The Super-Kamiokande Collaboration

# Dinucleon and Nucleon Decay to Two-Body Final States with no Hadrons in
Super-Kamiokande

S. Sussman<EMAIL_ADDRESS>Department of Physics, Boston University, Boston, MA
02215, USA  K. Abe  Kamioka Observatory, Institute for Cosmic Ray Research,
University of Tokyo, Kamioka, Gifu 506-1205, Japan  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  C. Bronner  Kamioka Observatory, Institute for Cosmic Ray Research,
University of Tokyo, Kamioka, Gifu 506-1205, Japan  Y. Hayato  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  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  M. Ikeda  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  K. Iyogi  Kamioka Observatory, Institute for Cosmic Ray
Research, University of Tokyo, Kamioka, Gifu 506-1205, Japan  J. Kameda
Kamioka Observatory, Institute for Cosmic Ray Research, University of Tokyo,
Kamioka, Gifu 506-1205, Japan  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  Y. Kato  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  Y. Kishimoto  Kamioka Observatory, Institute for Cosmic
Ray Research, University of Tokyo, Kamioka, Gifu 506-1205, Japan  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  Ll. Marti  Kamioka Observatory, Institute for
Cosmic Ray Research, University of Tokyo, Kamioka, Gifu 506-1205, Japan  M.
Miura  S. Moriyama  Kamioka Observatory, Institute for Cosmic Ray Research,
University of Tokyo, Kamioka, Gifu 506-1205, Japan  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  T. Mochizuki  Kamioka Observatory, Institute for Cosmic Ray Research,
University of Tokyo, Kamioka, Gifu 506-1205, Japan  M. Nakahata  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  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  Y. Nakajima  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  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  Y. Nakano  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  S. Nakayama  Kamioka Observatory, Institute for Cosmic
Ray Research, University of Tokyo, Kamioka, Gifu 506-1205, Japan  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  T. Okada  K. Okamoto  A. Orii  G. Pronost
Kamioka Observatory, Institute for Cosmic Ray Research, University of Tokyo,
Kamioka, Gifu 506-1205, Japan  H. Sekiya  M. Shiozawa  Kamioka Observatory,
Institute for Cosmic Ray Research, University of Tokyo, Kamioka, Gifu
506-1205, Japan  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  Y. Sonoda  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  A. Takeda  Kamioka Observatory, Institute for Cosmic Ray
Research, University of Tokyo, Kamioka, Gifu 506-1205, Japan  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  A. Takenaka  Kamioka Observatory, Institute for Cosmic Ray Research,
University of Tokyo, Kamioka, Gifu 506-1205, Japan  H. Tanaka  Kamioka
Observatory, Institute for Cosmic Ray Research, University of Tokyo, Kamioka,
Gifu 506-1205, Japan  T. Yano  Kamioka Observatory, Institute for Cosmic Ray
Research, University of Tokyo, Kamioka, Gifu 506-1205, Japan  R. Akutsu
Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research,
University of Tokyo, Kashiwa, Chiba 277-8582, Japan  T. Kajita  Research
Center for Cosmic Neutrinos, Institute for Cosmic Ray Research, University of
Tokyo, Kashiwa, Chiba 277-8582, Japan  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  Y.
Nishimura  Research Center for Cosmic Neutrinos, Institute for Cosmic Ray
Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan  K. Okumura
Research Center for Cosmic Neutrinos, Institute for Cosmic Ray Research,
University of Tokyo, Kashiwa, Chiba 277-8582, Japan  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  R. Wang  J. Xia  Research Center for Cosmic Neutrinos, Institute for
Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan  L.
Labarga  P. Fernandez  Department of Theoretical Physics, University Autonoma
Madrid, 28049 Madrid, Spain  F. d. M. Blaszczyk  Department of Physics, Boston
University, Boston, MA 02215, USA  C. Kachulis  Department of Physics, Boston
University, Boston, MA 02215, USA  E. Kearns  Department of Physics, Boston
University, Boston, MA 02215, USA  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  J. L.
Raaf  Department of Physics, Boston University, Boston, MA 02215, USA  J. L.
Stone  Department of Physics, Boston University, Boston, MA 02215, USA  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  S. Berkman  Department of Physics and
Astronomy, University of British Columbia, Vancouver, BC, V6T1Z4, Canada  J.
Bian  N. J. Griskevich  W. R. Kropp  S. Locke  S. Mine  P. Weatherly
Department of Physics and Astronomy, University of California, Irvine, Irvine,
CA 92697-4575, USA  M. B. Smy  H. W. Sobel  Department of Physics and
Astronomy, University of California, Irvine, Irvine, CA 92697-4575, USA  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  V. Takhistov  also at Department of Physics
and Astronomy, UCLA, CA 90095-1547, USA.  Department of Physics and Astronomy,
University of California, Irvine, Irvine, CA 92697-4575, USA  K. S. Ganezer
Deceased.  J. Hill  Department of Physics, California State University,
Dominguez Hills, Carson, CA 90747, USA  J. Y. Kim  I. T. Lim  R. G. Park
Department of Physics, Chonnam National University, Kwangju 500-757, Korea  B.
Bodur  Department of Physics, Duke University, Durham NC 27708, USA  K.
Scholberg  C. W. Walter  Department of Physics, Duke University, Durham NC
27708, USA  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  O. Drapier  M. Gonin  J. Imber  Th. A.
Mueller  P. Paganini  Ecole Polytechnique, IN2P3-CNRS, Laboratoire Leprince-
Ringuet, F-91120 Palaiseau, France  T. Ishizuka  Junior College, Fukuoka
Institute of Technology, Fukuoka, Fukuoka 811-0295, Japan  T. Nakamura
Department of Physics, Gifu University, Gifu, Gifu 501-1193, Japan  J. S. Jang
GIST College, Gwangju Institute of Science and Technology, Gwangju 500-712,
Korea  K. Choi  J. G. Learned  S. Matsuno  Department of Physics and
Astronomy, University of Hawaii, Honolulu, HI 96822, USA  R. P. Litchfield  A.
A. Sztuc  Y. Uchida  M. O. Wascko  Department of Physics, Imperial College
London , London, SW7 2AZ, United Kingdom  N. F. Calabria  M. G. Catanesi  R.
A. Intonti  E. Radicioni  Dipartimento Interuniversitario di Fisica, INFN
Sezione di Bari and Università e Politecnico di Bari, I-70125, Bari, Italy  G.
De Rosa  Dipartimento di Fisica, INFN Sezione di Napoli and Università di
Napoli, I-80126, Napoli, Italy  A. Ali  G. Collazuol  F. Iacob  Dipartimento
di Fisica, INFN Sezione di Padova and Università di Padova, I-35131, Padova,
Italy  L. Ludovici  INFN Sezione di Roma and Università di Roma “La Sapienza”,
I-00185, Roma, Italy  S. Cao  M. Friend  T. Hasegawa  T. Ishida  T. Kobayashi
T. Nakadaira  High Energy Accelerator Research Organization (KEK), Tsukuba,
Ibaraki 305-0801, Japan  K. Nakamura  High Energy Accelerator Research
Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan  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  Y. Oyama  K. Sakashita  T. Sekiguchi  T. Tsukamoto  High Energy
Accelerator Research Organization (KEK), Tsukuba, Ibaraki 305-0801, Japan  KE.
Abe  Department of Physics, Kobe University, Kobe, Hyogo 657-8501, Japan  M.
Hasegawa  Y. Isobe  H. Miyabe  T. Sugimoto  Department of Physics, Kobe
University, Kobe, Hyogo 657-8501, Japan  A. T. Suzuki  Department of Physics,
Kobe University, Kobe, Hyogo 657-8501, Japan  Y. Takeuchi  Department of
Physics, Kobe University, Kobe, Hyogo 657-8501, Japan  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  Y. Ashida  T. Hayashino  S. Hirota  M. Jiang  T. Kikawa  M. Mori
Department of Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan  KE.
Nakamura  Department of Physics, Kyoto University, Kyoto, Kyoto 606-8502,
Japan  T. Nakaya  Department of Physics, Kyoto University, Kyoto, Kyoto
606-8502, Japan  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  R. A. Wendell  Department
of Physics, Kyoto University, Kyoto, Kyoto 606-8502, Japan  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  L. H. V. Anthony  N. McCauley  A. Pritchard  K. M. Tsui  Department of
Physics, University of Liverpool, Liverpool, L69 7ZE, United Kingdom  Y.
Fukuda  Department of Physics, Miyagi University of Education, Sendai, Miyagi
980-0845, Japan  Y. Itow  Institute for Space-Earth Environmental Research,
Nagoya University, Nagoya, Aichi 464-8602, Japan  Kobayashi-Maskawa Institute
for the Origin of Particles and the Universe, Nagoya University, Nagoya, Aichi
464-8602, Japan  M. Murrase  Institute for Space-Earth Environmental Research,
Nagoya University, Nagoya, Aichi 464-8602, Japan  P. Mijakowski  National
Centre For Nuclear Research, 00-681 Warsaw, Poland  K. Frankiewicz  National
Centre For Nuclear Research, 00-681 Warsaw, Poland  C. K. Jung  X. Li  J. L.
Palomino  G. Santucci  C. Vilela  M. J. Wilking  C. Yanagisawa  also at
BMCC/CUNY, Science Department, New York, New York, USA.  Department of Physics
and Astronomy, State University of New York at Stony Brook, NY 11794-3800, USA
D. Fukuda  K. Hagiwara  H. Ishino  S. Ito  Department of Physics, Okayama
University, Okayama, Okayama 700-8530, Japan  Y. Koshio  Department of
Physics, Okayama University, Okayama, Okayama 700-8530, Japan  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  M. Sakuda  Y. Takahira  C. Xu  Department of Physics, Okayama
University, Okayama, Okayama 700-8530, Japan  Y. Kuno  Department of Physics,
Osaka University, Toyonaka, Osaka 560-0043, Japan  C. Simpson  Department of
Physics, Oxford University, Oxford, OX1 3PU, United Kingdom  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  D. Wark  Department of Physics, Oxford University, Oxford, OX1 3PU,
United Kingdom  STFC, Rutherford Appleton Laboratory, Harwell Oxford, and
Daresbury Laboratory, Warrington, OX11 0QX, United Kingdom  F. Di Lodovico  B.
Richards  S. Molina Sedgwick  School of Physics and Astronomy, Queen Mary
University of London, London, E1 4NS, United Kingdom  R. Tacik  Department of
Physics, University of Regina, 3737 Wascana Parkway, Regina, SK, S4SOA2,
Canada  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T2A3, Canada  S. B. Kim
Department of Physics, Seoul National University, Seoul 151-742, Korea  M.
Thiesse  L. Thompson  Department of Physics and Astronomy, University of
Sheffield, S3 7RH, Sheffield, United Kingdom  H. Okazawa  Department of
Informatics in Social Welfare, Shizuoka University of Welfare, Yaizu,
Shizuoka, 425-8611, Japan  Y. Choi  Department of Physics, Sungkyunkwan
University, Suwon 440-746, Korea  K. Nishijima  Department of Physics, Tokai
University, Hiratsuka, Kanagawa 259-1292, Japan  M. Koshiba  The University of
Tokyo, Bunkyo, Tokyo 113-0033, Japan  M. Yokoyama  Department of Physics,
University of Tokyo, Bunkyo, Tokyo 113-0033, Japan  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  A. Goldsack  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  Department of Physics,
Oxford University, Oxford, OX1 3PU, United Kingdom  K. Martens  M. Murdoch  B.
Quilain  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  Y. Suzuki  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  M. R. Vagins  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  Department of Physics and
Astronomy, University of California, Irvine, Irvine, CA 92697-4575, USA  M.
Kuze  Y. Okajima  T. Yoshida  Department of Physics,Tokyo Institute of
Technology, Meguro, Tokyo 152-8551, Japan  M. Ishitsuka  Department of
Physics, Faculty of Science and Technology, Tokyo University of Science, Noda,
Chiba 278-8510, Japan  J. F. Martin  C. M. Nantais  H. A. Tanaka  T. Towstego
Department of Physics, University of Toronto, ON, M5S 1A7, Canada  M. Hartz
A. Konaka  P. de Perio  TRIUMF, 4004 Wesbrook Mall, Vancouver, BC, V6T2A3,
Canada  S. Chen  L. Wan  Department of Engineering Physics, Tsinghua
University, Beijing, 100084, China  A. Minamino  Faculty of Engineering,
Yokohama National University, Yokohama, 240-8501, Japan

###### Abstract

Using 0.37 megaton$\cdot$years of exposure from the Super-Kamiokande detector,
we search for 10 dinucleon and nucleon decay modes that have a two-body final
state with no hadrons. These baryon and lepton number violating modes have the
potential to probe theories of unification and baryogenesis. For five of these
modes the searches are novel, and for the other five modes we improve the
limits by more than one order of magnitude. No significant evidence for
dinucleon or nucleon decay is observed and we set lower limits on the partial
lifetime of oxygen nuclei and on the nucleon partial lifetime that are above
$4\times 10^{33}$ years for oxygen via the dinucleon decay modes and up to
about $4\times 10^{34}$ years for nucleons via the single nucleon decay modes.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

One of the biggest unanswered questions about our universe is the origin of
the matter/antimatter asymmetry that we observe. Non-conservation of baryon
number, $\mathcal{B}$, is one of the three necessary conditions to create a
baryon asymmetry where none previously existed Sakharov (1967). Since
$\mathcal{B}$ is an accidental symmetry in the Standard Model (SM) of particle
physics, observation of $\mathcal{B}$ violation would imply new physics beyond
the Standard Model. Many theoretical extensions of the SM allow violation of
$\mathcal{B}$ and/or lepton number, $\mathcal{L}$, and predict experimentally
observable processes (see reviews in Babu  _et al._ (2013) and Nath and
Fileviez Perez (2007)). The searches for ten such $\mathcal{B}$-violating
processes via nucleon or dinucleon decay in Super-Kamiokande are detailed in
this Letter. Four of the eight dinucleon decay modes studied here have
$\Delta(\mathcal{B-L})=-2$, with two nucleons decaying to a lepton and an
antilepton. A scenario in which baryon asymmetry would remain after
$\Delta(\mathcal{B-L})=-2$ decays in the early universe is discussed in Ref.
Babu and Mohapatra (2012). Three of the eight dinucleon decay modes, with two
nucleons decaying to two antileptons, violate each of $\mathcal{B}$ and
$\mathcal{L}$ by two units, but conserve the quantity $(\mathcal{B-L})$. These
modes are interesting in the context of models such as Alberico  _et al._
(1985); Arnellos and Marciano (1982); Mohapatra and Senjanović (1982); Ajaib
_et al._ (2009), and are shown in Ref. Bramante  _et al._ (2015) to be
competitive with LHC measurements in probing the mass scale of new physics.
The final dinucleon decay mode and the two single-nucleon decay modes studied
here are radiative; these decay modes can arise in various models of grand
unification, but are often predicted to have suppressed decay rates Lucha
(1983); Lucha and Stremnitzer (1983). The radiative modes have similar
experimental signatures as the other modes studied; they also have similar
signatures to the previously searched $p\rightarrow e^{+}\pi^{0}$ and
$p\rightarrow\mu^{+}\pi^{0}$ modes, but have the benefit of higher detection
efficiency due to the lack of hadronic interactions.

The ten decay modes we search for in Super-Kamiokande data are characterized
by two back-to-back Cherenkov rings and no hadrons. The dinucleon decay modes
in these three categories are: (i) $pp\rightarrow e^{+}e^{+}$, $nn\rightarrow
e^{+}e^{-}$, $nn\rightarrow\gamma\gamma$, (ii) $pp\rightarrow e^{+}\mu^{+}$,
$nn\rightarrow e^{+}\mu^{-}$, $nn\rightarrow e^{-}\mu^{+}$, and (iii)
$pp\rightarrow\mu^{+}\mu^{+}$, $nn\rightarrow\mu^{+}\mu^{-}$. We classify the
modes as follows: (i) both rings are showering ($NN\rightarrow ee$), (ii) one
ring is showering and the other is non-showering ($NN\rightarrow e\mu$), and
(iii) both rings are non-showering ($NN\rightarrow\mu\mu$). Figure 1
illustrates how distinct these final states are seen in Super-Kamiokande, due
to their well-separated, bright rings. The nucleon decay modes with identical
experimental signatures, but lower invariant mass, are (i) $p\rightarrow
e^{+}\gamma$ and (ii) $p\rightarrow\mu^{+}\gamma$. We do not include the
search for dinucleon decays into tau leptons because there would be missing
momentum and some subsequent tau decay modes are hadronic.

Figure 1: (color online) An SK event display of a typical $pp\rightarrow
e^{+}\mu^{+}$ event shown in $\theta$-$\phi$ view. The non-showering ring
(from the $\mu^{+}$) is on the left and the showering ring (from the $e^{+}$)
is on the right. The energy of each ring is approximately 900 MeV.

The Super-Kamiokande (SK) water Cherenkov detector, with a fiducial volume of
22.5 kilotons, contains $1.2\times 10^{34}$ nucleons. SK lies one kilometer
under Mt. Ikenoyama in Japan’s Kamioka Observatory. The detector is
cylindrical with a diameter of 39.3 meters and a height of 41.4 meters,
optically separated into an inner and an outer region. Eight-inch
photomultiplier tubes (PMTs) line the outer detector facing outwards and serve
primarily as a veto for cosmic ray muons, and 20-inch PMTs face inwards to
measure Cherenkov light in the inner detector Fukuda  _et al._ (2003).

SK has collected data for four different detector periods, accumulating 91.5,
49.1, 31.8 and 199.3 kiloton$\cdot$years of exposure during SK-I, SK-II, SK-
III, and SK-IV, respectively. During SK-I, the inner detector photocathode
coverage was 40$\%$, but the SK-II period had a reduced photo-coverage of
19$\%$ after recovery from an accident. For SK-II, the remaining PMTs were
evenly distributed to maintain isotropic detector uniformity. SK-II efficiency
is only $\sim$2$\%$ lower than the other detector periods for these dinucleon
and nucleon decay searches because the rings still have many hits. In the
subsequent periods, SK-III and SK-IV, we restored the original photo-coverage
of 40$\%$. The SK-IV period benefited from an electronics upgrade described in
Ref. Yamada  _et al._ (2010): a “deadtime free” data acquisition system
enables SK-IV to detect the 2.2 MeV gamma ray emission from neutron capture on
hydrogen, which occurs about 200 $\mu$sec after the primary event.

For each dinucleon or nucleon decay mode studied, we simulated 100,000 signal
Monte Carlo (MC) events with vertices uniformly distributed throughout the
detector and final state particle momenta uniformly distributed in phase
space. Fermi motion, nuclear binding energy, and correlated decay are
simulated in the dinucleon and nucleon decay signal MC Abe  _et al._ (2017a);
Gustafson  _et al._ (2015). Unlike the atmospheric $\nu$ MC, where the Fermi
momentum distribution of the nucleons follows the Fermi gas model, the signal
MC Fermi momentum distribution follows a spectral function fit to electron-12C
scattering data Nakamura  _et al._ (1976). We address this difference between
signal and atmospheric $\nu$ event samples by computing the systematic
uncertainty in signal efficiency based on our choice of nuclear model.
Correlated decay is a hypothesized effect where the total mass and momentum
distributions are smeared out in a “tail” due to the correlated motion of a
nearby nucleon. For both nucleons and pairs of nucleons, we assume that 10$\%$
of such decays are affected by the correlated motion of an additional nucleon
Yamazaki and Akaishi (1999). Lepton rescattering within the nucleus is
negligible.

The atmospheric $\nu$ MC sample corresponds to an exposure of 500 years for
each of the four SK periods, 2000 years in total. Events in this sample are
weighted assuming two-flavor mixing as is done in recent dinucleon and nucleon
analyses Abe  _et al._ (2017a, b); Gustafson  _et al._ (2015). Details of the
cross-sections and flux modeling used in this sample are discussed in recent
SK nucleon decay analyses Abe  _et al._ (2017b, a). Event rates obtained from
this sample are normalized to the relevant SK detector livetime.

Details of the neutron simulation and neutron tagging algorithm used for both
the signal and atmospheric $\nu$ MC samples can be found in Ref. Abe  _et al._
(2017b). Neutron tagging can only be done for the SK-IV period; it reduces the
expected number of background events by about 50% for our searches, and
impacts signal efficiency by only a few percent.

Figure 2: (color online) Invariant mass vs. total momentum for several
dinucleon and nucleon decay modes after cut (A4). The left panels show signal
MC, where green corresponds to SK-IV nucleon decay MC and blue corresponds to
SK-IV dinucleon decay MC for the labeled modes. The signal MC distributions
for all SK periods look similar; only 10,000 signal MC events are plotted for
each mode in order to more clearly show the shape of the distribution. The
middle panels show atmospheric $\nu$ MC corresponding to 2000 years of SK
exposure, and the right panels show SK-I through SK-IV data. The marker size
is enlarged for data in the signal boxes. Figure 3: (color online) Total mass
($M_{tot}$) and total momentum ($P_{tot}$) projections for
$p\rightarrow\mu^{+}\gamma$ after cut (A4). The red histogram shows
atmospheric $\nu$ MC corresponding to 2000 years of SK exposure normalized to
SK-I through SK-IV data. The selection criteria are indicated by the vertical
blue lines.

Although the selection criteria for all ten modes are similar, the two single-
nucleon decay modes have more background due to their lower total mass. We
adapt our strategy, as is done in Ref. Abe  _et al._ (2017b), to perform a
two-box analysis which allows us to study free and bound protons separately.

The following selection criteria are applied to signal MC, atmospheric $\nu$
MC, and data:

  1. (A1)

Events must be fully contained in the inner detector with the event vertex
within the fiducial volume (two meters inward from the detector walls),

  2. (A2)

There must be two Cherenkov rings,

  3. (A3)

Both rings must be showering for the $pp\rightarrow e^{+}e^{+}$,
$nn\rightarrow e^{+}e^{-}$, $nn\rightarrow\gamma\gamma$ and $p\rightarrow
e^{+}\gamma$ modes; one ring must be showering and one ring must be non-
showering for the $pp\rightarrow e^{+}\mu^{+}$, $nn\rightarrow e^{+}\mu^{-}$,
$nn\rightarrow e^{-}\mu^{+}$ and $p\rightarrow\mu^{+}\gamma$ modes; both rings
must be non-showering for the $pp\rightarrow\mu^{+}\mu^{+}$,
$nn\rightarrow\mu^{+}\mu^{-}$ modes (see note in 111A note on cut (A3): Due to
the two photons in the final state of $nn\rightarrow\gamma\gamma$, there is no
visible energy in the first radiation-length of the two showers: this slightly
impacts particle identification. We are about $1\%$ less efficient at particle
identification for $nn\rightarrow\gamma\gamma$ than we are for similar
dinucleon decay modes $pp\rightarrow e^{+}e^{+}$ and $nn\rightarrow
e^{+}e^{-}$.),

  4. (A4)

There must be zero Michel electrons for the $pp\rightarrow e^{+}e^{+}$,
$nn\rightarrow e^{+}e^{-}$, $nn\rightarrow\gamma\gamma$ and $p\rightarrow
e^{+}\gamma$ modes; there must be less than or equal to one Michel electron
for the $pp\rightarrow e^{+}\mu^{+}$, $nn\rightarrow e^{+}\mu^{-}$,
$nn\rightarrow e^{-}\mu^{+}$ and $p\rightarrow\mu^{+}\gamma$ modes; there is
no Michel electron cut for the $pp\rightarrow\mu^{+}\mu^{+}$,
$nn\rightarrow\mu^{+}\mu^{-}$ modes (see note in 222The decay electron cut is
loose for the non-showering dinucleon decay modes because there is almost zero
background to eliminate.),

  5. (A5)

The reconstructed total mass, $M_{tot}$, should be $1600\leq M_{tot}\leq 2050$
MeV/c2 for the dinucleon decay modes; the reconstructed total mass should be
$800\leq M_{tot}\leq 1050$ MeV/c2 for the nucleon decay modes,

  6. (A6)

The reconstructed total momentum, $P_{tot}$, should be $0\leq P_{tot}\leq 550$
MeV/c for the dinucleon decay modes; for the nucleon decay modes, it should be
$100\leq P_{tot}\leq 250$ MeV/c for the event to be in the “High
$P_{\text{tot}}$” signal box and $0\leq P_{tot}\leq 100$ MeV/c for the event
to be in the “Low $P_{\text{tot}}$” signal box,

  7. (A7)

[SK-IV nucleon decay searches only] There must be zero tagged neutrons.

Figure 2 shows the distributions of signal MC events (left panels),
atmospheric neutrino background (middle), and data (right) as a function of
$P_{tot}$ versus $M_{tot}$ after cut (A4). The signal selection efficiencies
and background rates are summarized in Table 1 for each of the decay modes and
each of the SK running periods. The signal efficiency for the two nucleon
decay modes is $\sim 50\%$ for the “High $P_{\text{tot}}$”signal box and $\sim
28\%$ for the “Low $P_{\text{tot}}$” signal box for each SK period. It is
worth noting that these signal efficiencies are significantly higher than
those of the similar event signature in the $p\rightarrow\ell^{+}\pi^{0}$
decay mode searches. These differences are due to the fact that the $\pi^{0}$
undergoes nuclear effects before exiting the nucleus while the $\gamma$ does
not. For the eight dinucleon decay modes, the signal efficiency is
$\sim$$80\%$ for each SK period. Due to the high total mass required in (A5),
the modes are virtually background-free (as shown in the middle panels of Fig.
2).

Background estimates are done in one of the two following ways, depending on
the number of background events that fall in the signal box: (1) for signal
regions that contain more than 10 events from 2000 years of atmospheric $\nu$
MC, the background is estimated by the traditional method of counting the
number of events that fall inside the signal region; or, (2) for signal
regions that are nearly background-free, an extrapolation method is used to
estimate the expected background using the distribution of events nearby (but
outside) the signal region. The background extrapolation is done by measuring
the distance from the center of the signal box to the location of each nearby
event in mass-momentum parameter space, and then fitting an exponential to the
distribution of distances. Integration of the exponential function up to the
radius which approximates the signal box (250 units in mass-momentum parameter
space) gives the estimated background inside the signal region. A similar
estimation method was done in Ref. Shiozawa (1999). Background rate for
$p\rightarrow e^{+}\gamma$ “Low $P_{\text{tot}}$” is estimated by
extrapolation to be 0.089 events/Mton$\cdot$yr; we take double this value
(0.18 $\pm$ 0.18 events/Mton$\cdot$yr) as a conservative estimate of the
background rate for this decay mode. Similarly, we extrapolate for all of the
dinucleon decay modes, finding background rates of 0.008 ($NN\rightarrow ee$),
0.033 ($NN\rightarrow e\mu$), and 0.006 ($NN\rightarrow\mu\mu$)
events/Mton$\cdot$yr. We conservatively take the largest of these and double
it as our estimate of expected background for all of the dinucleon decay
modes: $0.07\pm 0.07$ events/Mton$\cdot$yr.

Decay mode | Efficiency (%) | Background (Events/livetime)
---|---|---
|  | SK-I | SK-II | SK-III | SK-IV | SK-I | SK-II | SK-III | SK-IV
$p\rightarrow e^{+}\gamma$ | High $P_{\text{tot}}$ | $51.0\pm 0.2$ | $49.5\pm 0.2$ | $50.8\pm 0.2$ | $50.6\pm 0.2$ | $0.01\pm 0.01$ | $0.02\pm 0.02$ | $<0.01$ | $0.07\pm 0.07$
Low $P_{\text{tot}}$ | $27.6\pm 0.1$ | $26.1\pm 0.1$ | $27.6\pm 0.1$ | $27.5\pm 0.1$ | $0.02\pm 0.02$ | $0.01\pm 0.01$ | $0.01\pm 0.01$ | $0.04\pm 0.04$
$p\rightarrow\mu^{+}\gamma$ | High $P_{\text{tot}}$ | $50.2\pm 0.2$ | $49.7\pm 0.2$ | $51.0\pm 0.2$ | $48.1\pm 0.2$ | $0.22\pm 0.14$ | $0.14\pm 0.11$ | $0.07\pm 0.07$ | $0.23\pm 0.14$
Low $P_{\text{tot}}$ | $29.1\pm 0.1$ | $28.3\pm 0.1$ | $29.0\pm 0.1$ | $29.4\pm 0.1$ | $0.02\pm 0.02$ | $0.01\pm 0.01$ | $<0.01$ | $0.02\pm 0.02$
$NN\rightarrow ee$ |  | $80.9\pm 0.1$ | $77.2\pm 0.1$ | $79.5\pm 0.1$ | $78.6\pm 0.1$ | $0.01\pm 0.01$ | $<0.01$ | $<0.01$ | $0.01\pm 0.01$
$NN\rightarrow e\mu$ |  | $84.1\pm 0.1$ | $83.7\pm 0.1$ | $83.4\pm 0.1$ | $81.7\pm 0.1$ | $0.01\pm 0.01$ | $<0.01$ | $<0.01$ | $0.01\pm 0.01$
$NN\rightarrow\mu\mu$ |  | $86.3\pm 0.1$ | $85.9\pm 0.1$ | $86.0\pm 0.1$ | $82.8\pm 0.1$ | $0.01\pm 0.01$ | $<0.01$ | $<0.01$ | $0.01\pm 0.01$
Table 1:  Summary of the number of expected background events (with
statistical uncertainty only) for the livetimes of SK-I (91.5
kiloton$\cdot$years), SK-II (49.1 kiloton$\cdot$years), SK-III (31.8
kiloton$\cdot$years), and SK-IV (199.3 kiloton$\cdot$years). The dinucleon
decay efficiency/background rate for a group of modes is the average of the
efficiencies/background rates for the individual modes (the efficiencies are
similar in the same group of modes.)

We find zero candidate events for the eight dinucleon decay modes. For the
nucleon decay mode $p\rightarrow e^{+}\gamma$, we also find zero candidate
events. We observe two candidate events during the SK-IV period for the
$p\rightarrow\mu^{+}\gamma$ decay mode in the “High $P_{tot}$” signal box when
$0.23\pm 0.14_{stat}\pm 0.07_{sys}$ events were expected. The Poisson
probability to see two or more events in the SK-IV livetime given an expected
rate of 0.23 events is 2.3%. One of the two candidates was previously found in
Ref. Abe  _et al._ (2017b). The other candidate is more ambiguous since it
lacks a Michel electron. This may be an indication that the event is due to a
$\nu_{e}n\rightarrow e^{-}p$ charged-current quasielastic interaction, where
the non-showering ring is due to a proton rather than a muon. Requiring a
Michel electron would have eliminated this event, however such a requirement
was not applied for the $p\rightarrow\mu^{+}\gamma$ mode in order to be
consistent with cut (A4) for the dinucleon decay mode. Fig. 3 shows the
agreement of data and atmospheric $\nu$ MC for $p\rightarrow\mu^{+}\gamma$.

Decay mode | Signal efficiency uncertainty (%) | Background rate uncertainty(%)
---|---|---
|  | Reconstruction | Correlated | Nuclear |
Decay | Model |
$p\rightarrow e^{+}\gamma$ | High $P_{\text{tot}}$ | 10.5 | 3.5 | 2.4 | 40.4
Low $P_{\text{tot}}$ | 8.1 | 2.9 | 5.3 | 100
$p\rightarrow\mu^{+}\gamma$ | High $P_{\text{tot}}$ | 10.3 | 3.5 | 3.7 | 31.0
Low $P_{\text{tot}}$ | 8.0 | 3.1 | 5.8 | 44.0
$NN\rightarrow ee$ |  | 5.7 | 8.0 | — | 100
$NN\rightarrow e\mu$ |  | 2.6 | 8.4 | — | 100
$NN\rightarrow\mu\mu$ |  | 4.4 | 8.7 | — | 100
Table 2:  Summary of the systematic uncertainties (percentage contribution) on
signal efficiency and background rate for the nucleon and dinucleon decay
searches. The uncertainties from SK-I to SK-IV are averaged by the live time.
$P(\Gamma|n_{i})\\!=\\!\int^{\lambda}\\!\int^{\epsilon}\\!\int^{b}\\!\frac{e^{-(\Gamma\lambda_{i}(\lambda)\epsilon_{i}(\epsilon)+b_{i}(b))}(\Gamma\lambda_{i}(\lambda)\epsilon_{i}(\epsilon)+b_{i}(b))^{n_{i}}}{n_{i}!}\\!P(\Gamma)\\!P(\lambda_{i}(\lambda)|\lambda_{i,0},\sigma_{\lambda_{i,0}})\\!P(\epsilon_{i}(\epsilon)|\epsilon_{i,0},\sigma_{\epsilon_{i,0}})\\!P(b_{i}(b)|b_{i,0},\sigma_{b_{i,0}})d\lambda\,d\epsilon\,db$
(1)

Table 2 summarizes the systematic uncertainties in the signal efficiency and
in the background rate for each of the nucleon and dinucleon decay modes. The
dominant contributions to uncertainty in the signal efficiency arise from
uncertainties in the areas of reconstruction, correlated decay, and nuclear
model. To assess the impact of differences in the reconstruction of data and
MC, for every variable used in the selection, we compute the percent shift of
the atmospheric $\nu$ MC distribution necessary to minimize its chi-square
against the corresponding data distribution. The cut value in the event
selection is then shifted by that percentage and applied to the signal MC to
recalculate the efficiency. The total systematic uncertainty due to
reconstruction is calculated by summing in quadrature the independent percent
changes in signal efficiency due to each percent-shifted cut. For nucleon
decay in the SK-IV period only, we also add in quadrature with the other
reconstruction uncertainties an additional 10% uncertainty due to neutron
tagging, as was done in Ref. Abe  _et al._ (2017b). This is the reason that
the reconstruction uncertainties for nucleon decay are $\sim$6% larger than
the corresponding uncertainties for dinucleon decay. To estimate the
uncertainty in the signal efficiency arising from uncertainties in correlated
decay, we assume 100% uncertainty on the correlated decay effect, reweight the
correlated decay events accordingly, and recalculate the signal efficiency,
taking the overall change in signal efficiency as the systematic uncertainty.
The nuclear model uncertainty is estimated as the percent change in signal
efficiency when the Fermi gas model is used to compute the true momentum of
the protons within the signal MC events instead of the spectral function fit
to data described earlier.

The systematic uncertainty on the rate of background events is conservatively
taken to be 100% for decay modes where the background events are scarce (all
dinucleon decay modes, and the $p\rightarrow e^{+}\gamma$ “Low
$P_{\text{tot}}$” nucleon decay). For the other nucleon decay signal regions,
we use an event-by-event database with uncertainty weights from 73 sources of
background systematic uncertainty including uncertainties in flux, cross
section and energy calibration, as described in the 2018 SK oscillation
analysis Abe  _et al._ (2018).

Lifetime limits are computed using a Bayesian method, assuming that the SK-I
through SK-IV datasets have correlated systematic uncertainties Tanabashi  _et
al._ (2018). For the nucleon decay modes, the systematic uncertainties of the
“High $P_{\text{tot}}$” and “Low $P_{\text{tot}}$” search boxes are treated as
independent datasets with fully correlated systematic uncertainties. The
conditional probability distribution for the decay rate is given by Eq. 1,
where $\Gamma$ is the decay rate and for dataset $i$, $\lambda_{i}$ is the
exposure (given in proton-years for nucleon decay and in oxygen-years for
dinucleon decay), $\epsilon_{i}$ is the efficiency, $b_{i}$ is the number of
background events, and $n_{i}$ is the number of candidate events. Since the
systematic errors are correlated for all datasets, integrating the prior
probability distribution up to $b$ in some dataset implies that we integrate
the prior distribution in dataset $i$ up to $b_{i}(b)$.

We assume a Gaussian prior distribution
$P(\lambda_{i}(\lambda)|\lambda_{i,0},\sigma_{\lambda_{i,0}})$ for
$\lambda_{i}$ with a mean value of $\lambda_{i,0}$ and
$\sigma_{\lambda_{i,0}}$ given by the $1\%$ percent systematic uncertainty in
exposure. We also assume Gaussian priors
$P(\epsilon_{i}(\epsilon)|\epsilon_{i,0},\sigma_{\epsilon_{i,0}})$ and
$P(b_{i}(b)|b_{i,0},\sigma_{b_{i,0}})$ for $\epsilon_{i}$ and $b_{i}$ with
standard deviations set to the total percent systematic uncertainties in
efficiency and background, respectively. To require a positive lifetime,
$P(\Gamma)$ is 1 for $\Gamma\geq 0$ and otherwise 0. We calculate the upper
bound of the decay rate $\Gamma_{\text{limit}}$ as in Eq. 2, with a 90%
confidence level (CL):

$\text{CL}=\frac{\int_{\Gamma=0}^{\Gamma_{\text{limit}}}\prod_{i=1}^{N}P(\Gamma|n_{i})d\Gamma}{\int_{\Gamma=0}^{\infty}\prod_{i=1}^{N}P(\Gamma|n_{i})d\Gamma}.$
(2)

Therefore we obtain the lower bound on the partial lifetime limit of a decay
mode: $\tau/\text{B}=1/\Gamma_{\text{limit}}$. Table 3 summarizes the partial
lifetime limits obtained for the ten decay modes studied, and these are also
shown in relation to previous measurements in Fig. 4.

Decay mode | Lifetime limit
---|---
per oxygen nucleus | per nucleon
($\times 10^{33}$ years) | ($\times 10^{34}$ years)
$pp\rightarrow e^{+}e^{+}$ | 4.2 | —
$nn\rightarrow e^{+}e^{-}$ | 4.2 | —
$nn\rightarrow\gamma\gamma$ | 4.1 | —
$pp\rightarrow e^{+}\mu^{+}$ | 4.4 | —
$nn\rightarrow e^{+}\mu^{-}$ | 4.4 | —
$nn\rightarrow e^{-}\mu^{+}$ | 4.4 | —
$pp\rightarrow\mu^{+}\mu^{+}$ | 4.4 | —
$nn\rightarrow\mu^{+}\mu^{-}$ | 4.4 | —
$p\rightarrow e^{+}\gamma$ | — | 4.1
$p\rightarrow\mu^{+}\gamma$ | — | 2.1
Table 3:  Summary of the partial lifetime limits for each of the ten dinucleon
and nucleon decay modes, including systematic uncertainties, at 90% CL.

We searched for the 10 dinucleon and nucleon decay modes characterized by a
two-body final state with no hadrons in the Super-Kamiokande data with an
accumulated exposure of 0.37 megaton$\cdot$years. No significant evidence for
dinucleon or nucleon decay was observed, and we set lower limits on the
partial lifetimes that are above $4\times 10^{33}$ years for the dinucleon
decay modes, $4.1\times 10^{34}$ years for $p\rightarrow e^{+}\gamma$, and
$2.1\times 10^{34}$ years for $p\rightarrow\mu^{+}\gamma$. For five of the
modes, the limits are novel, and the limits for all 10 modes are the most
stringent by over one order of magnitude.

Figure 4: (color online) The partial lifetime limits set by Super-Kamiokande
for these ten modes, compared with previous limits set by the Fréjus and IMB
detectors Berger  _et al._ (1991); McGrew  _et al._ (1999). Note that Fréjus
set dinucleon decay lifetime limits per iron nucleus rather than per oxygen
nucleus.

We gratefully acknowledge the cooperation of the Kamioka Mining and Smelting
Company. The Super-Kamiokande experiment has been built and operated from
funding by the Japanese Ministry of Education, Culture, Sports, Science and
Technology, the U.S. Department of Energy, and the U.S. National Science
Foundation.

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