights for each bin of log10 ðEvis Þ.

each bin of log10 ðEvis Þ. In this way, the upward anddownward samples are treated the same in the training
process, while the overall weight is still correct. Moreover,since the weights of the downward signal simulation are not
set to zero, the whole of signal MC statistics are preserved
for training.
During training, a testing data set is used as validation to
avoid overtraining. Figure 9 shows the neural network
output for background and signal with the training andtesting samples. The clear separation of signal and background in both samples demonstrates that the MLP learned
to separate signal from background. Also, the good agreement between training and testing samples shows that it isproperly trained.
The testing sample is also used to plot the efficiencies ofsignal selection and background rejection by cutting on NN
output, as shown in Fig. 10. By varying the cut on NN

5

Signal (test sample)

Signal (training sample)

Background (test sample)

Background (training sample)

0.4
0.6
Signal efficiency

0.8

1FIG. 10. Efficiencies of signal selection and backgroundrejection by cutting the NN output in SK-IV.

output, the efficiencies of signal selection and backgroundrejection can be changed. When selecting tau-like eventsfrom the events after reduction by requiring the NN output
be greater than 0.5, 76% of the signal events and only 28%of the background remain. Table I summarizes the breakdown of the interaction modes in different samples,
including the fraction for tau and non-tau-like samples.
Table II summarizes the decay modes of the largest
branching fractions and the fraction of tau-like events ineach mode. These efficiencies are only shown to assess the
TABLE I. The break down of interaction modes of bothbackground and expected signal shown in number of events insimulation scaled to SK-I through SK-IV live time. By cutting theNN output at 0.5, each mode is separated into tau-like(NN > 0.5) and non-tau-like (NN < 0.5).
Interaction mode
CC νe
CC νμ
CC ντ
NC

Non-tau-like

Tau-likeAll

3071.04231.949.1
291.8

1399.2
783.4
136.1
548.3

4470.25015.3185.2
840.1

Kolmogorov-Smirnov test: signal (background) probability = 0.8 ( 0.82)(1/N) dN / dx

4

TABLE II. Decay modes of tau leptons with branching ratio
adapted from [37], along with the fraction of tau-like events andthe product of branching ratio and tau-like ratio in each mode in
the Super-K simulation.

3

2

1

0

0

0.2

0.4

0.6

0.8

1

Neural Network Output

FIG. 9. Neural network output of training (filled histogram) andtest (dots) sample for background (magenta) and signal (gray)simulations in SK-IV.

Decay
mode

Branchingratio (%)

e− ν̄e ντμ− ν̄μ ντ
π − ντ
π − π 0 ντ
3πντOthers

17.83
17.41
10.83
25.52
18.29
10.12

052006-9

Tau-like Branching ratio × tau-likefraction (%)
fraction (%)
67.3  2.2
42.6  2.6
84.7  3.8
81.0  2.1
88.7  2.5
90.5  3.4

12.0  0.4
7.2  0.5
9.2  0.4
20.7  0.5
16.2  0.5
9.2  0.3

Z. LI et al.

PHYS. REV. D 98, 052006 (2018)
Signal PDF

TABLE III. Monte Carlo sample sizes for each stage of the
MLP. The same sample sizes are used for all SK running periods(I through IV).
Analysis∼25 000∼32 000

1500
1500∼6600 (100 yr)∼82 000 (100 yr)

performance of the neural network in selecting tau events.
No cut is used in the following analysis.
The analysis sample is finally processed with the trained
MLP. Unlike the training and testing processes, no information is given to the MLP regarding the composition of
the analysis samples as either signal or background. The
analysis sample processed with the trained MLP is used inthis analysis.
Table III summarizes the quantities of signal and background MC samples used for each stage. For each SK run
period, separate MLPs are trained, tested and analyzed.
Real data in each SK run period are processed with thecorresponding trained MLP.
VI. ANALYSIS

Neural Network Output

Testing

0.8
0.6
0.4
0.2
0
-1

-0.5

0

0.5

1

Cosine of the Zenith Angle
Background PDF
1Neural Network Output

SignalBackgroundTraining

1

0.8
0.6
0.4
0.2

A. Search for atmospheric tau neutrino appearance0

To search for atmospheric tau neutrino appearance, thedata is fit to a combination of the expected tau signal
resulting from neutrino oscillations and atmospheric neutrino background with neutrino oscillations. In order toextract maximum information from the sample, the analysis
uses a two-dimensional unbinned maximum likelihood fit
implemented in ROOFIT [38]. Using two-dimensional histograms of the neural network output and the reconstructed
zenith angle of the events, two-dimensional probability
distribution functions (PDFs) are built for background andtau signal. The probability density follows the normalizedbin contents in the histograms. Figure 11 is an example of the
2D distributions for oscillated signal on the top and background on the bottom. The horizontal axis of the plots is the
cosine of the reconstructed zenith angle as determined by
the energy-weighted sum of the ring directions in the event.
The vertical axis is the NN output, in which tau-like events
have a value close to 1 and non-tau-like events have a value
close to 0. The signal events (top panel) are primarily tau-likeand come from below (cosine of the zenith angle, Θ, less thanzero), while the background (bottom panel) is more non-taulike and come from all directions. The amount of signal andbackground events can be adjusted by varying the normalization of the distributions. Figure 12 shows a combination of
signal PDFs and background PDFs for SK-I to SK-IV withboth tau normalization and background normalization equal
to 1, with the data overlaid on the combined PDF.
The systematic errors used in this analysis are selectedfrom the systematic errors in the Super-K atmospheric

-1

-0.5

0

0.5

1

Cosine of the Zenith Angle

FIG. 11. Two-dimensional probability distribution likelihood as
a function of zenith angle and neural network output for tau (top)and background (bottom) built with SK-IV MC.

Neural Network Output

1
0.8
0.6
0.4
0.2
0
-1

-0.5

0

0.5

1

Cosine of Zenith Angle

FIG. 12. A combination of signal PDFs and background PDFs
with both tau normalization and background normalization equal
to 1, overlaid with every individual event shown in red dots for
SK-I to SK-IV data.

052006-10

MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION …
TABLE IV. Systematic errors used in the tau neutrino
appearance search that are common to all Super-K run
periods. The systematic errors are ordered by the maximum
fractional change in the bins of the two-dimensional event
distribution after shifting the systematic error by 1σ, and only
systematic errors with the maximum fractional change larger
than 2.5% are shown. The estimated 1σ error size is shown inpercentage.
Systematic error

σ (%)

NC/CC ratio
DIS q2 dependence for low W
Meson exchange current
1π axial coupling
DIS q2 dependence for high W
Coherent π cross section
Flux normalization (Eν > 1 GeV)
1π background scale factor
1π axial form factor
CCQE cross section
Single pion π 0 =π  ratioν̄μ =νμ ratio (Eν > 10 GeV)aν=ν̄ ratio (Eν > 10 GeV)b
DIS cross section (Eν < 10 GeV)FC multi-GeV normalizationν̄e =νe ratio (Eν > 10 GeV)cK=π ratio
Single meson cross section
Single-pion ν̄=ν ratio
Horizontal/vertical ratio
CCQE ν=ν̄ ratio
DIS cross section
Matter effect
Neutrino path length

20
10
10
10
10
100
15
10
10
10
40
15
5
10
5
8
10
20
10
1
10
5
6.8
10

a
Uncertainty linearly increases with log Eν from 6% (50 GeV)
to 40% (1 TeV).
b
Uncertainty linearly increases with log Eν from 5% (30 GeV)
to 30% (1 TeV).
c
Uncertainty linearly increases with log Eν from 8%(100 GeV) to 20% (1 TeV).

PHYS. REV. D 98, 052006 (2018)

neutrino analysis [17]. Only systematic errors for which the
maximum bin content changes in the two-dimensionalsignal/background histograms after shifting that systematic
error by 1σ is larger than 2.5% are considered. After thereevaluation, 28 systematic errors from atmospheric neutrino analysis are considered in the fit. The systematic
errors are summarized in Tables IV and V. In order tosimultaneously fit the systematic errors, a set of PDFs arebuilt for each systematic error with the same structure as
the PDFs for the signal and the background in Fig. 11. The
construction of PDFs for systematic errors is based on thethree-flavor oscillation framework in Ref. [39]. The framework is capable of estimating the resulting change in agiven event distribution after changing in a single systematic error. A two-dimensional histogram of NN output andreconstructed zenith angle is built which results from a 1σchange in each systematic error. An example of a twodimensional distribution for a 1σ change is shown in
Fig. 13. The size of a systematic error determines thenormalization of the distribution, which adjusts the size ofthe systematic error in the fit.
Since the uncertainties in oscillation parameters can alsochange the measured results and significance, a set of PDFs
is built for each oscillation parameter. The values of sin2 2θ23
and Δm232 are based on the Super-K atmospheric neutrino−3
oscillation analysis result (Δm232 ¼ 2.10þ0.12
eV2 ,−0.18 × 10
22
sin θ23 ¼ 0.5  0.13) [17]. The value of sin ð2θ13 Þ basedon the combined Daya Bay [8] and RENO [10] measurements of sin2 2θ13 ¼ 0.099  0.014. For this analysis, thevalue of δCP is set to be zero. Varying the value of δCP from 0
to 2π in the three-flavor oscillation formula results in lessthan 1% change in the number of fitted tau events. The
analysis is performed for both normal and inverted hierarchy.
The data is fitted to the sum of background PDF, signal PDF
and systematic PDFs varying the normalizations using a
ROOFIT-based [38] unbinned likelihood fit algorithm as

TABLE V. Systematic errors used in the tau neutrino appearance search that are dependent on Super-K run periods. The systematic
errors are ordered by the maximum fractional change in the bins of the two-dimensional event distribution after shifting the systematic
error by 1σ, and only systematic errors with the maximum fractional change larger than 2.5% are shown. The estimated 1σ error size isshown in percentage.

Multiring e-like background
Multiring PID
1-ring e-like background
Ring separation

Multi-GeV e-like
multi-GeV μ-like
Multi-GeV e-like
Multi-GeV μ-like
Multiring Multi-GeV e-like
Multiring Multi-GeV μ-like

SK-I σ (%)

SK-II σ (%)

SK-III σ (%)

SK-IV σ (%)

12.1
−2.9
6.5
13.2
3.7
1.7
−3.1
−4.1

11.1
−3.9
9.7
38.1
2.6
1.7
−1.9
−0.8

11.4
2.7
−4.9
26.7
1.3
1.0
−1.1
−2.1

11.6
−1.6
3.3
17.6
1.0
−1.2
0.9
2.4

052006-11

Z. LI et al.

PHYS. REV. D 98, 052006 (2018)

Neural Network Output

1
0.8
0.6
0.4
0.2
0
-1

-0.5

0

0.5

1

Cosine of Zenith Angle

FIG. 13. An example histogram of the PDF of DIS cross section
uncertainty for SK-IV background. The vertical axis is the outputof the NN, the horizontal axis is the cosine of the zenith angle
distribution.

Data ¼ PDFBG þ α × PDFtau þXϵi × PDFi ;ð5Þwhere α is the normalization of the tau signal, with 0
meaning no-tau appearance and 1 meaning the expected tau
appearance based on the neutrino oscillation parametersassumed in the simulation. The size of the ith systematic
error in the fit, ϵi , has a Gaussian univariate constraint in thefit. The PDFs of systematic errors are built separately forsignal and background, but are combined together in the fit
because the normalizations of both PDFs are adjusted by
the same normalization factor ϵi simultaneously.
The fit is performed jointly with all data periods being
fit at the same time. First, we present the results of the fit
assuming the normal hierarchy of neutrino mass splitting.
Relative to the expectation of unity, the tau normalization isfound to be 1.47  0.32 (stat þ syst) in the joint fit. This
corresponds to a significance level of 4.6σ of rejection the
hypothesis of no-tau-appearance. To estimate the statisticaluncertainty of the fitted tau normalization, the systematic
errors are excluded from the fit, and the tau normalization isfound to be 1.41  0.28. Therefore, the total uncertainty
is dominated by the statistical uncertainty. The measured
significance is larger than the expected significance of 3.3σ
because more events are measured than expected. The
number of tau events observed is evaluated after the fit byadding the tau events in the signal PDF rescaled by thefitted tau normalization and tau events in the systematic
PDFs rescaled by the fitted values of systematic errors. The
number of tau events is found to be 290.8 in the sample
selected for this analysis. After correcting for efficiency,
the observed number of fitted CC ντ events over theentire running periods is calculated to be 338.1  72.7
(stat þ syst), compared with an expectation of 224.5  57.2
(syst) interactions.
The fit is repeated with the inverted hierarchy whencalculating the oscillation probabilities, resulting in a

reduction in the expected number of θ13 induced
upward-going electron neutrino. Under the assumptionof inverted hierarchy, the fit results in a higher fitted valueof tau normalization, 1.57  0.31 and a correspondingly
higher significance of 5.0σ. The higher fitted tau normalization is due to the reduction in θ13 -induced upward-going
electron neutrinos when calculating the oscillation probabilities under the assumption of the inverted hierarchy.
In order to test the stability of our analysis to changes
in measured oscillation parameters, we also repeated
our analysis procedure with the values of oscillation
parameters in [37] and 2017 update [Δm232 ¼
ð2.45  0.05Þ × 10−3 eV2 , sin2 ðθ23 Þ ¼ 0.51  0.04, andsin2 θ13 ¼ 0.0210  0.0011]. Under the assumption of
the normal hierarchy, the tau normalization is fitted tobe 1.42  0.31. The result of fitted tau normalization isbasically unchanged with the updated oscillation parametervalues.
The results of the final combined fit are examinedgraphically by plotting the binned projections of the fitted
results. Figure 14 demonstrates the projection in zenith
angle for both tau-like (NN output > 0.5) and non-tau-like(NN output < 0.5) events, along with the projections in NN
output for both upward-going (cos Θ < −0.2) and downward-going (cos Θ > 0.2) events. In these plots, the signal
PDFs have been rescaled to the fitted normalization values,
and PDFs of systematic errors for signal and backgroundhave been rescaled by the fitted magnitudes of systematic
errors and added to the signal and the background respectively. The fitted tau signal is shown in gray. All distributions
have good agreement between data and MC simulations. In
these plots, the PDFs and data from all of the run periods are
combined.
B. Charged-current tau neutrino cross section
measurement
This sample of CC ντ interactions observed in Super-K
offers the opportunity to measure the CC ντ cross section.
By scaling the theoretical cross section in the MC simulations to match the data, we can measure the inclusive
charged-current tau neutrino cross section in water:
σ measured ¼ Sτ × hσ theory i;

ð6Þ

where Sτ is the factor that is used to scale the theoreticalcross section to match simulations and data. For thisanalysis, Sτ is the tau normalization measured in the search
for tau neutrino appearance in Sec. VI A. Therefore, themeasured CC ντ cross section is expressed as:
σ measured ¼ ð1.47  0.32Þ × hσ theory i;

ð7Þ

hσ theory i is the flux-averaged theoretical charged-current tau
neutrino cross sections used in the NEUT code.

052006-12

MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION …

PHYS. REV. D 98, 052006 (2018)

BG after fit

300

1000

Tau after fit
Data

800
Events

200
600
400

100

200

UpwardTau-like

0-1

-0.5

0

0.5

0

1

0

1200

Events

600

0.2

0.4

0.6

0.8

1

0.8

1

1000
800

400

600
400

200

200

Non tau-like
0
-1

-0.5

0
CosΘ

0.5

Downward

0

1

0

0.2

0.4 0.6
NN output

FIG. 14. Fit results, assuming the normal hierarchy, showing binned projections in the NN output and zenith angle distribution for taulike (NN > 0.5), upward-going [cos Θ < −0.2], non-tau-like (NN < 0.5) and downward-going [cos Θ > 0.2] events for both the twodimensional PDFs and data. The PDFs and data sets have been combined from SK-I through SK-IV. The fitted tau signal isshown in gray.

To calculate the flux-averaged theoretical cross section,
the differential CC ντ cross section as a function of neutrinoenergy is weighted with the energy spectrum of atmospheric tau neutrinos from neutrino oscillations. Because CCντ interactions are not distinguishable from CC ν̄τ interactions in Super-K, the theoretical cross section is a flux
average of ντ and ν̄τ cross sections. The flux-averaged
theoretical cross section, hσ theory i, is calculated asP

R dΦðEν ÞdEν σðEν ÞdEν;R dΦðEνÞντ ;ν̄τdEν dEνð8ÞνÞ
where dΦðEdEν is the differential flux of tau neutrinos as afunction of neutrino energy as shown in Fig. 2, and σðEν Þ
is the differential charged-current tau neutrino crosssections used in NEUT code as seen in Fig. 3. The rangeof the integral is determined to be between 3.5 and70 GeV from the tau neutrino energies in the simulation.
As shown in Fig. 15, the neutrinos have energies morethan 3.5 GeV in the CC ντ interactions because of theenergy threshold, and the expectation of CC ντ interactions with more than 70 GeV is less than one in theentire run period.
The flux-averaged theoretical charged-current tau neutrino cross section is calculated to be 0.64 × 10−38 cm2
between 3.5 and 70 GeV, and thus the measured fluxaveraged charged current tau neutrino cross section:

10

10

1

1

10-1

10-1

10-2

10-2

10-30

20
40
Neutrino Energy (GeV)60

Events per Bin

hσ theory i ¼ντ ;ν̄τ

ð9Þ

The measured cross section is shown together with thetheoretical cross sections and the MC simulations in
Fig. 15. The measured and theoretical cross section values
are consistent at the 1.5σ level.

Cross Section (10-38cm2)Pð0.94  0.20Þ × 10−38 cm2

10-3

FIG. 15. Measured flux-averaged charged-current tau neutrino
cross section (black), together with theoretical differential crosssections (ντ in red and ν̄τ in blue), flux-averaged theoretical crosssection (dashed gray) and tau events after selection in MC
simulations (gray histogram). The horizontal bar of the measurement point shows the 90% range of tau neutrino energies inthe simulation.

052006-13

Z. LI et al.

PHYS. REV. D 98, 052006 (2018)

C. Comparisons of charged-current tau neutrino
cross section measurement with previously
reported results
Because of the difficulties in tau neutrino production anddetection, charged-current tau neutrino cross sections havenot been well measured. DONUT [14] and OPERA [15]are the only two experiments that have directly observed
charged-current tau neutrino interactions, and DONUT is
the only experiment that reported a measurement of thecross section. The DONUT measurement was based onnine observed charged-current tau neutrino events with anestimated background of 1.5 events. In DONUT, 800 GeV
protons from the Fermilab Tevatron were used to produce
neutrino beam by colliding with a beam dump, and tau
neutrinos were produced via decays of charm mesons.
The mean energy of the detected tau neutrino interactions
was estimated to be 111 GeV, an energy at which deep
inelastic interactions are dominant. Assuming that the DIS
charged-current tau neutrino cross section had a linear
dependence on neutrino energy, DONUT measured theenergy-independent slope of the cross section, σ const , after
correcting for the kinematic effect of tau lepton mass fromthe standard model calculation:σðEÞ ¼ σ const · E · KðEÞ;ð10Þ

where σðEÞ is the charged-current cross section per nucleon
as a function of neutrino energy, σ const is the asymptotic
slope which is constant in σ=E for deep inelastic scattering,and KðEÞ is the kinematic effect of tau lepton mass.
DONUT measured σ const to be ð0.39  0.13  0.13Þ ×
10−38 cm2 GeV−1 in their final results paper [40].
DONUT was incapable of distinguishing the charge ofthe τ lepton, therefore, the measurement is an average ofthe ντ and ν̄τ cross sections assuming equal number of ντand ν̄τ in the neutrino flux.
We wish to compare the ντ cross section measured withatmospheric neutrinos by Super-K at relatively low energies to that measured by DONUT with a neutrino beam at
higher energies. We recalculate the DONUT value of σðEÞfrom Eq. (10) with the kinematic correction KðEÞ integrated over neutrino energies between 3.5 and 70 GeV andweighted to the world average ratio of cross sections
between νμ and ν̄μ [37]. The calculated DONUT valueof σðEÞ is then further weighted by the predicted ντ and ν̄τflux ratio of 1.11 for atmospheric neutrino tau appearance
at Super-K. The resulting σðEÞ is shown in Fig. 16. The
charged-current tau neutrino DIS cross section inferredfrom the DONUT published number and reweighted tolower energy is ð0.37  0.18Þ × 10−38 cm2 . This is smaller
than our measurement of ð0.94  0.20Þ × 10−38 cm2 , but
the measurements are not yet directly comparable. DONUT
measured the cross section with a neutrino beam that had amuch higher average energy than that of the tau neutrinosin the atmospheric neutrino flux at Super-K. Quasielastic

Cross Section (10-38 cm2)

10

1

10-1

NEUT CC ν τ cross section

NEUT CC ν τ cross section

DONUT CC ν τ cross section

DONUT CC ν τ cross section

10-2
SK measured CC tau cross section

10-3
0

20

40

60

Neutrino Energy (GeV)

FIG. 16. Comparison of the Super-K measured (marker witherror bars) and expected tau neutrino cross sections (solid lines)
with σðEÞ inferred from DONUT (dashed lines). The DONUT
cross section considers only DIS, and is digitized from [40].

scattering and resonant pion production is a small component of the DONUT measurement and was neglected intheir calculations. However, the tau neutrino flux at SuperK has a large component of neutrinos below 10 GeV, wherecharged-current quasi-elastic (CCQE) and resonant pion
production makes a significant contribution to the detected
event rate. We complete the comparison using the predicted
CC DIS fraction in the Super-K sample. According to ourMonte Carlo simulation, the fraction of DIS events inSuper-K CC tau neutrino sample is estimated to be 41%.
Therefore, the ντ DIS-only cross section determined bySuper-K atmospheric neutrinos is found to be ð0.40 
0.08Þ × 10−38 cm2 by scaling the measured cross section in
Eq. (9) by 41%. This resulting DIS-only cross section iscomparable and consistent with the DONUT measurementof the DIS ντ cross section extrapolated to lower neutrinoenergyVII. CONCLUSION
Using 5326 days of atmospheric neutrino data in SK-I
through SK-IV, Super-K measured the tau normalization
to be 1.47  0.32, excluding the hypothesis of no-tau
appearance with a significance of 4.6σ. A flux-averaged
charged current tau neutrino cross section is measured tobe ð0.94  0.20Þ × 10−38 cm2 for neutrino energy between
3.5 and 70 GeV in Super-K, to be compared with the fluxaveraged theoretical cross section of 0.64 × 10−38 cm2 .
Our result is consistent with the previous DONUT result,and is consistent with the Standard Model prediction towithin 1.5σ.
ACKNOWLEDGMENTS
We gratefully acknowledge the cooperation of the
Kamioka Mining and Smelting Company. The SuperKamiokande experiment has been built and operated from052006-14

MEASUREMENT OF THE TAU NEUTRINO CROSS SECTION …

PHYS. REV. D 98, 052006 (2018)

funding by the Japanese Ministry of Education, Culture,
Sports, Science and Technology, the U.S. Department ofEnergy, and the U.S. National Science Foundation. Someof us have been supported by funds from the National
Research Foundation of Korea NRF-2009-0083526(KNRC) funded by the Ministry of Science, ICT, andFuture Planning, the European Union H2020 RISE-

GA641540-SKPLUS, the Japan Society for the
Promotion of Science, the National Natural Science
Foundation of China under Grant No. 11235006, theNational Science and Engineering Research Council(NSERC) of Canada, the Scinet and Westgrid consortia
of Compute Canada, and the National Science Centre,
Poland (2015/17/N/ST2/04064, 2015/18/E/ST2/00758).

[1] Y. Fukuda et al. (Super-Kamiokande Collaboration), Phys.
Rev. Lett. 81, 1562 (1998).
[2] K. Abe et al. (Super-Kamiokande Collaboration), Phys.
Rev. Lett. 107, 241801 (2011).
[3] M. G. Aartsen et al., Nucl. Phys. B908, 161 (2016).
[4] Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett.
87, 071301 (2001).
[5] S. Fukuda et al. (Super-Kamiokande Collaboration), Phys.
Rev. Lett. 86, 5651 (2001).
[6] Q. R. Ahmad et al. (SNO Collaboration), Phys. Rev. Lett.
89, 011301 (2002).
[7] K. Eguchi et al. (KamLAND Collaboration), Phys. Rev.
Lett. 90, 021802 (2003).
[8] F. P. An et al. (Daya Bay Collaboration), Phys. Rev. Lett.
108, 171803 (2012).
[9] Y. Abe et al. (Double Chooz Collaboration), Phys. Rev.
Lett. 108, 131801 (2012).
[10] J. K. Ahn et al. (RENO Collaboration), Phys. Rev. Lett. 108,
191802 (2012).
[11] M. H. Ahn et al. (K2K Collaboration), Phys. Rev. Lett. 90,
041801 (2003).
[12] D. G. Michael et al. (MINOS Collaboration), Phys. Rev.
Lett. 97, 191801 (2006).
[13] K. Abe et al. (T2K Collaboration), Phys. Rev. D 85, 031103(2012).
[14] K. Kodama et al. (DONUT Collaboration), Phys. Lett. B
504, 218 (2001).
[15] N. Agafonova et al. (OPERA Collaboration), Phys. Rev.
Lett. 115, 121802 (2015).
[16] M. Honda, M. S. Athar, T. Kajita, K. Kasahara, and S.
Midorikawa, Phys. Rev. D 92, 023004 (2015).
[17] K. Abe et al. (Super-Kamiokande Collaboration), Phys.
Rev. Lett. 110, 181802 (2013).
[18] Y. Fukuda et al. (Super-Kamiokande Collaboration),
Advanced computing and analysis techniques in physics
research, in Proceedings of the 8th International Workshop,
ACAT 2002, Moscow, Russia, 2002 [Nucl. Instrum.
Methods Phys. Res., Sect. A 501, 418 (2003)].
[19] K. Abe et al., Nucl. Instrum. Methods Phys. Res., Sect. A
737, 253 (2014).
[20] S. Yamada et al. (Super-Kamiokande Collaboration), Real
time, in Proceedings of the 16th Conference on Computing
Applications in Nuclear and Plasma Sciences, RT2009,
Beijing, People’s Republic of China, 2009 [IEEE Trans.
Nucl. Sci. 57, 428 (2010)].

[21] A. D. Martin, M. G. Ryskin, and A. M. Stasto, Acta Phys.
Pol. B 34, 3273 (2003).
[22] R. Wendell et al., Prob3++, http://www.phy.duke.edu/∼raw22/public/Prob3++/.
[23] J. Beringer et al. (Particle Data Group), Phys. Rev. D 86,
010001 (2012).
[24] V. D. Barger, K. Whisnant, S. Pakvasa, and R. J. N. Phillips,
Phys. Rev. D 22, 2718 (1980).
[25] A. M. Dziewonski and D. L. Anderson, Phys. Earth Planet.
Inter. 25, 297 (1981).
[26] Y. Hayato, Neutrino interactions: From theory toMonte Carlo simulations, in Proceedings of the 45th
Karpacz Winter School in Theoretical Physics, LadekZdroj, Poland, 2009 [Acta Phys. Pol. B 40, 2477 (2009)].
[27] K. Hagiwara, K. Mawatari, and H. Yokoya, Nucl. Phys.
B668, 364 (2003); 701, 405(E) (2004).
[28] M. Glück, E. Reya, and A. Vogt, Eur. Phys. J. C 5, 461 (1998).
[29] A. Bodek and U. K. Yang, in Proceedings of the 1st
International Workshop on Neutrino-Nucleus Interactionsin the Few GeV Region (NuInt 01): Tsukuba, Japan, 2001
[Nucl. Phys. Proc. Suppl. 112, 70 (2002)].
[30] Z. Was and P. Golonka, Tau lepton physics, in Proceedingsof the 8th International Workshop, TAU’04, Nara, Japan,
2004 [Nucl. Phys. Proc. Suppl. 144, 88 (2005)].
[31] R. Brun, F. Bruyant, M. Maire, A. C. McPherson, and P.
Zanarini, GEANT 3 : user’s guide Geant 3.10, Geant 3.11
(1987), http://cds.cern.ch/record/1119728?ln=en.
[32] E. R. Davies, Machine Vision: Theory, Algorithms, Practicalities (Elsevier, New York, 2004).
[33] M. Shiozawa (Super-Kamiokande Collaboration), in Proceedings of the 3rd International Workshop on Ring
Imaging Cherenkov Detectors: Advances in Cherenkov
Light Imaging Techniques and Applications (RICH
1998): Ein-Gedi, Israel, 1998 [Nucl. Instrum. Methods
Phys. Res., Sect. A 433, 240 (1999)].
[34] Y. Ashie et al. (Super-Kamiokande Collaboration), Phys.
Rev. D 71, 112005 (2005).
[35] A. Hocker et al., in Proceedings of the 11th International
Workshop on Advanced Computing and Analysis Techniques in Physics Research (ACAT 2007): Amsterdam,
Netherlands, 2007 [Proc. Sci., ACAT2007 (2007) 040].
[36] T. Sjostrand, S. Mrenna, and P. Z. Skands, J. High Energy
Phys. 05 (2006) 026.
[37] C. Patrignani et al. (Particle Data Group), Chin. Phys. C 40,
100001 (2016).

052006-15

Z. LI et al.

PHYS. REV. D 98, 052006 (2018)[38] W. Verkerke and D. P. Kirkby, Statistical Problems in
Particle Physics, Astrophysics and Cosmology (PHYSTAT05): Proceedings, Oxford, UK, 2005, eConf C0303241,
MOLT007 (2003).

[39] R. Wendell et al. (Super-Kamiokande Collaboration), Phys.
Rev. D 81, 092004 (2010).
[40] K. Kodama et al. (DONUT Collaboration), Phys. Rev. D 78,
052002 (2008).

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