# Triangle Singularity as the Origin of the $a_{1}(1420)$

G. D. Alexeev  Joint Institute for Nuclear Research, 141980 Dubna, Moscow
region, Russia  M. G. Alexeev  Department of Physics, University of Torino,
10125 Turin, Italy  Torino Section of INFN, 10125 Turin, Italy  A. Amoroso
Department of Physics, University of Torino, 10125 Turin, Italy  Torino
Section of INFN, 10125 Turin, Italy  V. Andrieux  CERN, 1211 Geneva 23,
Switzerland  Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801-3080, USA  V. Anosov  Joint Institute for Nuclear
Research, 141980 Dubna, Moscow region, Russia  A. Antoshkin  Joint Institute
for Nuclear Research, 141980 Dubna, Moscow region, Russia  K. Augsten  Joint
Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  Czech
Technical University in Prague, 16636 Prague, Czech Republic  W. Augustyniak
National Centre for Nuclear Research, 02-093 Warsaw, Poland  C. D. R. Azevedo
Department of Physics, University of Aveiro, I3N, 3810-193 Aveiro, Portugal
B. Badełek  Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland
F. Balestra  Department of Physics, University of Torino, 10125 Turin, Italy
Torino Section of INFN, 10125 Turin, Italy  M. Ball  Helmholtz-Institut für
Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn, Germany  J. Barth
Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn,
Germany  R. Beck  Helmholtz-Institut für Strahlen- und Kernphysik, Universität
Bonn, 53115 Bonn, Germany  Y. Bedfer  IRFU, CEA, Université Paris-Saclay,
91191 Gif-sur-Yvette, France  J. Berenguer Antequera  Department of Physics,
University of Torino, 10125 Turin, Italy  Torino Section of INFN, 10125 Turin,
Italy  J. Bernhard  Institut für Kernphysik, Universität Mainz, 55099 Mainz,
Germany  CERN, 1211 Geneva 23, Switzerland  M. Bodlak  Faculty of Mathematics
and Physics, Charles University, 18000 Prague, Czech Republic  F. Bradamante
Trieste Section of INFN, 34127 Trieste, Italy  A. Bressan  Department of
Physics, University of Trieste, 34127 Trieste, Italy  Trieste Section of INFN,
34127 Trieste, Italy  V. E. Burtsev  Tomsk Polytechnic University, 634050
Tomsk, Russia  W.-C. Chang  Academia Sinica, Institute of Physics, Taipei
11529, Taiwan  C. Chatterjee  Department of Physics, University of Trieste,
34127 Trieste, Italy  Trieste Section of INFN, 34127 Trieste, Italy  M.
Chiosso  Department of Physics, University of Torino, 10125 Turin, Italy
Torino Section of INFN, 10125 Turin, Italy  A. G. Chumakov  Tomsk Polytechnic
University, 634050 Tomsk, Russia  S.-U. Chung  Also at: Department of Physics,
Pusan National University, Busan 609-735, Republic of Korea and Physics
Department, Brookhaven National Laboratory, Upton, NY 11973, USA.  Physik
Department, Technische Universität München, 85748 Garching, Germany  A.
Cicuttin  Also at: Abdus Salam ICTP, 34151 Trieste, Italy.  Trieste Section of
INFN, 34127 Trieste, Italy  P. M. M. Correia  Department of Physics,
University of Aveiro, I3N, 3810-193 Aveiro, Portugal  M. L. Crespo  Also at:
Abdus Salam ICTP, 34151 Trieste, Italy.  Trieste Section of INFN, 34127
Trieste, Italy  D. D’Ago  Department of Physics, University of Trieste, 34127
Trieste, Italy  Trieste Section of INFN, 34127 Trieste, Italy  S. Dalla Torre
Trieste Section of INFN, 34127 Trieste, Italy  S. S. Dasgupta  Matrivani
Institute of Experimental Research & Education, Calcutta-700 030, India  S.
Dasgupta  Trieste Section of INFN, 34127 Trieste, Italy  I. Denisenko  Joint
Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  O. Yu.
Denisov  Torino Section of INFN, 10125 Turin, Italy  S. V. Donskov  State
Scientific Center Institute for High Energy Physics of National Research
Center ”Kurchatov Institute”, 142281 Protvino, Russia  N. Doshita  Yamagata
University, Yamagata 992-8510, Japan  Ch. Dreisbach  Physik Department,
Technische Universität München, 85748 Garching, Germany  W. Dünnweber  Retired
from Ludwig-Maximilian-Universität, München, Germany.  R. R. Dusaev  Tomsk
Polytechnic University, 634050 Tomsk, Russia  A. Efremov  Deceased.  Joint
Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  P. D.
Eversheim  Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn,
53115 Bonn, Germany  P. Faccioli  LIP, 1649-003 Lisbon, Portugal  M. Faessler
Retired from Ludwig-Maximilian-Universität, München, Germany.  M. Finger
Faculty of Mathematics and Physics, Charles University, 18000 Prague, Czech
Republic  M. Finger Jr  Faculty of Mathematics and Physics, Charles
University, 18000 Prague, Czech Republic  H. Fischer  Physikalisches Institut,
Universität Freiburg, 79104 Freiburg, Germany  C. Franco  LIP, 1649-003
Lisbon, Portugal  J. M. Friedrich  Physik Department, Technische Universität
München, 85748 Garching, Germany  V. Frolov  Joint Institute for Nuclear
Research, 141980 Dubna, Moscow region, Russia  CERN, 1211 Geneva 23,
Switzerland  F. Gautheron  Institut für Experimentalphysik, Universität
Bochum, 44780 Bochum, Germany  Department of Physics, University of Illinois
at Urbana-Champaign, Urbana, IL 61801-3080, USA  O. P. Gavrichtchouk  Joint
Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  S.
Gerassimov  Lebedev Physical Institute, 119991 Moscow, Russia  Physik
Department, Technische Universität München, 85748 Garching, Germany  J. Giarra
Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany  I. Gnesi
Department of Physics, University of Torino, 10125 Turin, Italy  Torino
Section of INFN, 10125 Turin, Italy  M. Gorzellik  Physikalisches Institut,
Universität Freiburg, 79104 Freiburg, Germany  A. Grasso  Department of
Physics, University of Torino, 10125 Turin, Italy  Torino Section of INFN,
10125 Turin, Italy  A. Gridin  Joint Institute for Nuclear Research, 141980
Dubna, Moscow region, Russia  M. Grosse Perdekamp  Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA  B.
Grube  Physik Department, Technische Universität München, 85748 Garching,
Germany  A. Guskov  Joint Institute for Nuclear Research, 141980 Dubna, Moscow
region, Russia  D. von Harrach  Institut für Kernphysik, Universität Mainz,
55099 Mainz, Germany  R. Heitz  Department of Physics, University of Illinois
at Urbana-Champaign, Urbana, IL 61801-3080, USA  F. Herrmann  Physikalisches
Institut, Universität Freiburg, 79104 Freiburg, Germany  N. Horikawa  Also at:
Chubu University, Kasugai, Aichi 487-8501, Japan.  Nagoya University, 464
Nagoya, Japan  N. d’Hose  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-
Yvette, France  C.-Y. Hsieh  Also at: Department of Physics, National Central
University, 300 Jhongda Road, Jhongli 32001, Taiwan.  Academia Sinica,
Institute of Physics, Taipei 11529, Taiwan  S. Huber  Physik Department,
Technische Universität München, 85748 Garching, Germany  S. Ishimoto  Also at:
KEK, 1-1 Oho, Tsukuba, Ibaraki 305-0801, Japan.  Yamagata University, Yamagata
992-8510, Japan  A. Ivanov  Joint Institute for Nuclear Research, 141980
Dubna, Moscow region, Russia  T. Iwata  Yamagata University, Yamagata
992-8510, Japan  M. Jandek  Czech Technical University in Prague, 16636
Prague, Czech Republic  V. Jary  Czech Technical University in Prague, 16636
Prague, Czech Republic  R. Joosten  Helmholtz-Institut für Strahlen- und
Kernphysik, Universität Bonn, 53115 Bonn, Germany  P. Jörg  Present address:
Physikalisches Institut, Universität Bonn, 53115 Bonn, Germany  Physikalisches
Institut, Universität Freiburg, 79104 Freiburg, Germany  E. Kabuß  Institut
für Kernphysik, Universität Mainz, 55099 Mainz, Germany  F. Kaspar  Physik
Department, Technische Universität München, 85748 Garching, Germany  A.
Kerbizi  Department of Physics, University of Trieste, 34127 Trieste, Italy
Trieste Section of INFN, 34127 Trieste, Italy  B. Ketzer  Helmholtz-Institut
für Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn, Germany  G. V.
Khaustov  State Scientific Center Institute for High Energy Physics of
National Research Center ”Kurchatov Institute”, 142281 Protvino, Russia  Yu.
A. Khokhlov  Also at: Moscow Institute of Physics and Technology, Moscow
Region, 141700, Russia.  State Scientific Center Institute for High Energy
Physics of National Research Center ”Kurchatov Institute”, 142281 Protvino,
Russia  Yu. Kisselev  Deceased.  Joint Institute for Nuclear Research, 141980
Dubna, Moscow region, Russia  F. Klein  Physikalisches Institut, Universität
Bonn, 53115 Bonn, Germany  J. H. Koivuniemi  Institut für Experimentalphysik,
Universität Bochum, 44780 Bochum, Germany  Department of Physics, University
of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA  V. N. Kolosov
State Scientific Center Institute for High Energy Physics of National Research
Center ”Kurchatov Institute”, 142281 Protvino, Russia  K. Kondo Horikawa
Yamagata University, Yamagata 992-8510, Japan  I. Konorov  Lebedev Physical
Institute, 119991 Moscow, Russia  Physik Department, Technische Universität
München, 85748 Garching, Germany  V. F. Konstantinov  State Scientific Center
Institute for High Energy Physics of National Research Center ”Kurchatov
Institute”, 142281 Protvino, Russia  A. M. Kotzinian  Also at: Yerevan Physics
Institute, Alikhanian Br. Street, 0036 Yerevan, Armenia.  Torino Section of
INFN, 10125 Turin, Italy  O. M. Kouznetsov  Joint Institute for Nuclear
Research, 141980 Dubna, Moscow region, Russia  A. Koval  National Centre for
Nuclear Research, 02-093 Warsaw, Poland  Z. Kral  Faculty of Mathematics and
Physics, Charles University, 18000 Prague, Czech Republic  F. Krinner  Physik
Department, Technische Universität München, 85748 Garching, Germany  Y.
Kulinich  Department of Physics, University of Illinois at Urbana-Champaign,
Urbana, IL 61801-3080, USA  F. Kunne  IRFU, CEA, Université Paris-Saclay,
91191 Gif-sur-Yvette, France  K. Kurek  National Centre for Nuclear Research,
02-093 Warsaw, Poland  R. P. Kurjata  Institute of Radioelectronics, Warsaw
University of Technology, 00-665 Warsaw, Poland  A. Kveton  Faculty of
Mathematics and Physics, Charles University, 18000 Prague, Czech Republic  K.
Lavickova  Faculty of Mathematics and Physics, Charles University, 18000
Prague, Czech Republic  S. Levorato  Trieste Section of INFN, 34127 Trieste,
Italy  CERN, 1211 Geneva 23, Switzerland  Y.-S. Lian  Also at: Department of
Physics, National Kaohsiung Normal University, Kaohsiung County 824, Taiwan.
Academia Sinica, Institute of Physics, Taipei 11529, Taiwan  J. Lichtenstadt
School of Physics and Astronomy, Tel Aviv University, 69978 Tel Aviv, Israel
P.-J. Lin  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France
R. Longo  Department of Physics, University of Illinois at Urbana-Champaign,
Urbana, IL 61801-3080, USA  V. E. Lyubovitskij  Also at: Institut für
Theoretische Physik, Universität Tübingen, 72076 Tübingen, Germany.  Tomsk
Polytechnic University, 634050 Tomsk, Russia  A. Maggiora  Torino Section of
INFN, 10125 Turin, Italy  A. Magnon  Matrivani Institute of Experimental
Research & Education, Calcutta-700 030, India  N. Makins  Department of
Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080,
USA  N. Makke  Also at: Abdus Salam ICTP, 34151 Trieste, Italy.  Trieste
Section of INFN, 34127 Trieste, Italy  G. K. Mallot  CERN, 1211 Geneva 23,
Switzerland  Physikalisches Institut, Universität Freiburg, 79104 Freiburg,
Germany  A. Maltsev  Joint Institute for Nuclear Research, 141980 Dubna,
Moscow region, Russia  S. A. Mamon  Tomsk Polytechnic University, 634050
Tomsk, Russia  B. Marianski  Deceased.  National Centre for Nuclear Research,
02-093 Warsaw, Poland  A. Martin  Department of Physics, University of
Trieste, 34127 Trieste, Italy  Trieste Section of INFN, 34127 Trieste, Italy
J. Marzec  Institute of Radioelectronics, Warsaw University of Technology,
00-665 Warsaw, Poland  J. Matoušek  Department of Physics, University of
Trieste, 34127 Trieste, Italy  Trieste Section of INFN, 34127 Trieste, Italy
T. Matsuda  University of Miyazaki, Miyazaki 889-2192, Japan  G. Mattson
Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL
61801-3080, USA  G. V. Meshcheryakov  Joint Institute for Nuclear Research,
141980 Dubna, Moscow region, Russia  M. Meyer  Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA  IRFU,
CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette, France  W. Meyer  Institut
für Experimentalphysik, Universität Bochum, 44780 Bochum, Germany  Yu. V.
Mikhailov  State Scientific Center Institute for High Energy Physics of
National Research Center ”Kurchatov Institute”, 142281 Protvino, Russia  M.
Mikhasenko  Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn,
53115 Bonn, Germany  CERN, 1211 Geneva 23, Switzerland  E. Mitrofanov  Joint
Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  N.
Mitrofanov  Joint Institute for Nuclear Research, 141980 Dubna, Moscow region,
Russia  Y. Miyachi  Yamagata University, Yamagata 992-8510, Japan  A. Moretti
Department of Physics, University of Trieste, 34127 Trieste, Italy  Trieste
Section of INFN, 34127 Trieste, Italy  A. Nagaytsev  Joint Institute for
Nuclear Research, 141980 Dubna, Moscow region, Russia  C. Naim  IRFU, CEA,
Université Paris-Saclay, 91191 Gif-sur-Yvette, France  D. Neyret  IRFU, CEA,
Université Paris-Saclay, 91191 Gif-sur-Yvette, France  J. Nový  Czech
Technical University in Prague, 16636 Prague, Czech Republic  W.-D. Nowak
Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany  G. Nukazuka
Yamagata University, Yamagata 992-8510, Japan  A. S. Nunes  Present address:
Brookhaven National Laboratory, Brookhaven, USA  LIP, 1649-003 Lisbon,
Portugal  A. G. Olshevsky  Joint Institute for Nuclear Research, 141980 Dubna,
Moscow region, Russia  M. Ostrick  Institut für Kernphysik, Universität Mainz,
55099 Mainz, Germany  D. Panzieri  Also at: University of Eastern Piedmont,
15100 Alessandria, Italy.  Torino Section of INFN, 10125 Turin, Italy  B.
Parsamyan  Department of Physics, University of Torino, 10125 Turin, Italy
Torino Section of INFN, 10125 Turin, Italy  S. Paul  Physik Department,
Technische Universität München, 85748 Garching, Germany  H. Pekeler
Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn,
Germany  J.-C. Peng  Department of Physics, University of Illinois at Urbana-
Champaign, Urbana, IL 61801-3080, USA  M. Pešek  Faculty of Mathematics and
Physics, Charles University, 18000 Prague, Czech Republic  D. V. Peshekhonov
Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  M.
Pešková  Faculty of Mathematics and Physics, Charles University, 18000 Prague,
Czech Republic  N. Pierre  Institut für Kernphysik, Universität Mainz, 55099
Mainz, Germany  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-Yvette,
France  S. Platchkov  IRFU, CEA, Université Paris-Saclay, 91191 Gif-sur-
Yvette, France  J. Pochodzalla  Institut für Kernphysik, Universität Mainz,
55099 Mainz, Germany  V. A. Polyakov  State Scientific Center Institute for
High Energy Physics of National Research Center ”Kurchatov Institute”, 142281
Protvino, Russia  J. Pretz  Present address: III. Physikalisches Institut,
RWTH Aachen University, 52056 Aachen, Germany  Physikalisches Institut,
Universität Bonn, 53115 Bonn, Germany  M. Quaresma  Academia Sinica, Institute
of Physics, Taipei 11529, Taiwan  LIP, 1649-003 Lisbon, Portugal  C. Quintans
LIP, 1649-003 Lisbon, Portugal  G. Reicherz  Institut für Experimentalphysik,
Universität Bochum, 44780 Bochum, Germany  C. Riedl  Department of Physics,
University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA  T.
Rudnicki  Faculty of Physics, University of Warsaw, 02-093 Warsaw, Poland  D.
I. Ryabchikov  State Scientific Center Institute for High Energy Physics of
National Research Center ”Kurchatov Institute”, 142281 Protvino, Russia
Physik Department, Technische Universität München, 85748 Garching, Germany  A.
Rybnikov  Joint Institute for Nuclear Research, 141980 Dubna, Moscow region,
Russia  A. Rychter  Institute of Radioelectronics, Warsaw University of
Technology, 00-665 Warsaw, Poland  V. D. Samoylenko  State Scientific Center
Institute for High Energy Physics of National Research Center ”Kurchatov
Institute”, 142281 Protvino, Russia  A. Sandacz  National Centre for Nuclear
Research, 02-093 Warsaw, Poland  S. Sarkar  Matrivani Institute of
Experimental Research & Education, Calcutta-700 030, India  I. A. Savin  Joint
Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia  G.
Sbrizzai  Department of Physics, University of Trieste, 34127 Trieste, Italy
Trieste Section of INFN, 34127 Trieste, Italy  H. Schmieden  Physikalisches
Institut, Universität Bonn, 53115 Bonn, Germany  A. Selyunin  Joint Institute
for Nuclear Research, 141980 Dubna, Moscow region, Russia  L. Sinha  Matrivani
Institute of Experimental Research & Education, Calcutta-700 030, India  M.
Slunecka  Faculty of Mathematics and Physics, Charles University, 18000
Prague, Czech Republic  J. Smolik  Joint Institute for Nuclear Research,
141980 Dubna, Moscow region, Russia  A. Srnka  Institute of Scientific
Instruments of the CAS, 61264 Brno, Czech Republic  D. Steffen  CERN, 1211
Geneva 23, Switzerland  Physik Department, Technische Universität München,
85748 Garching, Germany  M. Stolarski  LIP, 1649-003 Lisbon, Portugal  O.
Subrt  CERN, 1211 Geneva 23, Switzerland  Czech Technical University in
Prague, 16636 Prague, Czech Republic  M. Sulc  Technical University in
Liberec, 46117 Liberec, Czech Republic  H. Suzuki  Also at: Chubu University,
Kasugai, Aichi 487-8501, Japan.  Yamagata University, Yamagata 992-8510, Japan
P. Sznajder  National Centre for Nuclear Research, 02-093 Warsaw, Poland  S.
Tessaro  Trieste Section of INFN, 34127 Trieste, Italy  F. Tessarotto  Trieste
Section of INFN, 34127 Trieste, Italy  CERN, 1211 Geneva 23, Switzerland  A.
Thiel  Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn,
53115 Bonn, Germany  J. Tomsa  Faculty of Mathematics and Physics, Charles
University, 18000 Prague, Czech Republic  F. Tosello  Torino Section of INFN,
10125 Turin, Italy  A. Townsend  Department of Physics, University of Illinois
at Urbana-Champaign, Urbana, IL 61801-3080, USA  V. Tskhay  Lebedev Physical
Institute, 119991 Moscow, Russia  S. Uhl  Physik Department, Technische
Universität München, 85748 Garching, Germany  B. I. Vasilishin  Tomsk
Polytechnic University, 634050 Tomsk, Russia  A. Vauth  Present address:
Universität Hamburg, 20146 Hamburg, Germany  Physikalisches Institut,
Universität Bonn, 53115 Bonn, Germany  CERN, 1211 Geneva 23, Switzerland  B.
M. Veit  Institut für Kernphysik, Universität Mainz, 55099 Mainz, Germany
CERN, 1211 Geneva 23, Switzerland  J. Veloso  Department of Physics,
University of Aveiro, I3N, 3810-193 Aveiro, Portugal  B. Ventura  IRFU, CEA,
Université Paris-Saclay, 91191 Gif-sur-Yvette, France  A. Vidon  IRFU, CEA,
Université Paris-Saclay, 91191 Gif-sur-Yvette, France  M. Virius  Czech
Technical University in Prague, 16636 Prague, Czech Republic  M. Wagner
Helmholtz-Institut für Strahlen- und Kernphysik, Universität Bonn, 53115 Bonn,
Germany  S. Wallner  Physik Department, Technische Universität München, 85748
Garching, Germany  K. Zaremba  Institute of Radioelectronics, Warsaw
University of Technology, 00-665 Warsaw, Poland  P. Zavada  Joint Institute
for Nuclear Research, 141980 Dubna, Moscow region, Russia  M. Zavertyaev
Lebedev Physical Institute, 119991 Moscow, Russia  M. Zemko  Faculty of
Mathematics and Physics, Charles University, 18000 Prague, Czech Republic
CERN, 1211 Geneva 23, Switzerland  E. Zemlyanichkina  Joint Institute for
Nuclear Research, 141980 Dubna, Moscow region, Russia  Y. Zhao  Trieste
Section of INFN, 34127 Trieste, Italy  M. Ziembicki  Institute of
Radioelectronics, Warsaw University of Technology, 00-665 Warsaw, Poland

###### Abstract

The COMPASS experiment recently discovered a new isovector resonance-like
signal with axial-vector quantum numbers, the $a_{1}(1420)$, decaying to
$f_{0}(980)\pi$. With a mass too close to and a width smaller than the axial-
vector ground state $a_{1}(1260)$, it was immediately interpreted as a new
light exotic meson, similar to the $X$, $Y$, $Z$ states in the hidden-charm
sector. We show that a resonance-like signal fully matching the experimental
data is produced by the decay of the $a_{1}(1260)$ resonance into
$K^{\ast}(\to K\pi)\bar{K}$ and subsequent rescattering through a triangle
singularity into the coupled $f_{0}(980)\pi$ channel. The amplitude for this
process is calculated using a new approach based on dispersion relations. The
triangle-singularity model is fitted to the partial-wave data of the COMPASS
experiment. Despite having less parameters, this fit shows a slightly better
quality than the one using a resonance hypothesis and thus eliminates the need
for an additional resonance in order to describe the data. We thereby
demonstrate for the first time in the light-meson sector that a resonance-like
structure in the experimental data can be described by rescattering through a
triangle singularity, providing evidence for a genuine three-body effect.

Quantum chromodynamics is generally accepted as the fundamental quantum-field
theory of the strong interaction. How exactly the spectrum of bound states
(hadrons) emerges from the underlying interaction between quarks and gluons
is, however, not yet understood. The main difficulty is the rise of the strong
coupling at the low-energy scale relevant for hadrons, which makes the theory
unsolvable with perturbative methods. Although the constituent-quark model
Tanabashi  _et al._ (2018); Ebert  _et al._ (2009); Eichten  _et al._ (1978)
describes many of the observed mesons, it seems that the spectrum is notably
richer: there is growing experimental evidence for bound states beyond the
constituent-quark model. Such states are commonly called exotic Ketzer (2012);
Meyer and Swanson (2015); Esposito  _et al._ (2017); Guo  _et al._ (2018);
Olsen  _et al._ (2018); Rodas  _et al._ (2019). In addition to mapping out the
full spectrum predicted by models and, more recently, by lattice gauge theory
Dudek  _et al._ (2013), the search for such exotic states drives the current
interest in hadron spectroscopy.

The study of single-diffractive reactions with a high-energy meson beam, as
performed by the COMPASS experiment at the CERN SPS Abbon  _et al._ (2007,
2015), is a natural way to investigate meson excitations (for a recent review,
see Ketzer  _et al._ (2020)). In such reactions, at high energies commonly
described by the exchange of a Pomeron $I\\!\\!P$, the incoming beam particle
is excited by the strong interaction with a proton target. Regge theory then
allows us to factorize off the target vertex, such that we only consider the
beam vertex. Although the produced excited system immediately decays, the
reaction products unveil the properties of the excitation. An unprecedented
amount of data comprising almost $50\,$ million events for the reaction
$\pi^{-}+p\to\pi^{-}\pi^{-}\pi^{+}+p$ were used by COMPASS to perform a
detailed analysis of $\pi_{J}$ and $a_{J}$ mesons with isospin $I=1$, negative
$G$-parity, and positive $C$-parity implied by $G=C(-1)^{I}$. The partial-wave
analysis (PWA) technique in connection with the isobar model was used to
separate excitations with different quantum numbers, see Ketzer  _et al._
(2020); Adolph  _et al._ (2017) for details. Individual waves are labeled
$J^{PC}\,M^{\epsilon}\,\xi\pi\,L$, where $J$ is the total angular momentum of
the 3-pion system, $P$ the spatial and $C$ the charge-conjugation parity. The
quantum number $M$ labels the projection of the spin $J$ onto the direction of
the beam in the rest frame of $\pi^{-}\pi^{-}\pi^{+}$, and $\epsilon$
indicates the reflection symmetry with respect to the production plane. At the
high center-of-momentum energies of the experiment, the reflectivity quantum
number $\epsilon$ corresponds to the naturality of the exchanged particle and
is hence always positive for Pomeron exchange. The orbital angular momentum
between the neutral system of two pions (isobar) and the remaining pion is
denoted by $L$. The symbol $\xi$ labels the assumed isobar, i.e. the
interaction amplitude in the neutral $\pi\pi$-subchannel.

A PWA including $88$ waves in total was performed separately in 100 bins of
the $3\pi$ invariant mass $m_{3\pi}$ and 11 bins of the reduced 4-momentum
transfer squared $t^{\prime}$ (see Eq. (6) in Adolph  _et al._ (2017)). The
results are summarized in Fig. 1(a), where we show the intensities of selected
waves as a function of $m_{3\pi}$, summed over all $t^{\prime}$ bins.

\begin{overpic}[width=190.79385pt]{pict/Fig1a.pdf}
\put(30.0,62.0){(a)}\end{overpic}\begin{overpic}[width=225.48424pt]{pict/Fig1b.pdf}
\put(80.0,55.0){(b)}\end{overpic}

Figure 1: (a) Intensities of selected waves from the PWA of the reaction
$\pi^{-}+p\to\pi^{-}\pi^{-}\pi^{+}+p$ Adolph  _et al._ (2017). The inset shows
an enlarged view of the $1^{++}0^{+}\,f_{0}\pi\,P$-wave. The colored bar on
the left indicates the contributions of the different waves to the total
intensity. (b) Diagrams showing possible contributions to the $\rho(770)\pi$
and $f_{0}(980)\pi$ production amplitudes. The Pomeron is labeled $I\\!\\!P$,
$a_{1}$ refers to the axial-vector ground state $a_{1}(1260)$, and $a_{2}$ to
the tensor ground state $a_{2}(1320)$. The framed diagram shows the dominant
contribution to the $a_{1}(1420)$ signal via the triangle diagram.

Among many important observations, an exotic resonance-like signal with
quantum numbers $J^{PC}=1^{++}$ was found in the
$1^{++}\,0^{+}\,f_{0}\pi\,P$-wave as a clear peak at $1.4\,\mathrm{GeV}/c^{2}$
Adolph  _et al._ (2015) (see inset of Fig. 1(a)). The resonance-like behavior
was corroborated by the observed phase motion, i.e. a mass-dependent relative
phase with respect to several other reference waves. Extensive studies, also
using the “freed-isobar” method Krinner  _et al._ (2018), undoubtedly
confirmed the signal and proved that it was not an artifact of any particular
isobar parametrization Adolph  _et al._ (2017). Following the PDG convention,
the signal was called $a_{1}(1420)$ according to its quantum numbers
$I^{G}(J^{PC})=1^{-}(1^{++})$. It was immediately realized that it could not
be an ordinary quark-model meson resonance: (i) with about
$150\,\mathrm{MeV}/c^{2}$, its width is much smaller than that of the axial-
vector ground state $a_{1}(1260)$ of about $500\,\mathrm{MeV}/c^{2}$; (ii) the
signal is separated from the ground state by only about
$150\,\mathrm{MeV}/c^{2}$, whereas the energy difference between different
radial excitation levels is typically $400\,\mathrm{MeV}/c^{2}$ as estimated
based on the slope of the radial excitation trajectory Chen  _et al._ (2015a);
Anisovich  _et al._ (2000); (iii) so far, the $a_{1}(1420)$ was seen only in
the $f_{0}(980)\pi$ final state.

Various interpretations followed the $a_{1}(1420)$ observation Mikhasenko  _et
al._ (2015); Aceti  _et al._ (2016); Chen  _et al._ (2015b); Gutsche  _et al._
(2017); Basdevant and Berger (2015), requiring or not a new resonance.
Resonances are consistently introduced in general scattering theory Gribov
_et al._ (2009), where the reaction amplitude is an analytic function of the
total energy squared $s$ that is regarded as a complex number; they are found
as poles on the unphysical sheet of the complex $s$-plane attached to the real
axis from below. In explanations involving either diquark-antidiquark
molecules or tightly bound tetraquarks, the observed signal, i.e. peak and
phase motion, is caused by a pole-type singularity located on the closest
sheet. Alternatively, a so-called triangle-singularity (TS) mechanism Gribov
_et al._ (2009); Eden (1971); Guo  _et al._ (2020) was proposed as the
mechanism behind the $a_{1}(1420)$ signal Mikhasenko  _et al._ (2015). Here, a
logarithmic branch point caused by a coupled-channel effect, particularly by
the $K^{\ast}\bar{K}$-$f_{0}\pi$ interaction, is located near the physical
region on the closest unphysical sheet. The other proposed model Basdevant and
Berger (2015, 1977) that does not require a new resonance pole, combines
resonant and nonresonant production mechanisms resulting in a peak in the
$1^{++}\,0^{+}\,f_{0}\pi\,P$-wave. However, the generated phase motion is at
the position of the $a_{1}(1260)$ resonance, which is inconsistent with
observation.

In this Letter, we interpret the COMPASS data in terms of the triangle-
singularity model based on a new method for the calculation of the amplitude.
The calculation implements the proposal of Mikhasenko  _et al._ (2019)
exploiting the unitarity and analyticity properties of the amplitude. The new
model goes beyond Mikhasenko  _et al._ (2015) by incorporating spin in a more
systematic way and allowing us to address higher-order rescattering effects.
To our knowledge, this is the first time that the TS model, mimicking a
resonance signal, is fitted successfully to experimental data in the light-
meson sector describing both intensity and phase motion simultaneously.
Comparable studies in the heavy-quark sector, see e.g. Szczepaniak (2015); Guo
_et al._ (2018); Nakamura and Tsushima (2019), were performed on a much
smaller statistical basis.

The dynamics of a hadronic three-body system is commonly understood in terms
of quasi-two-body interactions with subchannel resonances $\xi$ decaying
further into pairs of final-state particles. Often, however, the same final
state can be obtained through several decay chains when the two-particle
interaction is non-negligible for different particle pairs Herndon  _et al._
(1975); Aitchison (1965). Different decay chains are coherent, hence they
interfere. The unitarity of the scattering matrix enforces a consistency
relation between the different chains Khuri and Treiman (1960); Pasquier and
Pasquier (1968, 1969). This relation makes the line shapes of the resonances
in a particle pair in a system of three particles dependent on the dynamics in
the other pairs Aitchison and Brehm (1979); Niecknig and Kubis (2015);
Niecknig  _et al._ (2012); Danilkin  _et al._ (2015). An equivalent way of
describing this interrelation between pair-wise interactions is to state that
the cross-channel two-body resonances in the $\pi\pi$, $K\pi$ and $\bar{K}K$
systems rescatter to one another, thereby modifying the original undistorted
line shapes. In addition, the probabilities for a three-body resonance
decaying to one or another channel may be redistributed due to final-state
interaction Schmid (1967); Szczepaniak (2016). The latter effect is strongly
enhanced for certain kinematic conditions Landau (1959); Gribov  _et al._
(2009) and produces the observed resonance-like signal in the case considered
here.

We find that the presence of the $K^{\ast}(892)$ resonance (hereafter referred
to as $K^{\ast}$) in the $K\bar{K}\pi$ channel drastically affects the
$f_{0}(980)\pi$ channel, since the rescattering between $K^{\ast}\to
K\pi\,P$-wave and $K\bar{K}\to f_{0}(980)\to\pi\pi\,S$-wave occurs with all
intermediate particles being almost on their mass shell for $m_{3\pi}\approx
1.4\,$GeV$/c^{2}$, i.e. slightly above the $K^{\ast}\bar{K}$ threshold
Mikhasenko  _et al._ (2015). This effect does not disturb the narrow line
shape of the $f_{0}(980)$, but it leads to a significant redistribution of the
$a_{1}(1260)$ decay probabilities. The originally negligible
$f_{0}(980)\pi\,P$-wave decay channel is populated by the rescattering from
the $K^{\ast}\bar{K}$ decay locally around $1.4\,$GeV$/c^{2}$.

Our calculation of the TS amplitude is reminiscent of the Khuri-Treiman (KT)
equation first developed in 1960 Khuri and Treiman (1960); Pasquier and
Pasquier (1968, 1969): the dispersion relation and two-body unitarity are used
to connect the isobar amplitude with the partial-wave projection of the cross
channels. By $F_{W}^{\\{1\\}}$ we denote the production amplitude of a three-
particle system $(123)$ with a given set of quantum numbers $W$, the invariant
mass squared $s\equiv m_{3\pi}^{2}$, and the isobar formed by particles $2$
and $3$ (hereafter labeled $\\{1\\}$ using indices in curly brackets) with
invariant mass squared $\sigma_{\\{1\\}}\equiv(p_{2}+p_{3})^{2}$. We write the
dispersion relation for the kinematic-singularity-free amplitude
$F_{W}^{\\{1\\}}(s,\sigma_{\\{1\\}})/K^{\\{1\\}}_{W}(s,\sigma_{\\{1\\}})$ with
$K^{\\{1\\}}_{W}$ being the break-up momentum for the $f_{0}\pi$ system
required for the $P$-wave (see Supplemental Material):

$\displaystyle
F_{W}^{\\{1\\}}(s,\sigma_{\\{1\\}})=K^{\\{1\\}}_{W}(s,\sigma_{\\{1\\}})\bigg{(}C_{W}^{\\{1\\}}(s)$
(1)
$\displaystyle\quad+\frac{1}{2\pi}\sum_{\omega}\int_{\sigma_{\text{th},W}}^{\infty}\frac{\rho_{\omega}(\sigma)\hat{F}_{W,\omega}^{\\{1\\}}(s,\sigma)}{K^{\\{1\\}}_{W}(s,\sigma)(\sigma-\sigma_{\\{1\\}}-i\varepsilon)}\text{d}\sigma\bigg{)}.$

Here, the indices $W$ and $\omega$ refer to the full set of quantum numbers
labeling a given wave. For the $f_{0}\pi\,P$-wave considered in this paper,
the term $C_{W}^{\\{1\\}}(s)$ parametrizes the three-body production dynamics
and the decay into the given final state $W$. It includes the direct
production of the $a_{1}(1260)$ resonance and a term for the nonresonant
production that is further described below. The sum runs over all possible
cross channels with quantum numbers $\omega$. In the dispersion integral,
$\rho_{\omega}(\sigma)$ is the 2-body phase-space factor, and
$\hat{F}_{W,\omega}^{\\{1\\}}(s,\sigma)$ is the projection of the cross
channel ${\\{3\\}}$, i.e. the isobar formed by particles $1$ and $2$, with
quantum numbers $\omega$ onto channel $\\{1\\}$ with quantum numbers $W$. We
do not expect isobars in channel $\\{2\\}$, formed by, e.g., $K^{-}\pi^{-}$,
$K^{0}\pi^{-}$, or $\pi^{-}\pi^{-}$. We note an important difference to the
original KT equation. The latter constrains the subchannel dynamics in the
three-body system. The total invariant mass of the system is treated as a
fixed parameter in the model. In 1965, Aitchison suggested that this
parametric dependence is actually physical and represents the three-body
interaction Aitchison (1965). In Mikhasenko  _et al._ (2019), the authors
demonstrated that the KT kernel can be used to separate the genuine three-body
dynamics from the final-state interaction. Correspondingly, in Eq. (1) the
direct decay of $a_{1}(1260)$ enters in $C_{W}^{\\{1\\}}(s)$, while the
$s$-dependent dispersion integral adds the rescattering corrections. Assuming
that modifications of the line shapes of the cross-channel resonances due to
rescattering are negligible, we find that the $K^{\ast}\bar{K}$ channel
produces a narrow peak and a strong phase motion at the mass of the
$a_{1}(1420)$ due to the TS being very close to the physical region, while all
other possible rescattering corrections, which we investigated, manifest
themselves in a broad bump and a slow phase motion similar to the direct decay
and the nonresonant background (see Supplemental Material).

For a fit of the TS model to the COMPASS spin-density matrix elements pap ,
referred to below as the data points, we choose the three waves depicted in
Fig. 1(b), which constitute the dominant contributions to the $\rho\pi$ and
$f_{0}\pi$ production amplitudes: (i) the $1^{++}0^{+}\,\rho\pi\,S$-wave
describes the source of the rescattering process, since its largest
contribution comes from the $a_{1}(1260)$. This wave also contains a
significant contribution from nonresonant “Deck”-like processes Ascoli  _et
al._ (1973); (ii) the $1^{++}0^{+}\,f_{0}\pi\,P$-wave contains the
$a_{1}(1420)$ signal; (iii) the $2^{++}1^{+}\,\rho\pi\,D$-wave exhibits a
clean $a_{2}(1320)$-resonance and is included in order to fix the relative
phases and stabilize the fit. In general, there are two components for each
wave in the model: a resonance amplitude, i.e. a propagator that contains a
pole (in this case either the $a_{1}(1260)$ or the $a_{2}(1320)$), and a
component with $t$-channel $\pi$ exchange accounting for nonresonant
processes. We parametrize the $a_{1}(1260)$ propagator by a relativistic
Breit-Wigner (BW) amplitude with energy-dependent width saturated by the
$\rho\pi$ decay channel Aghasyan  _et al._ (2018). For the resonance part of
the $\rho\pi\,D$-wave we employ the $a_{2}(1320)$ propagator parametrized by a
BW amplitude with dynamical width including the $\rho\pi$ ($80\%$) and
$\eta\pi$ ($20\%$) channels, as discussed in Aghasyan  _et al._ (2018). The
nonresonant background is added coherently to each wave. We use an empirical
parametrization given by
$(m_{3\pi}/m_{0}-1)^{b}\>\exp[-(c_{0}+c_{1}t^{\prime}+c_{2}t^{\prime
2})\tilde{p}^{2}]$, where $\tilde{p}$ is an effective break-up momentum for
the decay into $\xi\pi$ at the given $m_{3\pi}$ value, taking into account the
finite width of the isobar $\xi$ and the Bose symmetry of the system, and
$m_{0}=0.5\,\mathrm{GeV}/c^{2}$ (see Eqs. (27) and (29) in Aghasyan  _et al._
(2018)). For the model calculations, the $t^{\prime}$ value is fixed to the
lower edge of the respective bin. For the $f_{0}\pi\,P$-wave, the resonance
part of the production amplitude is modified by the $K^{\ast}\bar{K}\to
f_{0}\pi$-rescattering via the TS. As the direct decay of the $a_{1}(1260)$ to
the $f_{0}\pi$ final state has a very slow phase motion and a similar shape as
the phenomenological parametrization of the nonresonant part due to the
limited phase space (see Supplemental Material for details), the fit cannot
distinguish between the two components. Therefore, this additional component
is only considered for systematic studies.

The free parameters of the model, i.e. the $t^{\prime}$-dependent complex
couplings, the background parameters $b$ and $c_{i}$, as well as the
$t^{\prime}$-independent BW parameters, are determined by a fit to the COMPASS
data points in $m_{3\pi}$ and $t^{\prime}$ bins. We note that there are no
free parameters influencing the line shape of the TS amplitude, while the
strength and the background parameters are adjusted in the fit. As explained
in more detail in Aghasyan  _et al._ (2018), the data points $y_{i}$ to be
fitted are the intensity and the real and imaginary parts of the interference
terms for the 3 selected waves inside the chosen $m_{3\pi}$ ranges (indicated
in Fig. 2) for all 11 $t^{\prime}$ bins. The fit is performed by minimizing
the sum of the squared differences between data points $y_{i}$ and model
prediction $\hat{y}_{i}$, weighted by the inverse squared statistical
uncertainties:

$\displaystyle\mathcal{R}^{2}=\sum_{i}\frac{(y_{i}-\hat{y}_{i})^{2}}{\sigma_{i}^{2}}.$
(2)

Figure 2 shows the result of the TS model fit in the lowest $t^{\prime}$ bin,
selecting only the $f_{0}\pi\,P$-wave (full lines). The fit results for all
three waves in all 11 $t^{\prime}$-bins can be found in Supplemental Material.
Figure 2(a) shows the intensity of the $f_{0}\pi\,P$-wave and Figure 2(b) the
relative phase to the $\rho\pi\,S$-wave, both as a function of $m_{3\pi}$. The
resonance-like behavior of the TS amplitude is most evident from the circle in
the Argand diagram in panel (c). The nonresonant background (green arrows)
helps to slightly adjust the position of the circle. Since the phase of the
background component does not change with $m_{3\pi}$, all green arrows are
parallel.

\begin{overpic}[width=140.92517pt]{pict/Fig2a.pdf}
\put(18.0,75.0){(a)}\end{overpic}\begin{overpic}[width=140.92517pt]{pict/Fig2b.pdf}
\put(18.0,75.0){(b)}\end{overpic}\begin{overpic}[width=140.92517pt]{pict/Fig2c.pdf}
\put(18.0,75.0){(c)}\end{overpic}

Figure 2:  Results of the fit with the TS model (solid lines) and the BW model
(dashed lines) to the wave with the $a_{1}(1420)$ resonance-like signal. The
fit range is indicated by the color saturation of data points and lines. (a)
intensity of the $1^{++}\,0^{+}\,f_{0}\pi\,P$-wave. The complete fit model
(red) is decomposed into its signal (blue) and background (green)
contributions. (b) Relative phase between the $1^{++}0^{+}\,\rho\pi\,S$-wave
and the $1^{++}\,0^{+}\,f_{0}\pi\,P$-wave. (c) Argand diagram. The red dots on
the TS-model curve correspond to the indicated $m_{3\pi}$ values in units of
$\mathrm{GeV}/c^{2}$.

In order to evaluate the quality of the TS model fit, we also perform a fit to
the data using a simple BW description of the $a_{1}(1420)$ signal instead of
the TS amplitude. This is accomplished by replacing the TS parametrization of
the $f_{0}\pi\,P$-wave by a relativistic BW amplitude with free mass and width
parameters assuming the $a_{1}(1420)$ being a genuine new resonance. We use a
constant-width parametrization since further decay modes of this hypothetical
new particle are unknown. Figure 2 shows that the fits with the BW model
(dashed) and the TS model (solid) are of very similar quality. Both models are
capable of describing the intensities as well as the corresponding
interference terms. For a quantitative comparison, one can use the quantity
defined in Eq. (2). The biggest contribution comes from the $\rho\pi\,S$ and
$\rho\pi\,D$-waves. Since the description of these two waves is very similar
in both fit models, we can omit them for the comparison of the fit quality. In
addition, we can exclude one of the two remaining phases of the interferences,
since they depend linearly on one another. Defining
$\mathcal{R}^{2}_{\text{red}}$ as the reduced weighted sum of the remaining
residuals squared divided by the number of degrees of freedom, where only the
fit parameters specific to the $f_{0}\pi\,P$-wave are taken into account, we
arrive at a value of $\mathcal{R}^{2}_{\text{red,TS}}=4.8$ for the TS and
$\mathcal{R}^{2}_{\text{red,BW}}=5.2$ for the BW model. The values indicate
that the two fits have comparable quality. The advantage of the TS model is
that it has two fit parameters less, since it does not require a new particle
with corresponding mass and width.

To study the stability of the result, we investigate a wide range of sources
of systematic uncertainties, both with respect to changes of the model and to
changes of the data points. We perform fits where the data points $y_{i}$ are
varied according to systematic studies for the PWA in bins of mass and
$t^{\prime}$, published in Adolph  _et al._ (2017). These include using a
smaller wave set, removing negative reflectivity waves, relaxing the event
selection, using a model with relaxed coherence assumption (see Adolph  _et
al._ (2017) for details) or changing the parametrization of the $f_{0}(980)$.
In an additional study, we use the result of a statistical reanalysis of
Adolph  _et al._ (2017) applying the bootstrap technique Efron and Tibshirani
(1986). Also, we consider several variations in the fit model for the TS: (i)
a fit with non-Bose-symmetrized phase space; (ii) neglecting the spins of the
particles involved (similar to Mikhasenko  _et al._ (2015)); (iii) including
the excitations $a_{1}(1640)$ and $a_{2}(1700)$ in the $\rho\pi\,S$ and
$\rho\pi\,D$ waves, respectively (mass and width fixed to the values from the
PDG Tanabashi  _et al._ (2018)); and (iv) varying mass and width of the
$K^{\ast}$ resonance according to their uncertainties Tanabashi  _et al._
(2018) in order to estimate the effect of further rescattering. The TS model
systematically yields a slightly smaller $\mathcal{R}^{2}_{\text{red}}$ than
the BW model (see Supplemental Material).

In summary, we have shown that the recently discovered resonance-like signal
$a_{1}(1420)$ can be fully explained by the decay of the ground state
$a_{1}(1260)$ into $K^{\ast}\bar{K}$ and subsequent rescattering through a
triangle singularity into the observed final state $f_{0}(980)\pi$ without the
need of a new genuine $a_{1}$ resonance. The effect of the triangle
singularity, which is expected to be present, is sufficient to explain the
observation.

###### Acknowledgements.

Acknowledgments.—We gratefully acknowledge the support of the CERN management
and staff as well as the skills and efforts of the technicians of our
collaborating institutions. We would like to thank Bastian Kubis for useful
discussions on the scalar form factor. This work was made possible by the
financial support of our funding agencies: MEYS Grant No. LG13031 (Czech
Republic); FP7 HadronPhysics3 Grant No. 283286 (European Union); CEA, P2I, and
P.-J. L. was supported by ANR (France) with P2IO LabEx (ANR-10-LBX-0038) in
the framework ”Investissements d’Avenir” (ANR-11-IDEX-003-01); BMBF
Collaborative Research Project 05P2018— COMPASS, W. D. and M. F. were
supported by the DFG Cluster of Excellence Origin and Structure of the
Universe” (uni ), M. G. was supported by the DFG Research Training Group
Programmes 1102 and 2044 (Germany); B. Sen Fund (India); Academy of Sciences
and Humanities (Israel); INFN (Italy); MEXT and JSPS, Grants No. 8002006, No.
20540299, and No. 18540281, Daiko and Yamada Foundations (Japan); NCN Grant
No. 2017/26/M/ST2/00498 (Poland); FCT Grants No. CERN/FIS-PAR/0007/2017 and
No. CERN/FIS-PAR/ 0022/2019 (Portugal); CERN-RFBR Grant No. 12-02-91500,
Presidential Grant No. NSh-999.2014.2 (Russia); MST (Taiwan); and NSF (U.S.).

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## Supplemental material

This supplemental material includes additional information on the systematic
studies performed as well as details of the amplitude calculation.

##  I Other triangles and the direct decay

Figure 3 shows the $m_{3\pi}$ dependence of the $f_{0}(980)\pi$ isobar
production amplitude for different individual cross channels $\omega$.

\begin{overpic}[width=216.81pt]{pict/Fig3a.pdf}
\put(15.0,60.0){(a)}\end{overpic}\begin{overpic}[width=212.47617pt]{pict/Fig3b.pdf}
\put(15.0,60.0){(b)}\end{overpic}

Figure 3: (a) Intensities and (b) phases of the $f_{0}\pi\,P$ amplitude
produced from different sources. The $m_{3\pi}$ dependence is shown for the
$f_{0}(980)$ subchannel invariant mass fixed to the nominal resonance mass of
$990\,\mathrm{MeV}/c^{2}$. See text and Refs. Adolph  _et al._ (2017);
Aghasyan  _et al._ (2018) for details on the calculations and the descriptions
of the parameterizations of the isobar and resonance amplitudes. The
intensities in panel (a) are relative to the maximum intensity of the
$K^{\ast}\bar{K}$ channel, with the couplings in each vertex set to unity.
Note that the intensity of the $K^{\ast}\bar{K}$ triangle graph (blue lines)
was scaled down by a factor of $20$. In panel (b) we also include the phase
from Ref. Basdevant and Berger (2015) (purple line). Its inflection point is
at a mass of approximately $1.25\,\mathrm{GeV}/c^{2}$. See text for details on
the different curves.

The blue lines represent the TS in the $K\bar{K}\pi$ channel with the
$K^{\ast}$ resonance. The full blue line is the result of the new partial-wave
projection method, Eq. (1) of the main text, taking into account the spins of
all particles involved. The dashed blue line labeled “(scalar)”, as well as
all other dashed lines, are obtained by calculations using the Feynman method
from Ref. Mikhasenko  _et al._ (2015) assuming that all particles are
spinless. The curves shown in dashed gray correspond to the rescattering
effects of the various $\pi\pi$ resonances in the cross channel. For the
calculation we assume that modifications of the lineshapes of the cross-
channel resonances due to rescattering are negligible. It can be seen that the
$K^{\ast}\bar{K}$ channel produces a narrow peak and a strong phase motion at
the mass of the $a_{1}(1420)$ due to the TS being very close to the physical
region, while all other channels including the direct decay and the non-
resonant background (Bgd) manifest themselves in a broad bump and a slow phase
motion. As the direct decay of the $a_{1}(1260)$ to the $f_{0}\pi$ final state
has a very slow phase motion and a similar shape as the phenomenological
parameterization of the non-resonant part (compare the red and green curves in
Fig. 3), the fit cannot distinguish between the two components. The direct
decay of the $a_{1}(1260)$ to $f_{0}\pi$ can hence be omitted.

##  II Systematic studies

We investigate several sources of systematic uncertainties including
variations of the fit model and uncertainties of the mass-independent PWA. We
perform fits where the COMPAS data points of the main text are varied
according to systematic studies for the PWA in bins of mass and $t^{\prime}$
(see Ref. Adolph  _et al._ (2017)). Figure 4 compares the quantity
$\mathcal{R}^{2}_{\text{red}}$, as defined in Eq. (2) of the main text, for
the TS model and the model using a simple Breit-Wigner amplitude for the
$a_{1}(1420)$, for the various studies. These include using a smaller wave set
(“53 waves”), removing negative-reflectivity waves (“no neg. waves”), relaxing
the event selection (“coarse ev. sel.”), using rank 2 instead of rank 1 (“rank
2”), and changing the parameterization of the $f_{0}(980)$
(“$f_{0}(980)\,\mathrm{BW}$”, “$(\pi\pi)_{S}\ K_{1}$”). In an additional
study, we use the result of a statistical re-analysis of Adolph  _et al._
(2017) applying the bootstrap technique Efron and Tibshirani (1986)
(“bootstrap”). Also, we consider several variations of the fit model for the
TS: (i) a fit with non-Bose-symmetrized phase space (“non-sym. ph. sp.”); (ii)
neglecting the spins of the particles involved (similar to Ref. Mikhasenko
_et al._ (2015), “scalar TS”); (iii) including the excitations $a_{1}(1640)$
and $a_{2}(1700)$ in the $\rho\pi\,S$ and $\rho\pi\,D$ waves, respectively
(masses and widths fixed to the values from the PDG Tanabashi  _et al._
(2018), “excited res.”); and (iv) we estimate the effect of further
rescattering by varying mass and width of the $K^{\ast}$ resonance according
to their uncertainties Tanabashi  _et al._ (2018) (“$K^{*}$ parameters”). We
see from Fig. 4 that the TS model systematically yields a slightly smaller
$\mathcal{R}^{2}_{\text{red}}$ than the Breit-Wigner model.

Figure 4:  $\mathcal{R}_{\mathrm{red,BW}}^{2}$ for the Breit-Wigner model vs.
$\mathcal{R}_{\mathrm{red,TS}}^{2}$ for the TS model. See main text for
details on their definition. The main fit shown in Fig. 2 in the main text, is
represented by the red cross, the gray dashed line indicates
$\mathcal{R}_{\mathrm{red,BW}}^{2}=\mathcal{R}_{\mathrm{red,TS}}^{2}$. Blue
crosses correspond to systematic studies using different data points and green
dots show the fit results with a modified model of the $f_{0}\pi\,P$ wave.
Fits with a modified lineshape of the $K^{*}$ resonance are shown by the
orange diamonds. The result of the bootstrap analysis is shown by the filled
ellipses which cover $68\,\%$, and, $95\,\%$ of the obtained sample,
respectively; the fit to the bootstrap-sample mean is represented by the brown
point.

##  III The cross-channel projection integral

The contribution to the dispersion term for the
$J^{PC}=1^{++}\,f_{0}\,\pi\,P$-wave in Eq. (1) that manifests the peaking
structure from TS is the $K^{*}\bar{K}\,S$-wave. It reads:

$\displaystyle
F_{f_{0}\pi\,P}^{\\{1\\}}(s,\sigma_{\\{1\\}})=k(s,\sigma_{{\\{1\\}}})\bigg{(}C_{W}^{\\{1\\}}(s)+\frac{1}{2\pi}\int_{4m_{K}^{2}}^{\infty}\frac{\rho_{K\bar{K}}(\sigma)\hat{F}_{f_{0}\pi\,P,K^{*}\bar{K}\,S}^{\\{1\\}}(s,\sigma)}{k(s,\sigma)(\sigma-\sigma_{\\{1\\}}-i\varepsilon)}\,\mathrm{d}\sigma\bigg{)}\frac{1}{\sqrt{1+k^{2}(s,\sigma_{{\\{1\\}}})R^{2}}}\,,$
(3)

where $k(s,\sigma)=\sqrt{\lambda(s,\sigma,m_{\pi}^{2})/(4s)}$, the phase space
factor
$\rho_{K\bar{K}}(\sigma)=\sqrt{\lambda(\sigma,m_{K}^{2},m_{K}^{2})}/(8\pi\sigma)$,
and $\lambda$ is the Källén function,
$\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2(xy+yz+zx)$. To limit the growth of the
$P$-wave amplitude, we add a customary Blatt-Weisskopf barrier factor Blatt
and Weisskopf (1952); Von Hippel and Quigg (1972) with the size parameter
$R=5\,$GeV-1. The projection integral reads:

$\displaystyle\hat{F}_{f_{0}\pi\,P,K^{*}\bar{K}\,S}^{\\{1\\}}(s,\sigma_{1})=\frac{\sigma_{1}}{\lambda(s,\sigma_{1},m_{\pi}^{2})\sqrt{\lambda(\sigma_{1},m_{K}^{2},m_{K}^{2})}}\int_{\sigma_{3}^{-}(s,\sigma_{1})}^{{\sigma_{3}^{+}(s,\sigma_{1})}}\frac{Q(\sqrt{s},\sqrt{\sigma_{1}},\sqrt{\sigma_{3}})}{(\sqrt{\sigma_{3}}+\sqrt{s}-m_{3})(\sqrt{\sigma_{3}}+\sqrt{s}+m_{3})}\frac{\mathrm{d}\sigma_{3}}{D_{K^{\ast}}(\sigma_{3})}\,,$
(4)

where we integrate over the phase space of the $K\bar{K}\pi$ system with
$\sigma_{3}=m^{2}(K\pi)$ and $\sigma_{1}=m^{2}(K\bar{K})$. The integration
limits as calculated as follows:

$\displaystyle\sigma_{3}^{\pm}(s,\sigma_{1})=m_{K}^{2}+m_{\pi}^{2}+(s-\sigma_{1}-m_{\pi}^{2})/2\pm\sqrt{\lambda(s,\sigma_{1},m_{\pi}^{2})\lambda(\sigma_{1},m_{K}^{2},m_{K}^{2})}/(2\sigma_{1})\,.$
(5)

Beyond the physical region, we use analytic continuation with the $s\to
s+i\epsilon$ prescription Bronzan and Kacser (1963). The function $Q(x,y,z)$
is polynomial in $x$, $y$, and $z$ and arises from the product of Wigner
d-functions (see Eq. (B12) in Ref. Mikhasenko  _et al._ (2018)),

$\displaystyle Q(x,y,z)$
$\displaystyle=m_{1}^{4}m_{3}^{2}+m_{1}^{4}x^{2}+2m_{1}^{4}xz+m_{1}^{4}z^{2}-m_{1}^{2}m_{2}^{2}m_{3}^{2}-3m_{1}^{2}m_{2}^{2}x^{2}-2m_{1}^{2}m_{2}^{2}xz$
(6)
$\displaystyle\quad+m_{1}^{2}m_{2}^{2}z^{2}+m_{1}^{2}m_{3}^{2}x^{2}+4m_{1}^{2}m_{3}^{2}xz-
m_{1}^{2}m_{3}^{2}y^{2}+m_{1}^{2}m_{3}^{2}z^{2}-2m_{1}^{2}x^{3}z$
$\displaystyle\quad+m_{1}^{2}x^{2}y^{2}-2m_{1}^{2}x^{2}z^{2}-2m_{1}^{2}xy^{2}z-2m_{1}^{2}xz^{3}-m_{1}^{2}x^{4}$
$\displaystyle\quad-3m_{1}^{2}y^{2}z^{2}-m_{1}^{2}z^{4}-m_{2}^{2}m_{3}^{2}x^{2}+m_{2}^{2}m_{3}^{2}y^{2}+m_{2}^{2}x^{4}-m_{2}^{2}x^{2}y^{2}$
$\displaystyle\quad-3m_{2}^{2}x^{2}z^{2}+2m_{2}^{4}x^{2}+2m_{2}^{2}xy^{2}z-2m_{2}^{2}x^{3}z-m_{2}^{2}y^{2}z^{2}+m_{3}^{2}x^{2}z^{2}$
$\displaystyle\quad-
m_{3}^{2}y^{2}z^{2}+x^{4}z^{2}-3x^{2}y^{2}z^{2}+x^{2}z^{4}-2xy^{2}z^{3}+2x^{3}z^{3}+2y^{4}z^{2}+y^{2}z^{4}\,,$

with $m_{1}=m_{\pi}$, and $m_{2}=m_{3}=m_{K}$. The term
$D_{K^{*}}(\sigma_{3})$ is the denominator of the $K\pi$ scattering amplitude,
i.e.

$\displaystyle
D_{K^{*}}(\sigma_{3})=m_{K^{*}}^{2}-\sigma_{3}-m_{K^{*}}\Gamma_{K^{*}}(\Sigma_{K^{\ast}}(\sigma_{3})-\text{Re}\,\Sigma(m_{K^{\ast}}^{2}))/\text{Im}\,\Sigma_{K^{\ast}}(m_{K^{\ast}}^{2})\,.$
(7)

We ensure the correct analytic structure using the Chew-Mandelstam function
$\Sigma(\sigma_{3})$ for the absorptive term Tanabashi  _et al._ (2018);
Basdevant and Berger (1977). Note that
$\text{Im}\,\Sigma(\sigma_{3})=\sqrt{\lambda(\sigma_{3},m_{\pi}^{2},m_{K}^{2})}/(16\pi\sigma_{3})$.

##  IV Fit result for all $t^{\prime}$ bins

Figures 5 – 15 show the spin-density matrix elements (SDMEs) of the three
waves selected for the model fit (data points) and the fit results of the TS
model for all $t^{\prime}$-bins. The full fit model (red) is decomposed into
signal (blue) and background (green) as described in the main text. The
intensities are plotted on the diagonal and the complex phase of the
interference parts on the off-diagonal. The 3 rows as well as the 3 columns
correspond to the $1^{++}0^{+}\rho\pi S$, $1^{++}0^{+}f_{0}\pi P$, and
$2^{++}1^{+}\rho\pi D$-waves.

Figure 5: Spin-density matrix elements of the 3 waves selected for the TS
model fit and the corresponding fit results, both shown for the first (lowest)
bin of $t^{\prime}$. The SDMEs as a function of $m_{3\pi}$ are visualized in
the form of a $3\times 3$ upper-triangular matrix of graphs with the partial-
wave intensities on the diagonal and the relative phases between the partial
waves on the off-diagonal. Black crosses correspond to the result of the PWA
in bins of $m_{3\pi}$ and $t^{\prime}$ from Ref. Adolph  _et al._ (2017) with
statistical uncertainties indicated by vertical lines. The data are overlaid
by the TS model curve (red), the contributions from signal (blue) and non-
resonant background (green). The $m_{3\pi}$ fit range is indicated by the
color saturation of data points and fit result. Regions indicated by lower
color saturation were not included in the fit; the model curves in these
regions are extrapolations. Figure 6: Same as Fig. 5, but for $t^{\prime}$-bin
2. Figure 7: Same as Fig. 5, but for $t^{\prime}$-bin 3. Figure 8: Same as
Fig. 5, but for $t^{\prime}$-bin 4. Figure 9: Same as Fig. 5, but for
$t^{\prime}$-bin 5. Figure 10: Same as Fig. 5, but for $t^{\prime}$-bin 6.
Figure 11: Same as Fig. 5, but for $t^{\prime}$-bin 7. Figure 12: Same as Fig.
5, but for $t^{\prime}$-bin 8. Figure 13: Same as Fig. 5, but for
$t^{\prime}$-bin 9. Figure 14: Same as Fig. 5, but for $t^{\prime}$-bin 10.
Figure 15: Same as Fig. 5, but for $t^{\prime}$-bin 11.


# Towards Understanding Underwater Weather Events in Rivers Using Autonomous
Surface Vehicles ††thanks: We greatly acknowledge the support of the NSF IUCRC
1939132 and the University of Pennsylvania’s University Research Foundation
Award.

Alice K. Li, Yue Mao, Sandeep Manjanna, Sixuan Liu,
Jasleen Dhanoa, Bharg Mehta,
Victoria M. Edwards, Fernando Cladera Ojeda, M. Ani Hsieh  GRASP Lab
University of Pennsylvania
Philadephia, USA  Maël Le Men, Eric Sigg,
Douglas J. Jerolmack, Hugo N. Ulloa  Department of Earth and Environmental
Science
University of Pennsylvania
Philadelphia, USA

###### Abstract

Climate change has increased the frequency and severity of extreme weather
events such as hurricanes and winter storms. The complex interplay of floods
with tides, runoff, and sediment creates additional hazards — including
erosion and the undermining of urban infrastructure — consequently impacting
the health of our rivers and ecosystems. Observations of these underwater
phenomena are rare, because satellites and sensors mounted on aerial vehicles
cannot penetrate the murky waters. Autonomous Surface Vehicles (ASVs) provides
a means to track and map these complex and dynamic underwater phenomena. This
work highlights preliminary results of high-resolution data gathering with
ASVs, equipped with a suite of sensors capable of measuring physical and
chemical parameters of the river. Measurements were acquired along the lower
Schuylkill River in the Philadelphia area at high-tide and low-tide
conditions. The data will be leveraged to improve our understanding of changes
in bathymetry due to floods; the dynamics of mixing and stagnation zones and
their impact on water quality; and the dynamics of suspension and resuspension
of fine sediment. The data will also provide insight into the development of
adaptive sampling strategies for ASVs that can maximize the information gain
for future field experiments.

###### Index Terms:

autonomous surface vehicles, extreme weather events, water quality analysis,
adaptive sampling

##  I Introduction

Many of the largest cities have thrived because of their proximity to
coastlines or rivers. The proximity to these large bodies of water provide a
plethora of advantages such as drinking water, hydropower, shipping ports,
transportation, waste disposal, and recreation [1, 2, 3, 4, 5]. Urban
development, however, has negatively impacted these bodies of water, e.g.,
increased pollution, interference with the delivery of water, sediment, and
nutrients, among others [6, 7, 8, 9, 10, 11, 12]. In addition, increasingly
intense rainfall events driven by climate change have altered their
hydroclimates, creating larger floods and longer dry spells. Flooding can
alter the shape and size of river channels [13, 14, 15]. For tidally-
influenced rivers like the lower Schuylkill River in Philadelphia, PA, rising
sea levels further enhance flood risk and ecosystem disruption [16]. Such
flooding can cause erosion that undermines infrastructures such as bridges and
piers [17], as well as the re-suspension of PCB-contaminated legacy sediments
[18].

The movement of water, solutes and sediments through estuaries is governed by
a complex interplay among river currents, tides, buoyancy, and bathymetry [19,
20]. Mud deposition resulting from changes in flow or water chemistry [21] can
shrink the channel cross section, enhancing flood risk [17, 22] and
suffocating bottom-dwelling organisms [23]. These “underwater weather” events,
as described in [24], a term we use to summarize flow and sediment dynamics
occurring under the water surface, are therefore critically important for
understanding how to protect river health. Yet, observations of the relevant
meteorological variables, if available, are typically sparse in terms of both
spatial and temporal coverage. This is because sensors are mounted either in
stationary locations, or on large boats where surveys are infrequent. Based on
the need to improve our understanding of a river’s response to extreme weather
events, the goal of this work is to document underwater weather events in the
lower Schuylkill River in Philadelphia for the first time. Data gathered will
be used to determine the cumulative effects these weather events have on
infrastructure and water quality. In this work, we leverage the mobility of
ASVs to conduct high resolution surveys of the river structure and its
dynamics. This data will also be used to develop long term adaptive sampling
strategies for ASVs, data-driven modeling of the system dynamics, and gaining
new physical insights.

##  II Related Work

Autonomous surface vehicles are increasingly being used to acquire a variety
of in-situ water measurements from oceans, rivers, and lakes. Hitz et al.
provide the design of a customized ASV used to monitor toxic algae blooms
[25]. Their custom-made ASV and winch system weights
$120\text{\,}\mathrm{k}\mathrm{g}$ and allows for lake measurements at depths
of $130\text{\,}\mathrm{m}$. The ASV detailed in [26] acquires water samples
up to $50\text{\,}\mathrm{m}$ in depth, allowing for measurements of physical
water parameters along the water column. In [27], chlorophyll density is
mapped, as an exploring ASV determines regions of interest for a data
collecting ASV to acquire water samples. Partial pressures and pH sensors were
used on board Wave Glider autonomous surface vehicles to sample in the central
California upwelling system [28]. Dunbabin et al. uses a
$16\text{\,}\mathrm{f}\mathrm{t}$ long solar powered ASV for water quality and
greenhouse gas monitoring [29]. Other works detailing ASVs with sensor suites
for water quality and bathymetry measurements can be found in [30, 31, 32, 33,
34, 35] and in a review [36]. In [37], Velasco et al. designed an ASV with a
similar sensor suite to that described in the current work. However, they
focus on highlighting their robust mechanical design for operating around
strong waves of the north coastline of Peru. Most works also focusing on the
mechanical design of their ASVs opt for a catamaran design, similar to that of
the ASVs in the current work, allowing for operation in shallow waters and
increased stability against waves [38, 39]. Yang et al. use unmanned surface
vehicles to understand the impacts that extreme weather has on seabeds
surrounding Taiwan; however, their vehicles are remote controlled rather than
autonomous [40]. Li et al. use surface vehicles to acquire data to better
understand and predict floods in the case of extreme weather events in water
channels. In their work, they only measure vertical profiles of horizontal
flow velocity with an ADCP sensor [41].

As evidenced by the literature, there is growing interest in using ASVs to
acquire water quality and at depth measurements. Nevertheless, to the authors’
knowledge, none directly addresses leveraging autonomy for improving our
understanding of river and sediment dynamics resulting from extreme weather
events, nor establish correlations between measured parameters. In this work,
we describe our on-going efforts towards the development of long-term
autonomous monitoring of underwater weather events in riverine environments.

Figure 1: Left Example paths taken by the ASVs, along and across the
Schuylkill River. Red represents an example lawn-mower coverage pattern by the
custom made ASV. Measurements acquired by this robot include pH, temperature,
nitrate, pressure, barometric pressure, oxidation reduction potential (ORP),
and chlorophyll-a fluorescence, sediment concentration, and bathymetry. Yellow
represents coverage of a larger region by the Clearpath Heron, acquiring
bathymetry measurements. Green represents the approximate path taken by the
custom made ASV across the river, for 2D riverbed reconstruction. Right Lawn-
mower pattern depicted by a yellow rectangle, and green path across the river.
Measurement locations are situated between two meanders in the river, possibly
giving rise to asymmetric riverbed shape. Figure background courtesy of Google
Maps.

(a) Clearpath Heron ASV with RTK base station.

(b) Customized lightweight ASV, which holds LISST-ABS, In-Situ Aqua Troll 600
sensor, and BlueRobotics bathymetry sensor.

(c) Sensor suite - LISST-ABS sensor (top) In-Situ Aqua Troll 600 sensor.
(bottom)

(d) Tritech Micron Bathymetry sensor on board the Clearpath Heron ASV.

Figure 2: Experimental setup.

##  III Experimental Methods

In this work, we employed two types of ASVs – a Clearpath Heron Autonomous
Surface Vehicle with a bathymetry sensor and a custom made ASV with a suite of
sensors – to carry out surveys in the Schuylkill river in Philadelphia. Figure
1 shows the paths traveled by the Clearpath Heron (yellow) and custom made ASV
(red). Figure 1 shows the region of coverage from a zoomed out view. The
custom made ASV, which had the sensor suite on board, to be described in
further detail below, surveyed a segment of the river under high-tide and low-
tide conditions, covering a spatial region of around $90\text{\,}\mathrm{m}$
$\times$ $40\text{\,}\mathrm{m}$ enclosed by GPS coordinates
$(39.94364,-75.19973)$, $(39.94403,-75.19943)$, $(39.94356,-75.19856)$, and
$(39.9432,-75.19883)$. The Clearpath Heron, with greater battery capacity
covered a larger spatial region of around $100\text{\,}\mathrm{m}$ $\times$
$75\text{\,}\mathrm{m}$, and was used to acquire bathymetry data. The custom
made ASV was also used to measure bathymetry measurements perpendicular to the
flow of the river, to generate 2D cross-sectional plots.

The Clearpath Heron is a differentially driven ASV with a catamaran design. It
has a built-in GPS, a 6 DOF inertial measurement unit (IMU) and a camera
(Figure 2a). To improve GPS measurements, we have installed a NovAtel Real
Time Kinematic (RTK) GPS on board the ASV. Coupling this GPS with an RTK base
station provides localization of $\pm$ $1\text{\,}\mathrm{c}\mathrm{m}$
accuracy. The custom made ASV, Figure 2b, is also differentially driven, with
a catamaran design consisting of two pontoons and two T200 BlueRobotics
thrusters. It has an Intel NUC i7, a Pixhawk 4 controller, and on-board
Pixhawk GPS with accuracy $\pm$ $2\text{\,}\mathrm{m}$. A TP-link router along
with the NUC creates an ad-hoc network. This allows us to communicate with the
NUC to control the ASV via ROS over WiFi, while it is in the river. The ASV
can also be remotely operated via a
$2.4\text{\,}\mathrm{G}\mathrm{H}\mathrm{z}$ RC controller at a range of
around $500\text{\,}\mathrm{m}$.

The suite of sensors that are mounted on the custom ASVs include (see Figure
2c): (1) a BlueRobotics Ping Sonar Altimeter and an Echosounder bathymetry
sensor with accuracy of $0.5\text{\,}\%$ of the range, with sensing range of
$0.5\text{\,}\mathrm{m}$ \- $70\text{\,}\mathrm{m}$. This bathymetry sensor
samples at $10\text{\,}\mathrm{H}\mathrm{z}$. (2) In-Situ Aqua Troll 600
sensor, capable of measuring numerous physical and chemical parameters
including: pH, temperature, nitrate, pressure, barometric pressure, oxidation
reduction potential (ORP), and chlorophyll-a fluorescence, sampling at
$0.5\text{\,}\mathrm{H}\mathrm{z}$. (3) LISST-ABS sediment concentration
sensor, a light-scattering acoustic sensor for measuring particle
concentration, sampling at $1\text{\,}\mathrm{H}\mathrm{z}$. On board the
Clearpath Heron is a Tritech Micron Echosounder bathymetry sensor, capable of
measuring up to a $50\text{\,}\mathrm{m}$ water depth with a resolution of
$1\text{\,}\mathrm{m}\mathrm{m}$ (Figure 2d) is mounted at the rear of the
Clearpath Heron ASV. The use of non-identical bathymetry sensors is due to
availability.

For this work, we ran the ASVs autonomously along a boustrophedon pattern,
i.e., a lawn mower pattern, in the region of interest, as depicted in Figure
1. Physical and chemical measurements with corresponding GPS coordinates were
acquired to construct spatial maps of these phenomena. Measurements were
acquired under two tidal conditions, as data could potentially be extrapolated
to predict river conditions during floods if salient trends exist. Data taken
within $1.5$ hours of peak high and low tide conditions are assumed to be
influenced by the affects of high and low tidal conditions respectively. Low
tide occurred on $2022-08-09$ at $12:50$ PM at a height of
$0.5\text{\,}\mathrm{f}\mathrm{t}$, and high tide occurred on $2022-08-03$ at
$11:46$ AM at a height of $5.8\text{\,}\mathrm{f}\mathrm{t}$. To remove as
much variability from ambient temperature conditions, all data was collected
at similar times of the day – 2 hours within noon. 3 data sets were collected
for low tide and 2 data sets were collected for high tide. The number of data
sets were limited by weather and time for data acquisition. Ambient
temperatures for low and high tidal conditions were around
$33\text{\,}{}^{\circ}\mathrm{C}$ \- $34\text{\,}{}^{\circ}\mathrm{C}$ and
$31\text{\,}{}^{\circ}\mathrm{C}$ \- $32\text{\,}{}^{\circ}\mathrm{C}$
respectively. Point measurements were acquired to generate spatial maps of pH,
temperature, nitrate, pressure, barometric pressure, ORP, and chlorophyll-a
fluorescence, sediment concentration, and depth over the region. Only plots
with significant relevance and conclusive results are included.

Figure 3: 2D reconstruction of the riverbed.

Figure 4: A triangulation based bi-linear interpolation approach was used to
interpolate between point measurements acquired along the yellow path, shown
in Figure 1, to produce a spatially continuous map of depth data.

##  IV Results and Discussion

2D cross-sectional areas of the riverbed were constructed by guiding the
custom ASV perpendicular to the flow of the river. Two sets of data were
acquired and averaged, via a sliding window to generate Figure 4. The left
hand side of the plot is closest to the pier shown in Figure 1, while the
right hand side is closest to the bank. Figure 4 shows an interpolated spatial
plot representing depth across the river, using the Clearpath Heron, the
yellow path in 1. Given that low latitude represents data closer to the pier,
it is in agreement with Figure 4, which shows that the riverbed is deeper
closer to the pier. It should be noted that just before and just after the
pier, there is a bend in the river. Such meandering can give rise to the non-
symmetric riverbed structure – the greater velocities of the river at the
outer edge of the bend causes greater erosion, and therefore gives rise to
greater riverbed depth. While the riverbed depth should end at
$0\text{\,}\mathrm{m}$ on each side, we are limited in mobility close to the
banks, as the sensor heads are positioned around
$15\text{\,}\mathrm{c}\mathrm{m}$ below the water surface.

A triangulation based bi-linear interpolation approach was used to interpolate
between point measurements acquired along the paths shown in Figure 1, to
produce a spatially continuous map. The red path is that of the custom-made
ASV; the yellow is that of the Clearpath Heron. The final coverage regions are
not identical because the ASVs motions are affected by the river currents.
Since data rates for all sensors are different, data were synchronized to
match corresponding timestamps. The spatial measurements were converted from
GPS coordinates to meters, that covered a rectangular region of 90 m x 40 m.
They were also rotated for easier readability and alignment with the final
coordinate system.

Across all datasets, as shown in Figure 6, pH ranged from 7.73 - 8.35. pH
conditions at low-tide were higher than those at high-tide; however, for any
given run, the maximum deviation in pH value was 0.18, which is not
significant. Next, temperatures in Figure 7 ranged from
$28.3\text{\,}{}^{\circ}\mathrm{C}$ to $31.3\text{\,}{}^{\circ}\mathrm{C}$.
For both tidal conditions, river water temperatures remained lower than
ambient temperatures, staying between $30.2\text{\,}{}^{\circ}\mathrm{C}$ \-
$31.2\text{\,}{}^{\circ}\mathrm{C}$ during low-tide when ambient temperatures
were around $33\text{\,}{}^{\circ}\mathrm{C}$ \-
$34\text{\,}{}^{\circ}\mathrm{C}$, and between
$28.3\text{\,}{}^{\circ}\mathrm{C}$ \- $29.5\text{\,}{}^{\circ}\mathrm{C}$
during high-tide when ambient temperatures were at
$31\text{\,}{}^{\circ}\mathrm{C}$ \- $32\text{\,}{}^{\circ}\mathrm{C}$.
Nitrate concentration levels shown in Figure 5, were between
$116.6\text{\,}\mathrm{m}\mathrm{g}\mathrm{/}\mathrm{L}$ \-
$124.9\text{\,}\mathrm{m}\mathrm{g}\mathrm{/}\mathrm{L}$, with a maximum
deviation of $4.77\text{\,}\mathrm{m}\mathrm{g}\mathrm{/}\mathrm{L}$ for any
given run. In general, at high-tide, there were higher nitrate levels.
Pressure remained close to ambient, at 14.7psi, while barometric pressures
were around $7.74\text{\,}\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$ \-
$8.36\text{\,}\mathrm{m}\mathrm{m}\mathrm{H}\mathrm{g}$. ORP levels ranged
from $146\text{\,}\mathrm{m}\mathrm{V}$ \- $262\text{\,}\mathrm{m}\mathrm{V}$.
The most deviation in ORP for any run was $40.4\text{\,}\mathrm{m}\mathrm{V}$.
Chlorophyll-a Fluorescence in Figure 8 ranged from
$0.003\text{\,}\mathrm{R}\mathrm{F}\mathrm{U}$ and
$2.054\text{\,}\mathrm{R}\mathrm{F}\mathrm{U}$, with a maximum change for all
runs of $1.34\text{\,}\mathrm{R}\mathrm{F}\mathrm{U}$. It should be noted that
chlorophyll-a fluorescence measurements were higher on the day of
experimentation where water levels were observed to clearer and less murky.
The differences in murkiness levels were visible to the eye, however, Secchi
disks will be used in future work for more precise quantification of river
water visibility. Chlorophyll-a fluorescence measurements had the most
variability for this spatial region, while the remaining parameters (pH,
temperature, nitrate, pressure, barometric pressure, ORP, sediment
concentration) did not show significant variability across the region. This
suggests that if spatial relationships between these parameters are to be
studied in more detail in the future, ASVs should be deployed to cover a
larger region.

(a) High-tide at 11:47AM on 08-03-2022.

(b) High-tide at 1:34AM on 08-03-2022.

(c) Low-tide at 11:07AM on 08-09-2022.

(d) Low-tide at 12:03PM on 08-09-2022.

(e) Low-tide at 12:19PM on 08-09-2022.

Figure 5: Interpolated nitrate measurements.

(a) High-tide at 11:47AM on 08-03-2022.

(b) High-tide at 1:34PM on 08-03-2022.

(c) Low-tide at 11:07PM on 08-09-2022.

(d) Low-tide at 12:03PM on 08-09-2022.

(e) Low-tide at 12:19PM on 08-09-2022.

Figure 6: Interpolated pH measurements.

(a) High-tide at 11:47AM on 08-03-2022.

(b) High-tide at 1:34PM on 08-03-2022.

(c) Low-tide at 11:07PM on 08-09-2022.

(d) Low-tide at 12:03PM on 08-09-2022.

(e) Low-tide at 12:19PM on 08-09-2022.

Figure 7: Interpolated temperature measurements.

(a) High-tide at 11:47AM on 08-03-2022.

(b) High-tide at 1:34PM on 08-03-2022.

(c) Low-tide at 11:07PM on 08-09-2022.

(d) Low-tide at 12:03PM on 08-09-2022.

(e) Low-tide at 12:19PM on 08-09-2022.

Figure 8: Interpolated chlorophyll-a fluorescence measurements. Figure 9:
Correlations between pH and temperature, with Pearson correlation coefficient,
$R=0.957$. H represents high-tide, L represents low-tide. Figure 10:
Correlations between chlorophyll-a fluorescence and nitrate concentration,
with Pearson correlation coefficient, $R=-0.77$. Figure 11: Correlations
between temperature and nitrate concentration, with Pearson correlation
coefficient, $R=-0.903$.

Next, correlations between various pairs of data were also determined. The
Pearson correlation coefficient, denoted by R, was used for this. It measures
the linear relationship between two sets of data. $R$ ranges between -1 and
+1, where $R=1$ or $R=-1$ describes an exact linear relationship. If R is
positive, then an increase in the first parameter gives rise to an increase in
the second, while if R is negative, then an increase in the first parameter
gives rise to a decrease in the second. Figures 9, 10 and 11 illustrate such
relationships.

Figure 9 shows the relationship between temperature and pH. From this data,
$R=0.957$, pH increases with an increase in temperature. This relationship can
be explained as follows: with an increase in temperature comes a decrease in
dissolution of carbon dioxide into the water. More carbon dioxide introduced
in the water means greater river acidity. This means that with increased river
water temperature, we have greater pH – a more basic measurement. For
chlorophyll-a fluorescence vs nitrate concentration shown in Figure 10,
$R=-0.77$. Again, the relationship obtained is to as expected. Chlorophyll
measurement is a proxy for the existence of phytoplankton. Phytoplankton
consumes nitrates, so when algae blooms develop, the level of nitrates in the
river decreases. This gives rise to the negative correlation between the two
parameters. Finally, nitrate levels as a function of temperature are plotted
in Figure 11. For this, $R=-0.903$. This correlation agrees with the
relationship of that in rivers, where nitrate levels tend to be lower during
the winter and higher during summer months.

##  V Conclusions and Future Work

Autonomous Surface Vehicles were deployed in the Schuylkill river in
Philadelphia. A custom made ASV surveyed the river autonomously in a lawn-
mower pattern and measured pH, temperature, nitrate, pressure, barometric
pressure, ORP, and chlorophyll-a fluorescence, sediment concentration, and
bathymetry data over a $90\text{\,}\mathrm{m}$ $\times$
$40\text{\,}\mathrm{m}$ region. A Clearpath Heron was used to acquire
additional depth measurements across a $100\text{\,}\mathrm{m}$ $\times$
$75\text{\,}\mathrm{m}$ region. Data were acquired under high-tide and low-
tide conditions, on two different days, to allow for measurements during
similar times of the day, and therefore similar ambient temperatures.
Experimental results show that there are clear trends among the same tidal
conditions, namely, high-tide data and low-tide data. Collectively, when
analyzing all high-tide and low-tide data together, Pearson correlations
between pH, temperature, nitrate, and chlorophyll-a fluorescence meet
expectations, while correlations between remaining parameters are either not
conclusive with current data or do not exist. 2D reconstructions of the
riverbed indicate that there is some asymmetry in the river bed, due to the
meandering of the river before and after the region of measurement.

Future surveys will target river responses to floods, to understand regime
shifts in hydrodynamics and water chemistry and to document erosion hotspots
that threaten infrastructure. Long-term surveys will examine annual cycles of
sediment, salt, and flow in the river, and changes in the bathymetry that may
influence flood capacity of the Lower Schuylkill River. Experiments will also
be repeated under additional tidal conditions _i.e._ during hours between peak
high-tide and low-tide, and over larger spatial regions. In addition,
measurements will be acquired at various depths, to generate 3D
reconstructions of the parameters acquired in this study. This data would be
used to better understand the fluid flow and sediment dynamics in the river,
as well as develop adaptive sampling strategies.

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