The Belle Collaboration

#
Measurements of the branching fractions of $\Lambda_{c}^{+}\to p\eta$ and
$\Lambda_{c}^{+}\to p\pi^{0}$ decays at Belle

S. X. Li  Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and
Institute of Modern Physics, Fudan University, Shanghai 200443  C. P. Shen
Key Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute
of Modern Physics, Fudan University, Shanghai 200443  I. Adachi  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The
Graduate University for Advanced Studies), Hayama 240-0193  J. K. Ahn  Korea
University, Seoul 02841  H. Aihara  Department of Physics, University of
Tokyo, Tokyo 113-0033  D. M. Asner  Brookhaven National Laboratory, Upton, New
York 11973  T. Aushev  Higher School of Economics (HSE), Moscow 101000  R.
Ayad  Department of Physics, Faculty of Science, University of Tabuk, Tabuk
71451  V. Babu  Deutsches Elektronen–Synchrotron, 22607 Hamburg  S. Bahinipati
Indian Institute of Technology Bhubaneswar, Satya Nagar 751007  P. Behera
Indian Institute of Technology Madras, Chennai 600036  J. Bennett  University
of Mississippi, University, Mississippi 38677  F. Bernlochner  University of
Bonn, 53115 Bonn  M. Bessner  University of Hawaii, Honolulu, Hawaii 96822  V.
Bhardwaj  Indian Institute of Science Education and Research Mohali, SAS
Nagar, 140306  B. Bhuyan  Indian Institute of Technology Guwahati, Assam
781039  T. Bilka  Faculty of Mathematics and Physics, Charles University, 121
16 Prague  J. Biswal  J. Stefan Institute, 1000 Ljubljana  A. Bobrov  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  A. Bozek  H. Niewodniczanski Institute of
Nuclear Physics, Krakow 31-342  M. Bračko  University of Maribor, 2000 Maribor
J. Stefan Institute, 1000 Ljubljana  T. E. Browder  University of Hawaii,
Honolulu, Hawaii 96822  M. Campajola  INFN - Sezione di Napoli, 80126 Napoli
Università di Napoli Federico II, 80126 Napoli  L. Cao  University of Bonn,
53115 Bonn  D. Červenkov  Faculty of Mathematics and Physics, Charles
University, 121 16 Prague  M.-C. Chang  Department of Physics, Fu Jen Catholic
University, Taipei 24205  A. Chen  National Central University, Chung-li 32054
B. G. Cheon  Department of Physics and Institute of Natural Sciences, Hanyang
University, Seoul 04763  K. Chilikin  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991  H. E. Cho  Department of Physics
and Institute of Natural Sciences, Hanyang University, Seoul 04763  K. Cho
Korea Institute of Science and Technology Information, Daejeon 34141  Y. Choi
Sungkyunkwan University, Suwon 16419  S. Choudhury  Indian Institute of
Technology Hyderabad, Telangana 502285  D. Cinabro  Wayne State University,
Detroit, Michigan 48202  S. Cunliffe  Deutsches Elektronen–Synchrotron, 22607
Hamburg  S. Das  Malaviya National Institute of Technology Jaipur, Jaipur
302017  N. Dash  Indian Institute of Technology Madras, Chennai 600036  G. De
Nardo  INFN - Sezione di Napoli, 80126 Napoli  Università di Napoli Federico
II, 80126 Napoli  R. Dhamija  Indian Institute of Technology Hyderabad,
Telangana 502285  F. Di Capua  INFN - Sezione di Napoli, 80126 Napoli
Università di Napoli Federico II, 80126 Napoli  Z. Doležal  Faculty of
Mathematics and Physics, Charles University, 121 16 Prague  T. V. Dong  Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE) and Institute of
Modern Physics, Fudan University, Shanghai 200443  S. Eidelman  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  P.N. Lebedev Physical Institute of the Russian
Academy of Sciences, Moscow 119991  D. Epifanov  Budker Institute of Nuclear
Physics SB RAS, Novosibirsk 630090  Novosibirsk State University, Novosibirsk
630090  T. Ferber  Deutsches Elektronen–Synchrotron, 22607 Hamburg  D.
Ferlewicz  School of Physics, University of Melbourne, Victoria 3010  B. G.
Fulsom  Pacific Northwest National Laboratory, Richland, Washington 99352  R.
Garg  Panjab University, Chandigarh 160014  V. Gaur  Virginia Polytechnic
Institute and State University, Blacksburg, Virginia 24061  A. Garmash  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  A. Giri  Indian Institute of Technology
Hyderabad, Telangana 502285  P. Goldenzweig  Institut für Experimentelle
Teilchenphysik, Karlsruher Institut für Technologie, 76131 Karlsruhe  B. Golob
Faculty of Mathematics and Physics, University of Ljubljana, 1000 Ljubljana
J. Stefan Institute, 1000 Ljubljana  O. Grzymkowska  H. Niewodniczanski
Institute of Nuclear Physics, Krakow 31-342  K. Gudkova  Budker Institute of
Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State University,
Novosibirsk 630090  C. Hadjivasiliou  Pacific Northwest National Laboratory,
Richland, Washington 99352  O. Hartbrich  University of Hawaii, Honolulu,
Hawaii 96822  K. Hayasaka  Niigata University, Niigata 950-2181  H. Hayashii
Nara Women’s University, Nara 630-8506  M. T. Hedges  University of Hawaii,
Honolulu, Hawaii 96822  W.-S. Hou  Department of Physics, National Taiwan
University, Taipei 10617  C.-L. Hsu  School of Physics, University of Sydney,
New South Wales 2006  T. Iijima  Kobayashi-Maskawa Institute, Nagoya
University, Nagoya 464-8602  Graduate School of Science, Nagoya University,
Nagoya 464-8602  K. Inami  Graduate School of Science, Nagoya University,
Nagoya 464-8602  G. Inguglia  Institute of High Energy Physics, Vienna 1050
A. Ishikawa  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801  SOKENDAI (The Graduate University for Advanced Studies), Hayama
240-0193  R. Itoh  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  M. Iwasaki  Osaka City University, Osaka 558-8585  Y. Iwasaki
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801  W. W.
Jacobs  Indiana University, Bloomington, Indiana 47408  E.-J. Jang  Gyeongsang
National University, Jinju 52828  S. Jia  Key Laboratory of Nuclear Physics
and Ion-beam Application (MOE) and Institute of Modern Physics, Fudan
University, Shanghai 200443  Y. Jin  Department of Physics, University of
Tokyo, Tokyo 113-0033  C. W. Joo  Kavli Institute for the Physics and
Mathematics of the Universe (WPI), University of Tokyo, Kashiwa 277-8583  K.
K. Joo  Chonnam National University, Gwangju 61186  A. B. Kaliyar  Tata
Institute of Fundamental Research, Mumbai 400005  K. H. Kang  Kyungpook
National University, Daegu 41566  G. Karyan  Deutsches Elektronen–Synchrotron,
22607 Hamburg  Y. Kato  Graduate School of Science, Nagoya University, Nagoya
464-8602  T. Kawasaki  Kitasato University, Sagamihara 252-0373  H. Kichimi
High Energy Accelerator Research Organization (KEK), Tsukuba 305-0801  B. H.
Kim  Seoul National University, Seoul 08826  C. H. Kim  Department of Physics
and Institute of Natural Sciences, Hanyang University, Seoul 04763  D. Y. Kim
Soongsil University, Seoul 06978  K.-H. Kim  Yonsei University, Seoul 03722
S. H. Kim  Seoul National University, Seoul 08826  Y.-K. Kim  Yonsei
University, Seoul 03722  K. Kinoshita  University of Cincinnati, Cincinnati,
Ohio 45221  P. Kodyš  Faculty of Mathematics and Physics, Charles University,
121 16 Prague  T. Konno  Kitasato University, Sagamihara 252-0373  A. Korobov
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  S. Korpar  University of Maribor, 2000
Maribor  J. Stefan Institute, 1000 Ljubljana  E. Kovalenko  Budker Institute
of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State University,
Novosibirsk 630090  P. Križan  Faculty of Mathematics and Physics, University
of Ljubljana, 1000 Ljubljana  J. Stefan Institute, 1000 Ljubljana  R. Kroeger
University of Mississippi, University, Mississippi 38677  P. Krokovny  Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk State
University, Novosibirsk 630090  T. Kuhr  Ludwig Maximilians University, 80539
Munich  M. Kumar  Malaviya National Institute of Technology Jaipur, Jaipur
302017  K. Kumara  Wayne State University, Detroit, Michigan 48202  A. Kuzmin
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  Y.-J. Kwon  Yonsei University, Seoul
03722  K. Lalwani  Malaviya National Institute of Technology Jaipur, Jaipur
302017  J. S. Lange  Justus-Liebig-Universität Gießen, 35392 Gießen  I. S. Lee
Department of Physics and Institute of Natural Sciences, Hanyang University,
Seoul 04763  S. C. Lee  Kyungpook National University, Daegu 41566  C. H. Li
Liaoning Normal University, Dalian 116029  L. K. Li  University of Cincinnati,
Cincinnati, Ohio 45221  Y. B. Li  Peking University, Beijing 100871  L. Li
Gioi  Max-Planck-Institut für Physik, 80805 München  J. Libby  Indian
Institute of Technology Madras, Chennai 600036  K. Lieret  Ludwig Maximilians
University, 80539 Munich  D. Liventsev  Wayne State University, Detroit,
Michigan 48202  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801  M. Masuda  Earthquake Research Institute, University of Tokyo, Tokyo
113-0032  Research Center for Nuclear Physics, Osaka University, Osaka
567-0047  T. Matsuda  University of Miyazaki, Miyazaki 889-2192  D. Matvienko
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090  Novosibirsk
State University, Novosibirsk 630090  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991  M. Merola  INFN - Sezione di
Napoli, 80126 Napoli  Università di Napoli Federico II, 80126 Napoli  F.
Metzner  Institut für Experimentelle Teilchenphysik, Karlsruher Institut für
Technologie, 76131 Karlsruhe  R. Mizuk  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991  Higher School of Economics (HSE),
Moscow 101000  S. Mohanty  Tata Institute of Fundamental Research, Mumbai
400005  Utkal University, Bhubaneswar 751004  T. Mori  Graduate School of
Science, Nagoya University, Nagoya 464-8602  M. Nakao  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The Graduate
University for Advanced Studies), Hayama 240-0193  Z. Natkaniec  H.
Niewodniczanski Institute of Nuclear Physics, Krakow 31-342  A. Natochii
University of Hawaii, Honolulu, Hawaii 96822  L. Nayak  Indian Institute of
Technology Hyderabad, Telangana 502285  M. Niiyama  Kyoto Sangyo University,
Kyoto 603-8555  N. K. Nisar  Brookhaven National Laboratory, Upton, New York
11973  S. Nishida  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  H. Ono  Nippon Dental University, Niigata 951-8580  Niigata
University, Niigata 950-2181  Y. Onuki  Department of Physics, University of
Tokyo, Tokyo 113-0033  P. Pakhlov  P.N. Lebedev Physical Institute of the
Russian Academy of Sciences, Moscow 119991  Moscow Physical Engineering
Institute, Moscow 115409  G. Pakhlova  Higher School of Economics (HSE),
Moscow 101000  P.N. Lebedev Physical Institute of the Russian Academy of
Sciences, Moscow 119991  T. Pang  University of Pittsburgh, Pittsburgh,
Pennsylvania 15260  S. Pardi  INFN - Sezione di Napoli, 80126 Napoli  H. Park
Kyungpook National University, Daegu 41566  S.-H. Park  High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  S. Patra  Indian
Institute of Science Education and Research Mohali, SAS Nagar, 140306  S. Paul
Department of Physics, Technische Universität München, 85748 Garching  Max-
Planck-Institut für Physik, 80805 München  T. K. Pedlar  Luther College,
Decorah, Iowa 52101  R. Pestotnik  J. Stefan Institute, 1000 Ljubljana  L. E.
Piilonen  Virginia Polytechnic Institute and State University, Blacksburg,
Virginia 24061  T. Podobnik  Faculty of Mathematics and Physics, University of
Ljubljana, 1000 Ljubljana  J. Stefan Institute, 1000 Ljubljana  V. Popov
Higher School of Economics (HSE), Moscow 101000  E. Prencipe
Forschungszentrum Jülich, 52425 Jülich  M. T. Prim  University of Bonn, 53115
Bonn  M. Röhrken  Deutsches Elektronen–Synchrotron, 22607 Hamburg  A.
Rostomyan  Deutsches Elektronen–Synchrotron, 22607 Hamburg  N. Rout  Indian
Institute of Technology Madras, Chennai 600036  G. Russo  Università di Napoli
Federico II, 80126 Napoli  D. Sahoo  Tata Institute of Fundamental Research,
Mumbai 400005  Y. Sakai  High Energy Accelerator Research Organization (KEK),
Tsukuba 305-0801  SOKENDAI (The Graduate University for Advanced Studies),
Hayama 240-0193  S. Sandilya  Indian Institute of Technology Hyderabad,
Telangana 502285  A. Sangal  University of Cincinnati, Cincinnati, Ohio 45221
L. Santelj  Faculty of Mathematics and Physics, University of Ljubljana, 1000
Ljubljana  J. Stefan Institute, 1000 Ljubljana  T. Sanuki  Department of
Physics, Tohoku University, Sendai 980-8578  V. Savinov  University of
Pittsburgh, Pittsburgh, Pennsylvania 15260  G. Schnell  Department of Physics,
University of the Basque Country UPV/EHU, 48080 Bilbao  IKERBASQUE, Basque
Foundation for Science, 48013 Bilbao  J. Schueler  University of Hawaii,
Honolulu, Hawaii 96822  C. Schwanda  Institute of High Energy Physics, Vienna
1050  Y. Seino  Niigata University, Niigata 950-2181  K. Senyo  Yamagata
University, Yamagata 990-8560  M. E. Sevior  School of Physics, University of
Melbourne, Victoria 3010  M. Shapkin  Institute for High Energy Physics,
Protvino 142281  C. Sharma  Malaviya National Institute of Technology Jaipur,
Jaipur 302017  V. Shebalin  University of Hawaii, Honolulu, Hawaii 96822
J.-G. Shiu  Department of Physics, National Taiwan University, Taipei 10617
B. Shwartz  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  E. Solovieva  P.N. Lebedev
Physical Institute of the Russian Academy of Sciences, Moscow 119991  S.
Stanič  University of Nova Gorica, 5000 Nova Gorica  M. Starič  J. Stefan
Institute, 1000 Ljubljana  Z. S. Stottler  Virginia Polytechnic Institute and
State University, Blacksburg, Virginia 24061  M. Sumihama  Gifu University,
Gifu 501-1193  T. Sumiyoshi  Tokyo Metropolitan University, Tokyo 192-0397  W.
Sutcliffe  University of Bonn, 53115 Bonn  M. Takizawa  Showa Pharmaceutical
University, Tokyo 194-8543  J-PARC Branch, KEK Theory Center, High Energy
Accelerator Research Organization (KEK), Tsukuba 305-0801  Meson Science
Laboratory, Cluster for Pioneering Research, RIKEN, Saitama 351-0198  K.
Tanida  Advanced Science Research Center, Japan Atomic Energy Agency, Naka
319-1195  Y. Tao  University of Florida, Gainesville, Florida 32611  F.
Tenchini  Deutsches Elektronen–Synchrotron, 22607 Hamburg  K. Trabelsi
Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay  M. Uchida  Tokyo
Institute of Technology, Tokyo 152-8550  S. Uehara  High Energy Accelerator
Research Organization (KEK), Tsukuba 305-0801  SOKENDAI (The Graduate
University for Advanced Studies), Hayama 240-0193  T. Uglov  P.N. Lebedev
Physical Institute of the Russian Academy of Sciences, Moscow 119991  Higher
School of Economics (HSE), Moscow 101000  K. Uno  Niigata University, Niigata
950-2181  S. Uno  High Energy Accelerator Research Organization (KEK), Tsukuba
305-0801  SOKENDAI (The Graduate University for Advanced Studies), Hayama
240-0193  P. Urquijo  School of Physics, University of Melbourne, Victoria
3010  R. Van Tonder  University of Bonn, 53115 Bonn  G. Varner  University of
Hawaii, Honolulu, Hawaii 96822  A. Vossen  Duke University, Durham, North
Carolina 27708  C. H. Wang  National United University, Miao Li 36003  E. Wang
University of Pittsburgh, Pittsburgh, Pennsylvania 15260  M.-Z. Wang
Department of Physics, National Taiwan University, Taipei 10617  P. Wang
Institute of High Energy Physics, Chinese Academy of Sciences, Beijing 100049
S. Watanuki  Université Paris-Saclay, CNRS/IN2P3, IJCLab, 91405 Orsay  E. Won
Korea University, Seoul 02841  X. Xu  Soochow University, Suzhou 215006  B. D.
Yabsley  School of Physics, University of Sydney, New South Wales 2006  W. Yan
Department of Modern Physics and State Key Laboratory of Particle Detection
and Electronics, University of Science and Technology of China, Hefei 230026
S. B. Yang  Korea University, Seoul 02841  H. Ye  Deutsches
Elektronen–Synchrotron, 22607 Hamburg  J. Yelton  University of Florida,
Gainesville, Florida 32611  J. H. Yin  Korea University, Seoul 02841  C. Z.
Yuan  Institute of High Energy Physics, Chinese Academy of Sciences, Beijing
100049  Y. Yusa  Niigata University, Niigata 950-2181  Z. P. Zhang  Department
of Modern Physics and State Key Laboratory of Particle Detection and
Electronics, University of Science and Technology of China, Hefei 230026  V.
Zhilich  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090  V. Zhukova  P.N. Lebedev
Physical Institute of the Russian Academy of Sciences, Moscow 119991  V.
Zhulanov  Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090
Novosibirsk State University, Novosibirsk 630090

###### Abstract

We report measurements of the branching fractions of singly Cabibbo-suppressed
decays $\Lambda_{c}^{+}\to p\eta$ and $\Lambda_{c}^{+}\to p\pi^{0}$ using the
full Belle data sample corresponding to an integrated luminosity of 980.6 $\rm
fb^{-1}$. The data were collected by the Belle detector at the KEKB
$e^{+}$$e^{-}$ asymmetric-energy collider. A clear $\Lambda_{c}^{+}$ signal is
seen in the invariant mass distribution of $p\eta$. The fitted number of
signal events of the $\Lambda_{c}^{+}\to p\eta$ process is $7734\pm 263$; from
this, we measure the ratio of branching fractions ${\cal B}(\Lambda_{c}^{+}\to
p\eta)/{\cal B}(\Lambda_{c}^{+}\to pK^{-}\pi^{+})=(2.258\pm 0.077(\rm
stat.)\pm 0.122(\rm syst.))\times 10^{-2}$, from which we infer the branching
fraction ${\cal B}(\Lambda_{c}^{+}\to p\eta)=(1.42\pm 0.05(\rm stat.)\pm
0.11(\rm syst.))\times 10^{-3}$. In addition, no significant signal for
$\Lambda_{c}^{+}\to p\pi^{0}$ is found so an upper limit on the branching
fraction of ${\cal B}(\Lambda_{c}^{+}\to p\pi^{0})<8.0\times 10^{-5}$ at 90%
credibility level is set, more than three times better than the best current
upper limit.

##  I Introduction

Weak decays of charmed baryons are useful for testing many contradictory
theoretical models and methods, e.g. the flavor symmetry approach and heavy
quark effective theory lamcR1 ; lamcR2 ; lamcR3 ; lamcR4 . In contrast with
the decays of charmed mesons, the decays of some charmed baryons are helicity
suppressed, making the $W$-boson exchange favored lamcR5 . The understanding
of charmed baryons has progressed relatively slowly compared to that of
charmed mesons. The main reason is that the cross section for the generation
of charmed baryons is smaller than that of the mesons, so that some reactions
with small decay branching fractions are difficult to observe experimentally
lamcR6 ; lamcR7 ; lamcR8 . Although there have been many improved measurements
of the properties of charmed baryons, precision measurements of the decay
branching fractions still remain poor for many Cabibbo favored (CF) decay
modes and even worse for some decay modes dominated by Cabibbo suppression and
$W$-boson exchange pdg .

In theory, the singly Cabibbo-suppressed (SCS) decays
$\Lambda_{c}^{+}\rightarrow p\pi^{0}$ and $\Lambda_{c}^{+}\rightarrow p\eta$
proceed predominantly through internal $W$ emission and $W$ exchange. Typical
decay diagrams of two SCS decays are shown in Fig. 1. The internal $W$
emission involving an $s$ quark in Fig. 1(f) is allowed in
$\Lambda_{c}^{+}\rightarrow p\eta$ but absent in $\Lambda_{c}^{+}\rightarrow
p\pi^{0}$. The theoretical calculations predict the branching fraction of
$\Lambda_{c}^{+}\rightarrow p\eta$ at least an order of magnitude greater than
that of $\Lambda_{c}^{+}\rightarrow p\pi^{0}$ and give different assumption-
dependent results for the branching fractions of these SCS decays lamcR1 ;
lamcR3 ; lamcR9 ; lamcR10 ; lamcR11 . In contrast with the strong decays of
heavy-flavor mesons, the $W$-boson exchange mechanism plays an important role
in the decay of charmed baryons. Thus, measuring the branching fractions of
these two SCS decays will help elucidate the decay mechanism of charmed
baryons.

Figure 1: Feynman diagrams for the weak Cabibbo-suppressed decays
$\Lambda_{c}^{+}\to p\pi^{0}$ and $\Lambda_{c}^{+}\to p\eta$: (a, b) Internal
$W$ emission, (c, d, e) $W$ exchange, and (f) Internal $W$ emission for
$\Lambda_{c}^{+}\rightarrow p\eta$.

The first evidence for the decay $\Lambda_{c}^{+}\rightarrow p\eta$ with a
statistical significance of 4.2$\sigma$ and a branching fraction of
$\mathcal{B}(\Lambda_{c}^{+}\rightarrow p\eta)=(1.24\pm 0.30)\times 10^{-3}$
was reported by the BESIII Collaboration besResult . They found no significant
$\Lambda_{c}^{+}\rightarrow p\pi^{0}$ signal and set an upper limit on its
branching fraction $\mathcal{B}(\Lambda_{c}^{+}\rightarrow p\pi^{0})<2.7\times
10^{-4}$ at 90% confidence level besResult .

To improve the measurement precision, we measure the ratio of the branching
fractions of the two SCS processes with respect to the CF
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ decay mode:

$\frac{\mathcal{B}(\rm SCS)}{\mathcal{B}(\rm CF)}=\frac{N^{\rm obs}(\rm
SCS)}{\epsilon^{\rm MC}(\rm
SCS)\times\mathcal{B}(\pi^{0}/\eta\rightarrow\gamma\gamma)}\times\frac{\epsilon^{MC}(\rm
CF)}{N^{\rm obs}(\rm CF)},$ (1)

where $\mathcal{B}$, $\epsilon^{\rm MC}$, and $N^{\rm obs}$ are the branching
fraction, signal efficiency, and the fitted yield of signal events from data,
respectively. The value of the branching fraction of the CF decay is $(6.28\pm
0.32)\times 10^{-2}$ pdg . The values of
$\mathcal{B}(\pi^{0}\rightarrow\gamma\gamma)$ and
$\mathcal{B}(\eta\rightarrow\gamma\gamma)$ are $0.9882\pm 0.0003$ and
$0.3941\pm 0.0020$, respectively pdg .

##  II The data sample and the belle detector

The measurements of the two SCS branching fractions are based on a data sample
taken at or near the $\Upsilon(1S)$, $\Upsilon(2S)$, $\Upsilon(3S)$,
$\Upsilon(4S)$, and $\Upsilon(5S)$ resonances collected with the Belle
detector at the KEKB asymmetric-energy $e^{+}e^{-}$ collider KEKB . The
integrated luminosity of the data samples is 980.6 $\rm fb^{-1}$, including
711 $\rm fb^{-1}$ on the $\Upsilon(4S)$ resonance, 89.4 $\rm fb^{-1}$ off the
$\Upsilon(4S)$ resonance, 121.4 $\rm fb^{-1}$ on the $\Upsilon(5S)$ resonance,
and 58.8 $\rm fb^{-1}$ at the $\Upsilon(1S,~{}2S,~{}3S)$ resonances. The Belle
detector is a large-solid-angle magnetic spectrometer that consists of a
silicon vertex detector (SVD), a 50-layer central drift chamber (CDC), an
array of aerogel threshold Cherenkov counters (ACC), a barrel-like arrangement
of time-of-flight scintillation counters (TOF), and an electromagnetic
calorimeter comprised of CsI(Tl) crystals (ECL) located inside a
superconducting solenoid coil that provides a 1.5 T magnetic field. An iron
flux-return located outside of the coil is instrumented to detect $K_{L}^{0}$
mesons and to identify muons (KLM). The detector is described in detail
elsewhere Belle .

Signal MC samples of $e^{+}e^{-}\rightarrow c\bar{c}$;
$c\bar{c}\rightarrow\Lambda_{c}^{+}X$ with $X$ denoting anything;
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}/p\pi^{0}/p\eta$ are used to optimize
the selection criteria and estimate the reconstruction and selection
efficiency, and are generated under the $\Upsilon(4S)$ resonance condition
with pythia pythia  and EvtGen evtgen  and propagated with geant3 geant3  to
simulate the detector performance. The charged-conjugate modes are included
unless otherwise stated.

Inclusive MC samples of $\Upsilon(4S)\rightarrow B^{+}B^{-}/B^{0}\bar{B}^{0}$,
$\Upsilon(5S)\rightarrow B_{s}^{(*)}\bar{B}_{s}^{(*)}$, $e^{+}e^{-}\to
q\bar{q}$ $(q=u,~{}d,~{}s,~{}c)$ at $\sqrt{s}$ = 10.58 and 10.867 GeV, and
$\Upsilon(1S,~{}2S,~{}3S)$ decays corresponding to two times the integrated
luminosity of each data set are used to characterize the (potentially peaking)
backgrounds topoana .

##  III Event selection criteria

For charged-particle tracks, the distance of closest approach with respect to
the interaction point (IP) along the $z$ axis (parallel to the positron beam)
and in the transverse $r\phi$ plane is required to be less than 2.0 cm and 0.1
cm, respectively. In addition, each track is required to have at least one SVD
hit. Particle identification (PID) is used to discriminate the type of charged
hadron tracks:
$\mathcal{R}(h|h^{\prime})=\mathcal{L}(h)/(\mathcal{L}(h)+\mathcal{L}(h^{\prime}))$
is defined as the ratio of the likelihoods for the $h$ and $h^{\prime}$
hypotheses, where $\mathcal{L}(h)$ ($h$ = $\pi$, $K$, or $p$) is the combined
likelihood derived from the ACC, TOF, and CDC $dE/dx$ measurements pidcode .
$\mathcal{R}(p|\pi)>0.9$ and $\mathcal{R}(p|K)>0.9$ are required for protons.
$\mathcal{R}(K|p)>0.4$ and $\mathcal{R}(K|\pi)>0.9$ are required for charged
kaons. $\mathcal{R}(\pi|p)>0.4$ and $\mathcal{R}(\pi|K)>0.4$ are required for
charged pions. $\mathcal{R}(e)$, a likelihood ratio for $e$ and $h$
identification formed from ACC, CDC, and ECL information eidcode , is required
to be less than 0.9 for all charged tracks to remove electrons. For the
typical momentum range of our SCS decays, the identification efficiencies of
$p$, $K$, and $\pi$ are 81.7%, 79.6%, and 96.9%, respectively.

A $\Lambda_{c}^{+}$ candidate for the CF decay is reconstructed from three
tracks identified as $p$, $K$, and $\pi$, subject to a common-vertex fit. The
$\chi^{2}$ of the vertex fit is required to be less than 40 to reject
background from incorrect combinations. The scaled momentum of the
$\Lambda_{c}^{+}$, defined as $x_{p}={p^{*}}/{\sqrt{E^{2}_{\rm cm}/4-M^{2}}}$
speedoflight , is required to be greater than 0.53 for all $\Lambda_{c}^{+}$
candidates to suppress the combinatorial background, especially from $B$-meson
decays. Here, $E_{\rm cm}$ is the center-of-mass (CM) energy, while $p^{*}$
and $M$ are the momentum and invariant mass, respectively, of the
$\Lambda_{c}^{+}$ candidates in the CM frame. All of these optimized selection
criteria are taken from Ref. dcspkpi .

An ECL cluster not matching any track is identified as a photon candidate.
Each photon candidate is required to have a ratio of energy deposited in the
central $3\times 3$ square of ECL cells to that deposited in the enclosing
$5\times 5$ square of cells of $E9/E25>0.8$ to reject neutral hadrons. An
optimized figure-of-merit (FOM) study determines that the energy of photon
candidates must exceed 50 MeV and 110 MeV in the barrel and endcap regions of
the ECL, respectively, for both photons from $\pi^{0}\rightarrow\gamma\gamma$.
For the $\eta\rightarrow\gamma_{1}\gamma_{2}$ decay, the $\gamma_{1}$
($\gamma_{2}$) energy must exceed 220 (260) MeV, 480 (340) MeV, and 260 (360)
MeV in the barrel, forward, and backward endcaps, respectively. Two photon
candidates are combined to form a $\pi^{0}/\eta$ candidate and a mass-
constrained fit is performed for this candidate. The $\chi^{2}$ value of the
mass-constrained fit must be less than 7.5 and 4 for $\pi^{0}$ and $\eta$
candidates, respectively, to suppress the background in which the two-photon
invariant mass is far from $\pi^{0}$ and $\eta$ nominal masses pdg . The
momentum in the CM frame must be greater than 0.69 GeV/$c$ and 0.82 GeV/$c$
for $\pi^{0}$ and $\eta$ candidates, respectively. All these requirements are
optimized. An SCS $\Lambda_{c}^{+}$ candidate is reconstructed by combining a
proton candidate with a $\pi^{0}/\eta$ candidate. Again, $x_{p}$ for the
$\Lambda_{c}^{+}\rightarrow p\pi^{0}/p\eta$ candidates is required to exceed
0.53. After applying all the selection criteria, about 0.8%, 1.4%, and 1.7% of
the events in the signal region have multiple $\Lambda_{c}^{+}$ candidates for
the $pK^{-}\pi^{+}$, $p\eta$, and $p\pi^{0}$ decays, respectively.

The SCS signal region in data is optimized with the control sample of
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ as well as the $\Lambda_{c}^{+}$
mass sidebands to the hidden SCS signal region (i.e. the signal region is
blinded) by optimizing the ratio $S/\sqrt{S+B}$, where $S$ and $B$ are the
expected number of signal events for SCS decays in data and the number of
background events normalized to the signal region, respectively. $S$ is
obtained via

$\begin{split}S&=\epsilon^{\rm MC}(\Lambda_{c}^{+}\rightarrow
p\pi^{0}/p\eta)\times\frac{N^{\rm obs}(\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+})}{\epsilon^{\rm MC}(\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+})}\\\ &\times\frac{\mathcal{B}(\Lambda_{c}^{+}\rightarrow
p\pi^{0}/p\eta)\times\mathcal{B}(\pi^{0}/\eta\rightarrow\gamma\gamma)}{\mathcal{B}(\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+})},\end{split}$ (2)

where $N^{\rm obs}$ and $\epsilon^{\rm MC}$ are the fitted $\Lambda_{c}^{+}$
events of data and the detection efficiency of the signal MC sample,
respectively; $\mathcal{B}(\Lambda_{c}^{+}\rightarrow p\pi^{0}/p\eta)$ are the
branching fractions of $2.7\times 10^{-4}$ and $1.24\times 10^{-3}$ for
$\Lambda_{c}^{+}\rightarrow p\pi^{0}$ and $\Lambda_{c}^{+}\rightarrow p\eta$,
respectively besResult ; and $\mathcal{B}(\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+})$ is the branching fraction of the CF decay pdg .

##  IV efficiency estimation and fit results

With the final selection criteria applied, the invariant mass distributions of
$pK^{-}\pi^{+}$, $p\eta$, and $p\pi^{0}$ from data are shown in Figs. 2, 3,
and 4, respectively. From the study of the topology of inclusive MC samples
topoana , no peaking backgrounds contribute to these mass distributions in the
$\Lambda_{c}^{+}$ signal region.

Figure 2: Fit to the invariant mass distribution of $pK^{-}\pi^{+}$ from data.
Black dots with error bars represent the data; the pink dashed line, the blue
dash-dotted line, the green dashed line, and the red solid line represent the
background contribution, the core Gaussian, tail Gaussian, and the total fit,
respectively. Figure 3: Fit to the invariant mass distribution of $p\eta$ from
data. Black dots with error bars represent the data; the magenta dash-dotted
line, the blue dashed line, and the red solid line represent the background
component, the signal, and the total fit, respectively. Figure 4: Fit to the
invariant mass distribution of $p\pi^{0}$. Black dots with error bars
represent the data; the magenta dash-dotted line, the blue dashed line, and
the red solid line represent the background component, the signal, and the
total fit, respectively.

For the CF mode, we fit the invariant mass distribution of $pK^{-}\pi^{+}$
displayed in Fig. 2 from 2.15 to 2.42 GeV/$c^{2}$ using the binned maximum
likelihood fit with a bin width of 3 MeV/$c^{2}$. A double-Gaussian function
with the common mean value is used to model the signal events and a second-
order polynomial is used to model the background events. The parameters of the
signal and background shapes are free in the fit. The reduced $\chi^{2}$ value
of the fit is $\chi^{2}/{\rm ndf}=87/82=1.06$ and the fitted number of signal
events is 1476200 $\pm$ 1560, where $\rm ndf$ is the number of degrees of
freedom and the uncertainty is statistical only.

The Dalitz dalitz  distribution of $M^{2}(K^{-}\pi^{+})$ versus
$M^{2}(pK^{-})$ in the signal region from data is shown in Fig. 5. The signal
region is taken from 2.274 to 2.298 GeV/$c^{2}$. We divide this into
120$\times$120 pixels, with size 0.027 $\rm GeV^{2}$$/c^{4}$ for
$M^{2}(pK^{-})$ and 0.016 $\rm GeV^{2}$$/c^{4}$ for $M^{2}(K^{-}\pi^{+})$. The
number of background events has been subtracted using the normalized
sidebands. The sideband regions are defined to be (2.262, 2.274) GeV/$c^{2}$
and (2.298, 2.310) GeV/$c^{2}$.

Figure 5: Dalitz plot of the selected $\Lambda_{c}^{+}\rightarrow
pK^{-}\pi^{+}$ candidates.

A MC sample mixing four subchannels of CF decay weighted with the
corresponding branching fractions taken from Ref. pdg  is used to assess the
selection efficiency of the CF mode. The total number of reconstructed MC
signal events is normalized to that of signal candidates in data. We calculate
the overall efficiency using the efficiency of each pixel. The formula is
$\epsilon={\Sigma_{i}s_{i}}/{\Sigma_{j}(s_{j}/\epsilon_{j})}$, where
$\Sigma_{i}s_{i}$ is the number of signal candidates in data, $s_{j}$ and
$\epsilon_{j}$ are the number of signal events from data and the efficiency
from the MC sample for each pixel, respectively. The efficiency of one pixel
is obtained by dividing the number of events remaining in the signal MC sample
by the number of generated events. The weighted efficiency for each bin is
exhibited in Fig. 6 and the corrected efficiency for data is $(14.06\pm
0.01)$%.

Figure 6: Dalitz plot of efficiency distribution for
$\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+}$ decay.

An obvious $\Lambda_{c}^{+}$ signal peaking in the signal region of the
M($p\eta$) spectrum is observed. We use the binned maximum likelihood method
to fit the invariant mass distribution of $p\eta$ from 2.15 to 2.42
GeV/$c^{2}$ with 3 MeV/$c^{2}$ bin width. A combined Gaussian and Crystal Ball
(CB) cb  function with a common mean value models the signal, and a second-
order polynomial models the background. The parameters of the signal and
background line shapes are free in the fit. Figure 3 exhibits the distribution
of invariant mass of $p\eta$ and corresponding fit result. The reduced
$\chi^{2}$ of the fit is $\chi^{2}/{\rm ndf}=102/83=1.23$ and the fitted
number of signal events is $7734\pm 263$.

There is no significant excess observed in the signal region for
$\Lambda_{c}^{+}\rightarrow p\pi^{0}$. We fit the $M(p\pi^{0})$ with the
binned maximum likelihood method; the fit result is shown in Fig. 4. The
signal is modeled by a combined Gaussian and CB function with the a common
mean convolved with a Gaussian function; the background is described by a
second-order polynomial. The parameters of the signal are fixed to MC-derived
values and the convolving Gaussian with width 2.1 MeV accounts for the
difference in widths between data and MC for the $\Lambda_{c}^{+}\rightarrow
p\eta$ signal. The fitted number of signal events and the parameters of the
background polynomial are free in the fit. The fitting range is from 2.15 to
2.42 GeV/$c^{2}$ with a bin width of 3 MeV/$c^{2}$. The fitted number of
signal events is $11\pm 140$, which is consistent with zero. Thus, with a
uniform prior probability density function estimation of a Bayesian upper
limit is performed to obtain the 90% credibility level (C.L.) upper limit on
the branching fraction of $\Lambda_{c}^{+}\rightarrow p\pi^{0}$. The
likelihood function is integrated from zero to the value that gives 90% of the
total area. Before integrating, we include the systematic uncertainty
($\sigma_{\rm sys}$) described below by convolving the likelihood distribution
with a Gaussian whose width is equal to $\sigma_{\rm sys}$. An upper limit on
the branching fraction of $9.44\times 10^{-5}$ at 90% C.L. is set. The
likelihood distribution as a function of the branching fraction, with the
systematic uncertainty included, is displayed in Fig. 7.

Figure 7: The likelihood distribution as a function of the branching fraction
for $\Lambda_{c}^{+}\to p\pi^{0}$ with the systematic uncertainty included.
The blue arrow refers to the 90% C.L. upper limit on the branching fraction.

To estimate the efficiencies of the two SCS decays, we take the ratio of the
number of fitted signal events in the invariant mass distribution of
$p\pi^{0}/p\eta$ to that of generated events from signal MC samples as the
efficiency. We find $(8.28\pm 0.03)$% and $(8.89\pm 0.03)$% for
$\Lambda_{c}^{+}\rightarrow p\eta$ and $\Lambda_{c}^{+}\rightarrow p\pi^{0}$,
respectively. The uncertainties are statistical only.

##  V Systematic Uncertainties

Since the branching fraction is obtained from the ratio of the corresponding
quantities in Eq. 1, some systematic uncertainties for
$\Lambda_{c}^{+}\rightarrow p\pi^{0}/p\eta$ cancel. The sources of systematic
uncertainties include the fits of CF and SCS decays, PID, tracking efficiency,
photon efficiency, the uncertainties of branching fractions of CF and
$\pi^{0}/\eta\rightarrow\gamma\gamma$ decays, and the statistics of the signal
MC samples.

To estimate the uncertainties from the fits of CF and SCS decays, we modify
the signal and background functions, bin width, and the fit range and refit.
To evaluate the uncertainty from the signal function, the signal shape for
$\Lambda_{c}^{+}\to pK^{-}\pi^{+}$/$p\eta$ is fixed to that from the fit to
the MC sample, while that for $\Lambda_{c}^{+}\rightarrow p\pi^{0}$ is changed
from a Gaussian and CB combined function to a double CB function. The
uncertainty from the background line shape is assessed by using a first-order
polynomial. Furthermore, we change the bin width to 2 MeV/$c^{2}$ or 4
MeV/$c^{2}$, and adjust the fit range of invariant mass spectrum to estimate
the uncertainties from binning and fit range. The difference of branching
fractions between the refitted and nominal conditions is taken as the
uncertainty, which is 3.86% for $\Lambda_{c}^{+}\rightarrow p\pi^{0}$ and
2.85% for $\Lambda_{c}^{+}\rightarrow p\eta$, respectively.

The systematic uncertainties from PID and tracking efficiency of the proton
cancel in the branching-fraction ratio. Systematic uncertainties of 1.6% and
1.2% are assigned for the $K$ and $\pi$ identification efficiencies by
studying a low-background control sample of $D^{\ast}$, respectively. The
total systematic uncertainty from PID is 2.0%, the sum in quadrature of the
individual uncertainties for $K$ and $\pi$. From the study of the mid-to-high-
momentum track reconstruction efficiency in $D^{*}\rightarrow\pi D^{0}$ decay,
the uncertainty of the efficiency for each charged track is 0.35%, resulting
in a total uncertainty of 0.7% from tracking efficiency. We assign a 2%
systematic uncertainty due to the photon efficiency per photon according to a
study of radiative Bhabha events; the total systematic uncertainty from photon
reconstruction is thus 4%.

The systematic uncertainties from the branching fractions of CF and
$\pi^{0}/\eta\rightarrow\gamma\gamma$ are 5.1%, 0.034%, and 0.5% pdg ,
respectively.

The systematic uncertainty from the size of the signal MC sample is estimated
to be 0.34% and 0.35% for $\Lambda_{c}^{+}\to p\pi^{0}$ and
$\Lambda_{c}^{+}\to p\eta$ decays, respectively.

The systematic uncertainties are summarized in Table 1 and give in total 7.8%
and 7.4% for $\Lambda_{c}^{+}\rightarrow p\pi^{0}$ and
$\Lambda_{c}^{+}\rightarrow p\eta$, respectively, which are obtained by
assuming all uncertainties are independent and therefore added in quadrature.

Table 1: The sources of the relative systematic uncertainties (%) on the branching fractions of $\Lambda_{c}^{+}\rightarrow p\pi^{0}$ and $\Lambda_{c}^{+}\rightarrow p\eta$ decays. Source | $\Lambda_{c}^{+}\rightarrow p\pi^{0}$ | $\Lambda_{c}^{+}\rightarrow p\eta$
---|---|---
Fit of signal decay | 3.9 | 2.8
PID | 2.0 | 2.0
Tracking efficiency | 0.7 | 0.7
Photon efficiency | 4.0 | 4.0
$\mathcal{B}(\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+})$ | 5.1 | 5.1
$\mathcal{B}(\pi^{0}/\eta\rightarrow\gamma\gamma)$ | 0.0 | 0.5
Statistics of signal MC samples | 0.3 | 0.4
Total | 7.8 | 7.4

##  VI Conclusion

We observe the decay $\Lambda_{c}^{+}\rightarrow p\eta$. A significant
$\Lambda_{c}^{+}$ signal is observed in the invariant mass distribution of
$p\eta$ from data. Using the numbers of the fitted signal events of the
$\Lambda_{c}^{+}\rightarrow p\eta$ and $pK^{-}\pi^{+}$ models and the
reconstruction efficiencies, the measured ratio of
$\frac{\mathcal{B}(\Lambda_{c}^{+}\rightarrow
p\eta)}{\mathcal{B}(\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+})}=(2.258\pm
0.077(\rm stat.)\pm 0.122(\rm syst.))\times 10^{-2}$ is obtained via Eq.(1).
With the independently measured value of
$\mathcal{B}(\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+})$ pdg , we extract a
branching fraction of $\mathcal{B}(\Lambda_{c}^{+}\rightarrow p\eta)=(1.42\pm
0.05(\rm stat.)\pm 0.11(\rm syst.))\times 10^{-3}$, which is consistent with
both the latest published measurement of $(1.24\pm 0.30)\times 10^{-3}$
besResult , but with much improved precision, and theoretical predictions
within 1.3$\sigma$ lamcR10 ; lamcR11 .

We see no obvious signal excess in the distribution of $M(p\pi^{0})$ and so
set an upper limit on the ratio of the branching fractions
$\frac{\mathcal{B}(\Lambda_{c}^{+}\rightarrow
p\pi^{0})}{\mathcal{B}(\Lambda_{c}^{+}\rightarrow pK^{-}\pi^{+})}$ at 90% C.L.
of $1.273\times 10^{-3}$. From this, we extract an upper limit on the
branching fraction of $\mathcal{B}(\Lambda_{c}^{+}\rightarrow
p\pi^{0})<8.0\times 10^{-5}$ at 90% C.L., more than three times more stringent
than the best current upper limit of $2.7\times 10^{-4}$ besResult . The
measured $\mathcal{B}(\Lambda_{c}^{+}\rightarrow p\eta)$ is at least an order
of magnitude larger than $\mathcal{B}(\Lambda_{c}^{+}\rightarrow p\pi^{0})$,
which is consistent with the theoretical prediction of internal $W$-emission
mechanism involving an $s$ quark in $\Lambda_{c}^{+}\rightarrow p\eta$ lamcR10
.

##  VII ACKNOWLEDGMENTS

We thank the KEKB group for the excellent operation of the accelerator; the
KEK cryogenics group for the efficient operation of the solenoid; and the KEK
computer group, and the Pacific Northwest National Laboratory (PNNL)
Environmental Molecular Sciences Laboratory (EMSL) computing group for strong
computing support; and the National Institute of Informatics, and Science
Information NETwork 5 (SINET5) for valuable network support. We acknowledge
support from the Ministry of Education, Culture, Sports, Science, and
Technology (MEXT) of Japan, the Japan Society for the Promotion of Science
(JSPS), and the Tau-Lepton Physics Research Center of Nagoya University; the
Australian Research Council including grants DP180102629, DP170102389,
DP170102204, DP150103061, FT130100303; Austrian Science Fund (FWF); the
National Natural Science Foundation of China under Contracts No. 11435013, No.
11475187, No. 11521505, No. 11575017, No. 11675166, No. 11705209; No.
11761141009; No. 11975076; No. 12042509; Key Research Program of Frontier
Sciences, Chinese Academy of Sciences (CAS), Grant No. QYZDJ-SSW-SLH011; the
CAS Center for Excellence in Particle Physics (CCEPP); the Shanghai Pujiang
Program under Grant No. 18PJ1401000; the Ministry of Education, Youth and
Sports of the Czech Republic under Contract No. LTT17020; the Carl Zeiss
Foundation, the Deutsche Forschungsgemeinschaft, the Excellence Cluster
Universe, and the VolkswagenStiftung; the Department of Science and Technology
of India; the Istituto Nazionale di Fisica Nucleare of Italy; National
Research Foundation (NRF) of Korea Grant Nos. 2016R1D1A1B01010135,
2016R1D1A1B02012900, 2018R1A2B3003643, 2018R1A6A1A06024970,
2018R1D1A1B07047294, 2019K1A3A7A09033840, 2019R1I1A3A01058933; Radiation
Science Research Institute, Foreign Large-size Research Facility Application
Supporting project, the Global Science Experimental Data Hub Center of the
Korea Institute of Science and Technology Information and KREONET/GLORIAD; the
Polish Ministry of Science and Higher Education and the National Science
Center; the Ministry of Science and Higher Education of the Russian
Federation, Agreement 14.W03.31.0026; University of Tabuk research grants
S-1440-0321, S-0256-1438, and S-0280-1439 (Saudi Arabia); the Slovenian
Research Agency; Ikerbasque, Basque Foundation for Science, Spain; the Swiss
National Science Foundation; the Ministry of Education and the Ministry of
Science and Technology of Taiwan; and the United States Department of Energy
and the National Science Foundation.

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# MaxCUCL: Max$-$Consensus with Deterministic Convergence in Networks with
Unreliable Communication Links

Apostolos I. Rikos<EMAIL_ADDRESS>Themistoklis Charalambous
<EMAIL_ADDRESS>Karl H. Johansson<EMAIL_ADDRESS>Department
of Electrical and Computer Engineering, Division of Systems Engineering,
Boston University, Boston, US.  Department of Electrical and Computer
Engineering, University of Cyprus, Nicosia, Cyprus.  Division of Decision and
Control Systems, KTH Royal Institute of Technology,
and also affiliated with Digital Futures, SE-100 44 Stockholm, Sweden.

###### Abstract

In this paper, we present a novel distributed algorithm (herein called
MaxCUCL) designed to guarantee that $\max-$consensus is reached in networks
characterized by unreliable communication links (i.e., links suffering from
packet drops). Our proposed algorithm is the first algorithm that achieves
$\max-$consensus in a deterministic manner (i.e., nodes always calculate the
maximum of their states regardless of the nature of the probability
distribution of the packet drops). Furthermore, it allows nodes to determine
whether convergence has been achieved (enabling them to transition to
subsequent tasks). The operation of MaxCUCL relies on the deployment of
narrowband error-free feedback channels used for acknowledging whether a
packet transmission between nodes was successful. We analyze the operation of
our algorithm and show that it converges after a finite number of time steps.
Finally, we demonstrate our algorithm’s effectiveness and practical
applicability by applying it to a sensor network deployed for environmental
monitoring.

###### keywords:

Distributed algorithms, $\max$-consensus, unreliable communication links,
packet drops, narrowband feedback channels, deterministic convergence,
environmental monitoring.

††thanks: Corresponding author Apostolos I. Rikos.

, ,

##  1 Introduction

The relentless surge of interconnectivity in the modern world has led to a
growing need for effective control and coordination within networks of
multiple agents, such as groups of sensors [1] and mobile autonomous agents
[2]. A problem of particular interest in distributed control and coordination
is the consensus problem [3, 4, 5]. The objective of the consensus problem is
to develop distributed algorithms that can be used by a group of agents in
order to reach agreement to a common decision (or state). The agents engage in
the process to reach consensus with different initial states/data and
communicate locally via information exchange with neighboring agents with
which communication can be established. The concept of $\max-$consensus refers
to the process of achieving consensus to the maximum state among a set of
states (or data) held by different nodes in a network. Knowledge of the
maximum value of a state is of paramount importance for various applications
such as leader election [6], clock and data synchronization [7, 8],
termination mechanisms [9], distributed clustering [10], fault detection [11],
and resource allocation [12, 13, 14].

Related Work. The problem of $\max-$consensus has received significant
attention in the literature. Asynchronous operation has been a focal point in
several studies [15, 16, 14]. In [15] randomized algorithms with asynchronous
operations were explored, demonstrating convergence within finite time with
high probability. Furthermore, in [16], algorithms operating asynchronously
with transmissions affected by time delays were presented. Additionally, in
[14], asynchronous $\max-$consensus algorithms leveraging quantized
communication were proposed. Conversely, synchronous operation took precedence
in other works [17, 18, 19]. For example in [17] nodes operated synchronously
and achieved $\max-$consensus within finite time over static directed graphs.
In [18] scenarios involving synchronous algorithms with transmission delays
were examined to detect convergence of the underlying protocol. Furthermore,
in [19] max-plus algebra was employed to analyze the convergence rate of
synchronous $\max-$consensus algorithms for time-invariant networks. The
dynamics of time-varying networks were analyzed in [20, 21]. In [20], a family
of consensus algorithms was introduced for time-varying networks,
demonstrating connections to Bellman-Ford and mean hitting time iterations
under certain modifications. In a related manner, [21] focused on
$\max-$consensus over time-varying and state-dependent graphs, establishing
necessary and sufficient conditions for convergence. Unreliable networks were
the main focus in [22, 23, 24]. Specifically, [24] explored cases where edges
were absent with non-zero probabilities independently across edges and time,
[22] calculated tighter convergence bounds for two algorithms achieving max
consensus in asynchronous networks, and [23] considered scenarios where
messages were dropped based on non-zero probabilities. Additionally, prior
approaches on the $\max-$consensus problem considered also a diversity of
alternative perspectives (which however fall outside the specific scope of
this paper). For instance, in [25], the focus was on scenarios where the size
of the communication network underwent dynamic changes. In [26], two distinct
algorithms were proposed to address the tracking of time-varying maximum
states within networks, addressing scenarios where the maximum state evolves
over time. The work in [27] delved into the domain of privacy preservation
during maximum state calculations. Furthermore, the effects of additive
communication noise were studied in [24, 28], and max-plus algebra was
employed as a tool for analyzing the ergodic process.

Motivation. All aforementioned works typically offer probabilistic convergence
guarantees to the maximum state (lacking deterministic convergence guarantees)
or their convergence relies on specific conditions regarding network
disturbances. Hence, the challenge of achieving deterministic convergence to
the maximum state in networks with unreliable communication links (i.e.,
packet losses) remains an open question. Furthermore, in most works, nodes do
not possess a distributed mechanism to identify whether convergence to the
maximum state has been reached, thus limiting their ability to proceed with
subsequent operations.

Main Contributions. In this paper, we present the first distributed algorithm
(called MaxCUCL) addressing the challenge of achieving $\max-$consensus in a
deterministic manner in networks with unreliable communication links (see
Algorithm 1). More specifically, by leveraging the concept of narrowband
error-free feedback channels, we show that MaxCUCL not only enables nodes to
deterministically compute the maximum state in finite time, but also allows
them to distributively determine whether to terminate their operation despite
the presence of unreliable links (see Theorem 4.2). Finally, we present a
motivating application where we demonstrate the algorithm’s effectiveness and
practicality by applying it to a sensor network deployed for environmental
monitoring (see Section 5).

##  2 Preliminaries and Problem Formulation

Notation. The sets of real, rational, integer, and natural numbers are denoted
by $\mathds{R},\mathds{Q},\mathds{Z}$, and $\mathds{N}$, respectively. The
symbol $\mathds{Z}_{+}$ denotes the set of nonnegative integers and the symbol
$\mathds{N}_{0}$ denotes the set of natural numbers that also includes zero.
The $div$ operation represents the division of one number by another,
returning the whole number quotient. For example, $\alpha\ div\ \beta$ denotes
the division of the dividend $\alpha$ by the divisor $\beta$ resulting in the
largest whole number quotient. The $mod$ operation represents the modulo or
remainder when one number is divided by another. For example, in the
expression $\alpha\ mod\ \beta$ the dividend $\alpha$ is divided by the
divisor $\beta$, and the mod operation returns the remainder. The $max$
operation, also known as the maximum operation, returns the greater of two
given values. It can be represented as $max(\alpha,\beta)$ where $\alpha$ and
$\beta$ are the values being compared. The expected value of a random variable
$x$ is denoted as $\mathrm{E}[x]$, and represents the theoretical mean or
average outcome that would be obtained if the random variable were repeatedly
observed. For any real number $a\in\mathds{R}$, the floor $\lfloor a\rfloor$
denotes the greatest integer less than or equal to $a$ while the ceiling
$\lceil a\rceil$ denotes the least integer greater than or equal to $a$.

Communication Network. The communication network consists of $n$ ($n\geq 2$)
nodes communicating only with their immediate neighbors and can be captured by
a static directed graph (digraph). A static digraph is defined as
$\mathcal{G}_{d}=(\mathcal{V},\mathcal{E})$ with $n=|\mathcal{V}|$ nodes and
$m=|\mathcal{E}|$ edges. The set of nodes is defined as
$\mathcal{V}=\\{v_{1},v_{2},\dots,v_{n}\\}$, and the set of edges (self-edges
excluded) is defined as
$\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}-\\{(v_{j},v_{j})\ |\
v_{j}\in\mathcal{V}\\}$. A directed edge from node $v_{i}$ to node $v_{j}$ is
denoted by $(v_{j},v_{i})\in\mathcal{E}$, and captures the fact that node
$v_{j}$ can receive information from node $v_{i}$ (but not the other way
around). The subset of nodes that can directly transmit information to node
$v_{j}$ is called the set of in-neighbors of $v_{j}$ and is represented by
$\mathcal{N}_{j}^{-}=\\{v_{i}\in\mathcal{V}\;|\;(v_{j},v_{i})\in\mathcal{E}\\}$.
The subset of nodes that can directly receive information from node $v_{j}$ is
called the set of out-neighbors of $v_{j}$ and is represented by
$\mathcal{N}_{j}^{+}=\\{v_{l}\in\mathcal{V}\;|\;(v_{l},v_{j})\in\mathcal{E}\\}$.
The cardinality of $\mathcal{N}_{j}^{-}$ is called the in-degree of $v_{j}$
and is denoted by $\mathcal{D}_{j}^{-}$ (i.e.,
$\mathcal{D}_{j}^{-}=|\mathcal{N}_{j}^{-}|$). The cardinality of
$\mathcal{N}_{j}^{+}$ is called the out-degree of $v_{j}$ and is denoted by
$\mathcal{D}_{j}^{+}$ (i.e., $\mathcal{D}_{j}^{+}=|\mathcal{N}_{j}^{+}|$). The
diameter $D$ of a digraph is defined as the longest shortest path between any
two nodes $v_{j},v_{i}\in\mathcal{V}$ in the network.

###  2.1 System Model for Packet Dropping Links and Feedback Channels

In many real-world scenarios, distributed systems and networks often operate
in environments characterized by unreliable communication links [29, 30].
Understanding and addressing the challenges posed by such unreliable networks
is of utmost importance for ensuring the robustness and efficiency of
distributed algorithms.

Modelling Packet Dropping Links. In our setting, each communication link in
the distributed network is unreliable (i.e., it is subject to packet drops or
losses). Packet drops can occur due to network congestion, channel
impairments, or other factors. The probability of a packet being dropped (or
lost) at link $(v_{j},v_{i})$ is denoted by $q_{ji}$ (where $0\leq q_{ji}<1$),
and is assumed to be independent and identically distributed across different
links and time instances. The occurrence of packet drops is modeled by
utilizing a Bernoulli random variable. Specifically, we define $x_{k}(j,i)$ as
an indicator variable ($x_{k}(j,i)=1$ indicates a successful transmission from
node $v_{i}$ to node $v_{j}$ at time step $k$). The probability distribution
of $x_{k}(j,i)$ is given by

$Pr\\{x_{k}(j,i)=m\\}=\left\\{\begin{array}[]{ll}q_{ji},&\mbox{if $m=0$,}\\\
1-q_{ji},&\mbox{if $m=1$,}\end{array}\right.$ (1)

where $q_{ji}$ represents the probability of a packet being dropped at the
link $(v_{j},v_{i})$, and $1-q_{ji}$ represents the probability of a
successful transmission.

Modelling Feedback Channels. In various message transmission protocols such as
Transmission Control Protocol (TCP) and High-Level Data Link protocol, a
mechanism called Automatic Repeat reQuest (ARQ) is utilized [31, 32]. ARQ
serves as an error control protocol, ensuring reliable packet transmissions
over unreliable communication channels. The operation of ARQ relies on a
feedback signal transmitted from the data receiver to the data transmitter.
The feedback signal indicates whether a packet has been successfully received
or not. This feedback signal is transmitted via a narrowband feedback channel,
for which we make the following assumption.

######  Assumption 1.

For each directed link $(v_{j},v_{i})\in\mathcal{E}$ from node $v_{i}$ to node
$v_{j}$, there exists a narrowband error-free feedback channel from node
$v_{j}$ to node $v_{i}$.

The feedback channel introduced in Assumption 1 usually comprises messages of
$1$ bit (i.e., it is not used for data transmission). For this reason, it is
reasonable to assume that it is error-free [33]. Such a feedback mechanism is
necessary for acknowledging whether a packet has been received or not. In our
setting, the feedback mechanism is also utilized to notify each node in the
network whether the operation of the algorithm could be terminated or continue
to be executed.

###  2.2 Node Operation

At time step $k$, each node $v_{j}\in\mathcal{V}$ maintains the following
variables: $x_{j}[k]$ is the node’s state; $x^{old}_{j}$ is used for checking
whether the node’s state was updated during a set of $\max-$consensus
operations; $\text{dflag}_{j}$ is a flag that is used to notify nodes in the
network whether the operation of the algorithm is terminated or needs to be
continued; $R_{ji}$ is used from node $v_{j}$ to mark its incoming links from
which it received information at least once.

###  2.3 Problem Formulation

Consider a digraph $\mathcal{G}_{d}=(\mathcal{V},\mathcal{E})$. Each node
$v_{j}\in\mathcal{V}$ has an initial state $x_{j}[0]\in\mathds{R}$ and $q_{m}$
is the maximum state among the initial states defined as

$q_{m}=\max_{v_{j}\in\mathcal{V}}\\{x_{j}[0]\\}.$ (2)

Furthermore, let us assume that each communication link in the digraph
$\mathcal{G}_{d}$ is subject to packet drops (see Section 2.1).

In this paper, we aim to develop a distributed algorithm that allows nodes to
(i) calculate $q_{m}$ in a finite number of time steps and, (ii) terminate
their operation _simultaneously_ after they calculate $q_{m}$. More
specifically, we require that there exists a time step $k_{0}$ so that for
every $v_{j}\in\mathcal{V}$ we have

$\displaystyle x_{j}[k_{0}]=q_{m},$ (3a) $\displaystyle\text{nodes terminate
operation at time step}\ k_{0},$ (3b)

where $q_{m}$ is defined in (2).

##  3 Main Results

In this section, we present our MaxCUCL algorithm that solves the problem
described in Section 2.3. Before presenting our algorithm, we consider the
following assumptions which are important for the development of the results
in this paper.

######  Assumption 2.

We assume that the digraph $\mathcal{G}_{d}$ is strongly connected.

######  Assumption 3.

We assume that every node $v_{j}$ knows the diameter $D$ of the digraph
$\mathcal{G}_{d}$, or an upper bound on the diameter denoted as $D^{\prime}$
(i.e., $D^{\prime}\geq D$).

Assumption 2 means that for each pair of nodes $v_{j},v_{i}\in\mathcal{V}$,
$v_{j}\neq v_{i}$, there exists a directed path from $v_{i}$ to $v_{j}$, i.e.,
information from every node can reach any other node in the network.
Assumption 3 is important for coordinating the actions of nodes in the
network, and notifying them whether convergence has been achieved. Note that
the $\max-$consensus has been extensively used in the literature as part of
other processes [34]. Therefore, the need to know the network diameter $D$ is
for completing the current procedure and moving on to the next process.

The operation of MaxCUCL is based on two phases: Phase1 and Phase2. Phase1 is
executed in Iteration-step $1$ and lasts $D$ time steps. At the beginning of
this phase, each node $v_{j}$ marks each incoming link with the value one.
Then, for $D$ time steps, each node $v_{j}$ executes the $\max-$consensus
protocol. During the execution of $\max-$consensus, if node $v_{j}$ received
at least one message from in-neighbor $v_{i}\in\mathcal{N}^{-}_{j}$, it marks
the link $(v_{j},v_{i})$ with the value zero. Phase2 is executed in Iteration-
step $2$ and also lasts $D$ time steps. At the beginning of this phase, each
node $v_{j}$ checks the following:

  * _i)_

if during Phase1, there is at least one incoming link from which it did not
receive any message (i.e., there is at least one incoming link marked with
value one). Then, it sets its $\text{dflag}_{j}$ equal to one.

  * _ii)_

if during Phase1, its value was updated (i.e., increased), then it sets its
$\text{dflag}_{j}$ equal to one.

Then, for $D$ time steps it transmits its $\text{dflag}_{j}$ to its in-
neighbors. It receives the dflags of its out-neighbors and updates its own
$\text{dflag}_{j}$ to be equal to the maximum value between the stored and the
received dflags. Finally, at the end of Phase2 (i.e., after $D$ time steps),
if its $\text{dflag}_{j}$ is equal to zero, it terminates its operation.
Otherwise, if its $\text{dflag}_{j}$ is equal to one, it goes to Phase1 and
repeats the operation.

The algorithm is described in detail in Algorithm 1.

Algorithm 1 MaxCUCL: $\max-$Consensus for Unreliable Links

Input: Digraph $\mathcal{G}_{d}=(\mathcal{V},\mathcal{E})$ with
$n=|\mathcal{V}|$ nodes and $m=|\mathcal{E}|$ edges. Each node
$v_{j}\in\mathcal{V}$ has an initial state $x_{j}[0]\in\mathds{R}$.
Initialization: Each node $v_{j}\in\mathcal{V}$ sets $x^{old}_{j}=x_{j}[0]$,
$\text{dflag}_{j}=1$, and $R_{ji}=1$ for every $v_{i}\in\mathcal{N}_{j}^{-}$.
Iteration: For $k=0,1,2,\dots$, each node $v_{j}\in\mathcal{V}$, does the
following:

Phase1: if $k\ div\ D=2\alpha$ (where $\alpha\in\mathds{Z}_{+}$) then

  * •

if $k\ mod\ D$ = 0, then if $\text{dflag}_{j}=0$, then terminate operation;

  * •

else:

    * $1a)$

set $\text{dflag}_{j}=0$;

    * $1b)$

broadcast $x_{j}[k]$ to every $v_{l}\in\mathcal{N}_{j}^{+}$;

    * $1c)$

receive $y_{i}[k]$ from every $v_{i}\in\mathcal{N}_{j}^{-}$, and set

$x_{j}[k+1]=\max_{v_{i}\in\mathcal{N}_{j}^{-}}\\{x_{j}[k],w_{ji}[k]x_{i}[k]\\},$
(4)

where $w_{ji}[k]=1$ if node $v_{j}$ receives $y_{i}[k]$ from
$v_{i}\in\mathcal{N}_{j}^{-}$ at iteration $k$ (otherwise $w_{ji}[k]=0$);

    * $1d)$

if $w_{ji}[k]=1$, for in-neighbor $v_{i}\in\mathcal{N}_{j}^{-}$, then it sets
$R_{ji}=0$;

Phase2: if $k\ div\ D=2\alpha+1$ (where $\alpha\in\mathds{Z}_{+}$) then

  * $2a)$

if $R_{ji}=1$ for at least one $v_{i}\in\mathcal{N}_{j}^{-}$ set
$\text{dflag}_{j}=1$;

  * $2b)$

if $x_{j}[k]>x^{old}_{j}$ set $\text{dflag}_{j}=1$;

  * $2c)$

broadcast $\text{dflag}_{j}$ to every in-neighbor
$v_{i}\in\mathcal{N}_{j}^{-}$;

  * $2d)$

receive $\text{dflag}_{l}$ from every out-neighbor
$v_{l}\in\mathcal{N}_{j}^{+}$ and set

$\text{dflag}_{j}=\max_{v_{l}\in\mathcal{N}_{j}^{+}}\\{\text{dflag}_{j},\text{dflag}_{l}\\};$
(5)

  * $2e)$

set $x^{old}_{j}=x_{j}[k]$, and $R_{ji}=1$ for every
$v_{i}\in\mathcal{N}_{j}^{-}$;

  * $2f)$

set $k=k+1$ and go to Phase1.

Output: (3a), (3b) hold for every $v_{j}\in\mathcal{V}$.

We next provide an example to illustrate our algorithm’s operation.

######  Example 3.1.

Consider the strongly connected digraph in Fig. 1. This digraph consists the
set of nodes $\mathcal{V}=\\{v_{1},v_{2},v_{3}\\}$, and the set of edges
$\mathcal{E}=\\{(v_{2},v_{1}),(v_{1},v_{2}),$
$(v_{3},v_{2}),(v_{2},v_{3}),(v_{3},v_{1})\\}$. Nodes $v_{1}$, $v_{2}$,
$v_{3}$ have initial states $x_{1}[0]=5$, $x_{2}[0]=4$, $x_{3}[0]=3$,
respectively. Furthermore, the network digraph is $D=2$, and Assumptions 1, 2,
3, hold. During the initialization of MaxCUCL, each node $v_{j}$ sets
$x^{old}_{j}=x_{j}[0]$, $\text{dflag}_{j}=1$, and $R_{ji}=1$ for every
$v_{i}\in\mathcal{N}_{j}^{-}$.

Figure 1: Example of digraph for execution of MaxCUCL algorithm.

During the iteration of MaxCUCL at Phase1, at time step $k=0$ each node
$v_{j}$ check its $\text{dflag}_{j}$. Since all dflags are equal to one, the
operation is not terminated. Then, each node executes the $\max-$consensus
protocol for $D$ time steps. Let us assume that during $k\in\\{0,1\\}$ time
steps, link $(v_{3},v_{2})$ had two packet drops, while every other link had
successful transmissions. This means that the states of nodes become:
$x_{1}[2]=5$, $x_{2}[2]=5$, $x_{3}[2]=3$. Furthermore, $R_{32}=1$ and
$R_{ji}=0$, for every
$(v_{j},v_{i})\in\mathcal{E}\setminus\\{(v_{3},v_{2})\\}$. At Phase2, at time
step $k=2$, node $v_{3}$ sets $\text{dflag}_{3}=1$ because $R_{32}=1$.
Furthermore, node $v_{2}$ sets $\text{dflag}_{2}=1$ because
$x_{2}[2]>x_{2}^{old}$ (note that $x_{2}^{old}=4$). Then, for time steps
$k=\\{2,3\\}$, nodes will transmit their dflags to their in-neighbors, and all
dflags will become equal to one. Furthermore, each node $v_{j}$ will update
its $x_{j}^{old}$ value and its $R_{ji}$ value, for every
$v_{i}\in\mathcal{N}_{j}^{-}$.

At time step $k=4$, all dflags are equal to $1$. For this reason, every node
will repeat Phase1 (i.e., the operation is not terminated). MaxCUCL is not
terminated, because during time steps $k\in\\{0,1\\}$ (i.e., during Phase1)
(i) at least one link did not have any successful transmission (i.e., link
($v_{3},v_{2}$)), and (ii) at least one node’s state was increased (i.e., node
$v_{2}$).

During time steps $k\in\\{4,5\\}$ of Phase1, let us assume that every link had
successful transmissions. This means that the states of nodes become:
$x_{1}[6]=5$, $x_{2}[6]=5$, $x_{3}[6]=5$. Furthermore, $R_{ji}=0$ for every
$(v_{j},v_{i})\in\mathcal{E}$. At Phase2, at time step $k=6$, node $v_{3}$
sets $\text{dflag}_{3}=1$ because $x_{3}[6]>x_{3}^{old}$ (note that
$x_{3}^{old}=3$). Then, for time steps $k=\\{6,7\\}$, nodes will transmit
their dflags to their in-neighbors, and all dflags will become equal to one.
Furthermore, each node $v_{j}$ will update its $x_{j}^{old}$ value and its
$R_{ji}$ value, for every $v_{i}\in\mathcal{N}_{j}^{-}$.

At time step $k=8$, all dflags are equal to one, and every node will repeat
again Phase1. Note that MaxCUCL is not terminated, because during time steps
$k\in\\{4,5\\}$ (i.e., during Phase1) at least one node’s state was increased
(i.e., node $v_{3}$).

During time steps $k\in\\{8,9\\}$, of Phase1, let us assume that every link
had successful transmissions. This means that the states of nodes are:
$x_{1}[10]=5$, $x_{2}[10]=5$, $x_{3}[10]=5$. Furthermore, $R_{ji}=0$, for
every $(v_{j},v_{i})\in\mathcal{E}$. At Phase2, at time step $k=10$, we have
that $\text{dflag}_{j}=0$, for every $v_{j}\in\mathcal{V}$ (since for every
$v_{j}$, we have $x_{j}[10]=x_{j}^{old}$, and $R_{ji}=0$ for every
$v_{i}\in\mathcal{N}_{j}^{-}$). Then, for time steps $k=\\{10,11\\}$, nodes
will transmit their dflags to their in-neighbors. In this case, we have that
all dflags will remain equal to zero. Furthermore, each node $v_{j}$ will
update its $x_{j}^{old}$ value and its $R_{ji}$ value, for every
$v_{i}\in\mathcal{N}_{j}^{-}$.

Finally, at time step $k=12$ every node will check its dflag. Note that all
dflags are equal to zero because during time steps $k\in\\{8,9\\}$ (i.e.,
during Phase1) (i) every link had at least one successful transmission, and
(ii) the state of every node did not increase. For this reason, the operation
of MaxCUCL will be terminated. The states of the nodes during the algorithm’s
operation are shown in Fig. 2.

Figure 2: Node states plotted against the number of iterations during
execution of MaxCUCL for the digraph shown in Fig. 1.

##  4 Convergence Analysis

We now analyze the convergence of MaxCUCL algorithm. We first introduce a
theorem where we show that during the algorithm’s operation there exists a
time step $k_{\mathrm{terminal}}$ for which (3a), (3b) hold for every node
$v_{j}$ after a finite number of time steps. Then, we present a proposition we
calculate an explicit upper bound on time step $k_{\mathrm{terminal}}$.

######  Theorem 4.2.

Consider a strongly connected digraph
$\mathcal{G}_{d}=(\mathcal{V},\mathcal{E})$ with $n=|\mathcal{V}|$ nodes and
$m=|\mathcal{E}|$ edges. Each node $v_{j}\in\mathcal{V}$ has an initial state
$x_{j}[0]\in\mathds{R}$ and Assumptions 1, 2, and 3 hold. During the execution
of MaxCUCL algorithm (i.e., Algorithm 1), there exists a time step
$k_{\mathrm{terminal}}\in\mathds{N}$ for which (3a) and (3b) hold for every
node $v_{j}\in\mathcal{V}$. This information is known to every node
$v_{j}\in\mathcal{V}$, allowing them to terminate the algorithm in a
distributed fashion.

To enhance the clarity of the proof, we have organized it into three parts;
namely, Part1, Part2, and Part3. In Part1, we establish that our algorithm
converges for the simple case where every transmission from every link is
successful (i.e., from Section 2.1 we have $q_{ji}=0$, for every
$(v_{j},v_{i})\in\mathcal{E}$). Also, in Part1 we highlight the main
differences from the standard $\max-$consensus algorithm. In Part2, we show
that our algorithm converges to the correct maximum for the case where one
link has packet drops for $D$ consecutive time steps only (i.e., for every
time step of one execution of Phase1), while every other transmission from
that particular link (excluding these $D$ consecutive time steps) and every
other link is successful. In Part3 (which is the main argument for the proof),
we extend to the general case in which every link suffers from packet drops
(i.e., from Section 2.1 we have $0<q_{ji}<1$ for
$(v_{j},v_{i})\in\mathcal{E}$) and we show that our algorithm converges to the
maximum.

Part1: During the execution of MaxCUCL, each node performs $\max-$consensus
for time steps $k=0,...,D-1$. This ensures that by time step $D$, the state of
every node satisfies (3a) (i.e., $x_{j}[D]=q_{m}$). During time step $k=D$, at
least one node $v_{j}$ sets $\text{dflag}_{j}=1$ (because its state was
increased). Therefore, during time steps $k=D,...,2D-1$, every node $v_{j}$
sets $\text{dflag}_{j}=1$ (see Iteration Steps $2c$ and $2d$). At time step
$k=2D$, every node $v_{j}$ has $\text{dflag}_{j}=1$. For this reason, each
node performs $\max-$consensus for an additional $D$ time steps (i.e., for
time steps $k=2D,...,3D-1$). Following this, during time steps
$k=3D,...,4D-1$, each node $v_{j}$ sets $\text{dflag}_{j}=0$ (note that since
by time step $D$, the state of every node satisfies (3a), then no node
increased its state during time steps $k=2D,...,3D-1$). Finally, at time step
$4D$, each node $v_{j}$ checks its $\text{dflag}_{j}$. Since
$\text{dflag}_{j}=0$, then every node terminates its operation. Thus, for
Part1, our algorithm converges to the maximum after $4D$ time steps.

Discussion of Part1: Compared to the $\max-$consensus algorithm (see e.g.,
[17]), the MaxCUCL algorithm is initially executed for $2D$ time steps. Then,
it is executed for an additional number of $2D$ time steps every time (i) a
node updates its state, and/or (ii) a packet from a link is dropped for $D$
consecutive time steps (as it will be shown in Part2 below). To this regard,
in Part1 nodes terminated their operation at time step $4D$ since no node
updated its state and no packet was dropped during time steps $k=2D,...,3D-1$.

Part2: Without loss of generality, let us assume that link $(v_{l},v_{j})$
drops packets at time steps $2D,...,3D-1$. Furthermore, every other
transmission from every link is successful. During the operation of MaxCUCL,
every node will execute $\max-$consensus for time steps $k=0,...,D-1$. At time
step $D$, the state of every node fulfills (3a). During time step $k=D$, at
least one node $v_{j}$ sets $\text{dflag}_{j}=1$ (because its state was
increased). Therefore, during time steps $k=D,...,2D-1$, every node $v_{j}$
sets $\text{dflag}_{j}=1$. For this reason, every node will execute
$\max-$consensus for an additional number of $D$ time steps (i.e., for time
steps $k=2D,...,3D-1$). Note however that at time steps $2D,...,3D-1$ every
packet at link $(v_{l},v_{j})$ is dropped. This means that at time step
$k=3D$, node $v_{l}$ sets $\text{dflag}_{l}=1$, and every node
$v_{j}\in\mathcal{V}\setminus\\{v_{l}\\}$ sets $\text{dflag}_{j}=0$.
Consequently, for time steps $k=3D,...,4D-1$ every node $v_{j}\in\mathcal{V}$
will set $\text{dflag}_{j}=1$ (see Iteration Steps $2c$, $2d$). Since
$\text{dflag}_{j}=1$ for every $v_{j}$, every node will execute
$\max-$consensus for time steps $k=4D,...,5D-1$. Then, for time steps
$k=5D,...,6D-1$ every node $v_{j}$ will set $\text{dflag}_{j}=0$ (because
during time steps $k=4D,...,5D-1$, every transmission from every link was
successful, and no node increased its state). At time step $6D$, since
$\text{dflag}_{j}=0$ for every $v_{j}$, then every node will terminate its
operation. Therefore, for Part2, our algorithm will converge to the maximum
after $6D$ time steps.

Discussion of Part2: In Part2, the presence of $D$ consecutive packet drops at
link $(v_{l},v_{j})$ during time steps $2D,...,3D-1$ resulted in MaxCUCL being
executed for an extra $2D$ time steps compared to Part1. This additional
execution was necessary to account for the packet drops that occurred during
time steps $2D,...,3D-1$, even though $\max-$consensus had already been
reached at time step $D$.

Part3: In this part, suppose that there does not exist any node
$v_{l}\in\mathcal{N}$ that is not receiving any packet within $D$ consecutive
time steps from any of its in-neighbors $v_{j}\in\mathcal{N}^{-}$ (i.e., at
least on of the in-neighboring nodes will send a packet within $D$ steps).
Then, the only condition for the node $v_{l}$ to set its $\text{dflag}_{l}=1$
is if it changes its value. Notably, the union of all the graphs formed at
each of the $D$ time steps is a strongly connected graph, and as a result, if
a node has a different value than the other nodes, a dflag will be set to one
at the end of Phase2. Therefore, the nodes in the network should continue
executing MaxCUCL until they ensure that they calculate the maximum of their
states. ∎

In Theorem 4.2 we showed that our algorithm converges in a deterministic
fashion. In the following proposition, we establish a probabilistic bound on
the number of time steps needed for every node to calculate the maximum state
in the network.

######  Proposition 4.

The termination step $k_{\mathrm{terminal}}$ for MaxCUCL (i.e., Algorithm 1),
in which every node $v_{j}\in\mathcal{}V$ satisfies (3a), and (3b), is given
by

$\displaystyle k_{\mathrm{terminal}}=2D(k_{0}+k_{1}),$ (6)

with probability at least $1-\varepsilon$, where $D$ is the diameter of the
network,

$\displaystyle
k_{0}\geq\Bigl{\lceil}\frac{\log(1/\varepsilon_{1})}{\log(1/q_{\text{max}})}\Bigr{\rceil},$
(7)

and

$\displaystyle
k_{1}\geq\Bigl{\lceil}\frac{\log(1/\varepsilon_{2})}{\log(1/p_{1})}\Bigr{\rceil},$
(8)

and $\varepsilon=1-(1-\varepsilon_{1})^{D}(1-\varepsilon_{2})$, where
$\varepsilon_{1},\varepsilon_{2}>0$.

In Part3 of Theorem 4.2, we showed that MaxCUCL converges to the maximum state
after a finite number of time steps in a deterministic fashion. However, it is
possible to calculate an upper bound on the time steps required for achieving
convergence with high probability.

Let us consider the maximum packet drop probability among all links in the
network defined as
$q_{\text{max}}=\max_{(v_{j},v_{i})\in\mathcal{E}}\\{q_{ji}\\}$. Also, let us
define an epoch as the number of $D$ time steps. We calculate a time step
$k_{0}$ for which every link $(v_{j},v_{i})\in\mathcal{E}$ will have performed
at least one successful transmission with probability almost equal to one. To
achieve this, we solve the inequality

$1-q_{\text{max}}^{k_{0}}\geq 1-\varepsilon_{1},$ (9)

where $\varepsilon_{1}$ represents a probability almost equal to zero. In (9)
we solve for $k_{0}$ and take the ceiling of the result (because time steps
are discrete values). Thus, we obtain

$k_{0}\geq\Bigl{\lceil}\frac{\log(1/\varepsilon_{1})}{\log(1/q_{\text{max}})}\Bigr{\rceil}.$
(10)

Eq. (10) represents the time step $k_{0}$ at which every link will have
performed at least one successful transmission with probability
$(1-\varepsilon_{1})$ almost equal to one. Let us suppose that we execute
Phase1 of MaxCUCL for $Dk_{0}$ time steps ($k_{0}$ fulfills (10)). This means
that each link has at least one successful transmission during each one of the
$D$ epochs with probability $(1-\varepsilon_{1})^{D}$. Therefore, by setting
$\varepsilon_{1}$ small enough, we have that after $Dk_{0}$ time steps during
Phase1 of our algorithm each node fulfills (3a). Continuing the algorithm’s
operation, consider the maximum packet drop probability for $D$ consecutive
time steps defined as $q_{\text{max}}^{D}$. The probability $p_{1}$ that at
least one link will suffer packet drops for $D$ consecutive time steps is

$p_{1}\leq\sum_{\lambda=1}^{|\mathcal{E}|}\binom{|\mathcal{E}|}{\lambda}q_{\text{max}}^{\lambda
D}(1-q_{\text{max}}^{(|\mathcal{E}|-\lambda)D}),$ (11)

where $|\mathcal{E}|$ is the number of links in the network, $D$ is the
network diameter, and $q_{\text{max}}$ is the maximum packet drop probability
among all links. We now calculate a number of epochs $k_{1}$ for which no link
has $D$ consecutive packet drops. To achieve this we solve the inequality

$1-p_{1}^{k_{1}}\geq 1-\varepsilon_{2},$ (12)

where $\varepsilon_{2}$ is a probability almost equal to zero. Focusing on
(12) we have,

$k_{1}\geq\Bigl{\lceil}\frac{\log(1/\varepsilon_{2})}{\log(1/p_{1})}\Bigr{\rceil}.$
(13)

In (13) we have that after $k_{1}$ epochs of $D$ time steps each, no link will
have $D$ consecutive packet drops with probability at least
$1-\varepsilon_{2}$.

Let us now combine the results in (10) and (13). In (10) we have that after
$Dk_{0}$ time steps during Phase1 each link will do at least one successful
transmission with probability $(1-\varepsilon_{1})^{D}$. Therefore, after
$2Dk_{0}$ time steps during the execution of MaxCUCL each link will do at
least one successful transmission with probability $(1-\varepsilon_{1})^{D}$.
Note that $Dk_{0}$ are the required time steps of Phase1, $2Dk_{0}$ are the
required time steps of both Phase1 and Phase2. Furthermore, in (13) we have
that after $k_{1}$ epochs of $D$ time steps each, no link will have $D$
consecutive packet drops with probability at least $1-\varepsilon_{2}$. This
means that after $2Dk_{1}$ time steps during our algorithm’s operation no link
will have $D$ consecutive packet drops with probability at least
$1-\varepsilon_{2}$. Combining these results, we have that after
$2D(k_{0}+k_{1})$ time steps during our algorithm each node fulfills (3a), and
(3b) with probability at least $(1-\varepsilon_{1})^{D}(1-\varepsilon_{2})$.
This concludes our proof.

##  5 Application: Distributed Environmental Monitoring in Sensor Networks

We now present an application of the MaxCUCL algorithm. Our application
focuses on the scenario where a sensor network is deployed in an outdoor
environment, and sensor nodes are tasked with monitoring various environmental
parameters [35, 36]. These parameters may include temperature, humidity, air
quality, or other relevant factors; in our application, we will focus on
temperature monitoring. Specifically, in the scenario under consideration,
nodes aim to achieve consensus on the maximum observed temperature. This
information is pivotal in deciding whether specific environmental factors
require active control or intervention. Additionally, consensus needs to be
reached in finite time, since in real-world scenarios nodes may be operating
under resource constraints (e.g., battery-powered operation). For this reason
nodes need to conclude their operation successfully, in order to ensure that
the calculated consensus is not only accurate but also usable for subsequent
decision-making processes. Moreover, in realistic outdoor environments,
communication among sensor nodes is susceptible to challenges such as
unreliability and packet drops. These issues can stem from factors like signal
interference, obstacles in the terrain, or limited energy resources for
wireless communication. We will show that MaxCUCL serves as a valuable tool
for extracting critical information from the sensor network’s data, as it
allows nodes to achieve consensus in finite time on the maximum observed
temperature even in the presence of unreliable communication links.

During the environmental monitoring operation each sensor node continuously
measures the temperature of its surroundings, and stores the maximum locally
measured temperature. At the end of each day, sensor nodes execute the MaxCUCL
algorithm in order to calculate the maximum globally measured temperature in
the environment. In our scenario, nodes calculate the maximum globally
measured temperature at the end of a single day. Let us note however, that
nodes can repeat this operation (collecting measurements, storing the maximum
locally measured temperature and then, executing our algorithm to calculate
the maximum globally measured temperature) every single day in order to
calculate the maximum measured temperature for a specific time period (e.g., a
month).

In order to highlight the behavior and advantages of MaxCUCL algorithm, we
focus on the three following cases. In Case $1$, we evaluate our algorithm on
a network for environmental monitoring modeled as a random digraph of $20$
nodes. Each edge of the digraph was generated with probability $0.2$ and
diameter is $D=4$. The packet drop probability for each link
$(v_{j},v_{i})\in\mathcal{E}$ is set to $q_{ji}=0.9$. We plot the state
$x_{j}[k]$ of each node $v_{j}$ and the aggregate sum of $\text{dflag}_{j}$
against the number of executions of Phase1 and Phase2 (note that Phase1 and
Phase2 are executed sequentially, i.e., when Phase1 is executed, Phase2 is
executed immediately afterward). In Case $2$, we analyze how the network
diameter affects the operation of our algorithm. For this reason, we assess
the performance of our algorithm over environmental monitoring networks
modeled as randomly generated digraphs of $50$ nodes with diameters $D$ chosen
from the set $\\{3,5,7\\}$. Each edge of the digraphs was generated with
probability $0.2$, and the packet drop probability for each link
$(v_{j},v_{i})\in\mathcal{E}$ is set to $q_{ji}=0.9$. We plot the distribution
of the number of executions of Phase1 and Phase2 required for our algorithm to
converge. The results are averaged over $20$ executions to ensure robustness.
In Case $3$, we explore the influence of packet drop probabilities on the
convergence of MaxCUCL. We focus on an environmental monitoring network
modeled as a random digraph of $50$ nodes. Each edge of the digraph was
generated with probability $0.2$ and diameter is $D=4$. We execute our
algorithm for $100$ instances for packet drop probabilities set to
$q_{ji}=0.9,0.93,0.96$, and $0.99$ for each $(v_{j},v_{i})\in\mathcal{E}$. We
plot the distribution of the number of executions of Phase1 and Phase2
required for our algorithm to converge.

Figure 3: Execution of MaxCUCL over a random digraph of $20$ nodes with
diameter $D=4$.

Case $1$. In Fig. 3, the top figure shows that each node successfully
calculates the maximum value $q_{m}=9.81$ after $10$ executions of Phase1 and
Phase2, and then terminates its operation. This convergence is also reflected
in the bottom figure, where $\text{dflag}_{j}=0$ for every node $v_{j}$ after
$10$ executions. It is interesting to note that all nodes are able to
determine the maximum state in the network after $6$ executions of Phase1 and
Phase2. However, the presence of potential packet drops in the network extends
the execution of our algorithm. Specifically, after $6$ executions, the
aggregate sum of $\text{dflag}_{j}$ for each node $v_{j}$ is equal to $7$.
This means that for $7$ nodes, at least one incoming link had $D$ consecutive
packet drops during Phase1 (the probability of $D$ consecutive packet drops is
equal to $0.9^{4}$). As a result, an additional $4$ instances of Phase1 and
Phase2 are executed in our algorithm (until the $\text{dflag}_{j}$ for each
node $v_{j}$ becomes equal to $0$).

Figure 4: Required executions of Phase1 and Phase2 for MaxCUCL to converge,
for random digraphs with diameters $D=3,5,7$.

Case $2$. In Fig. 4, for $D=3$ we can see that our algorithm converges after
$17-23$ executions of Phase1 and Phase2 on average. However, it is important
to notice that increasing the diameter to $D=5$, results in a decrease of the
required executions of Phase1 and Phase2 for convergence. Specifically, for
$D=5$ our algorithm converges after $11-13$ executions of Phase1 and Phase2 on
average. This is mainly because the probability of $D$ consecutive packet
drops during Phase1 for $D=5$ decreases to $0.9^{5}$ compared to $0.9^{3}$ for
$D=3$. Finally, increasing the diameter, to $D=7$, further decreases the
required executions of Phase1 and Phase2 for convergence. This occurs because
the probability of $D$ consecutive packet drops during Phase1 further
decreases to $0.9^{7}$ for $D=7$.

Figure 5: Required executions of Phase1 and Phase2 for MaxCUCL to converge,
for $q_{ji}=0.9,0.93,0.96$, and $0.99$ for each $(v_{j},v_{i})\in\mathcal{E}$,
and diameter $D=4$.

Case $3$. In Fig. 5, for $q_{ji}=0.9$ for each link
$(v_{j},v_{i})\in\mathcal{E}$, we have that Phase1 and Phase2 are executed on
average for $14-17$ instances for our algorithm to converge. For
$q_{ji}=0.93$, the number of executions of Phase1 and Phase2 slightly
increases to $20-25$ instances. This is mainly because the probability of $D$
consecutive packet drops during Phase1 for $q_{ji}=0.93$ increases to
$0.93^{4}$ compared to $0.9^{4}$ for $q_{ji}=0.9$. For $q_{ji}=0.96$ the
number of executions of Phase1 further increases to $35-45$ (because the
probability of $D$ consecutive packet drops during Phase1 further increases to
$0.96^{4}$). Finally, for $q_{ji}=0.99$ for each link
$(v_{j},v_{i})\in\mathcal{E}$, the number of executions of Phase1 and Phase2
increases to $130-180$ instances (because $D$ consecutive packet drops during
Phase1 occur with probability $0.99^{4}$).

######  Remark 5.3.

In our analysis, we observed that as we increase the diameter $D$, the number
of executions of Phase1 and Phase2 decreases. Thus, increasing $D$ improves
the algorithm’s convergence. However, let us note that increasing the diameter
$D$ also affects the total number of time steps required for convergence of
our algorithm. As a result, although increasing the diameter $D$ results in
Phase1 and Phase2 being executed fewer times, each individual execution of
Phase1 and Phase2 lasts longer (i.e., because Phase1 and Phase2 are executed
for $D+D=2D$ time steps). This trade-off between the number of executions of
Phase1 and Phase2 and the duration of the execution of MaxCUCL, highlights the
importance of carefully considering the choice of diameter $D$ in applications
of where convergence speed and communication overhead are critical factors.

######  Remark 5.4.

Let us note here that another interesting application of our algorithm is for
the case where nodes aim to calculate the average observed temperature in a
finite time frame, with the added objective of terminating their operations
upon calculation completion. Calculation of the average temperature is
achieved via average consensus algorithms that operate over networks consisted
of unreliable communication links (e.g., see [37] and references therein). Our
proposed algorithm is able to operate in parallel with a robust average
consensus algorithm. This guarantees that nodes will be able to terminate
their operation in a finite time frame and the average calculated temperature
will be utilized for subsequent decision-making processes.

##  6 Conclusions and Future Directions

We introduced the MaxCUCL algorithm for achieving $\max-$consensus in finite
time over networks with unreliable communication links. We showed that our
algorithm enables nodes to calculate the maximum of their states
deterministically by relying on narrowband error-free feedback channels. It
also allows nodes to identify whether convergence has been achieved and
terminate the process. We analyzed the algorithm’s operation and established
its convergence. Finally, we presented a motivating application in sensor
networks for environmental monitoring where we demonstrated the effectiveness
and practical applicability of our proposed algorithm.

Our future research includes extending our algorithm to handle dynamic network
conditions and more complex tasks that could provide valuable insights into
its adaptability and scalability. Finally, developing mechanisms to ensure our
algorithm’s resilience against adversarial attacks and Byzantine faults (where
nodes may provide malicious or incorrect information) is an essential
direction to enhance its robustness and reliability.

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