BABAR-PUB-21/001

SLAC-PUB-17606

††thanks: Deceased††thanks: Deceased††thanks: Deceased††thanks:
Deceased††thanks: Deceased††thanks: Deceased

The BABAR Collaboration

#  Light meson spectroscopy from Dalitz plot analyses of $\eta_{c}$ decays to
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ produced in two-photon interactions

J. P. Lees  V. Poireau  V. Tisserand  Laboratoire d’Annecy-le-Vieux de
Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3, F-74941
Annecy-Le-Vieux, France  E. Grauges  Universitat de Barcelona, Facultat de
Fisica, Departament ECM, E-08028 Barcelona, Spain  A. Palano  INFN Sezione di
Bari, I-70126 Bari, Italy  G. Eigen  University of Bergen, Institute of
Physics, N-5007 Bergen, Norway  D. N. Brown  Yu. G. Kolomensky  Lawrence
Berkeley National Laboratory and University of California, Berkeley,
California 94720, USA  M. Fritsch  H. Koch  T. Schroeder  Ruhr Universität
Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany  R. Cheaibb
C. Heartyab T. S. Mattisonb J. A. McKennab R. Y. Sob Institute of Particle
Physics${}^{\,a}$; University of British Columbiab, Vancouver, British
Columbia, Canada V6T 1Z1  V. E. Blinovabc A. R. Buzykaeva V. P. Druzhininab V.
B. Golubevab E. A. Kozyrevab E. A. Kravchenkoab A. P. Onuchinabc S. I.
Serednyakovab Yu. I. Skovpenab E. P. Solodovab K. Yu. Todyshevab Budker
Institute of Nuclear Physics SB RAS, Novosibirsk 630090a, Novosibirsk State
University, Novosibirsk 630090b, Novosibirsk State Technical University,
Novosibirsk 630092c, Russia  A. J. Lankford  University of California at
Irvine, Irvine, California 92697, USA  B. Dey  J. W. Gary  O. Long  University
of California at Riverside, Riverside, California 92521, USA  A. M. Eisner  W.
S. Lockman  W. Panduro Vazquez  University of California at Santa Cruz,
Institute for Particle Physics, Santa Cruz, California 95064, USA  D. S. Chao
C. H. Cheng  B. Echenard  K. T. Flood  D. G. Hitlin  J. Kim  Y. Li  D. X. Lin
T. S. Miyashita  P. Ongmongkolkul  J. Oyang  F. C. Porter  M. Röhrken
California Institute of Technology, Pasadena, California 91125, USA  Z. Huard
B. T. Meadows  B. G. Pushpawela  M. D. Sokoloff  L. Sun  Now at: Wuhan
University, Wuhan 430072, China  University of Cincinnati, Cincinnati, Ohio
45221, USA  J. G. Smith  S. R. Wagner  University of Colorado, Boulder,
Colorado 80309, USA  D. Bernard  M. Verderi  Laboratoire Leprince-Ringuet,
Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France  D. Bettonia C.
Bozzia R. Calabreseab G. Cibinettoab E. Fioravantiab I. Garziaab E. Luppiab V.
Santoroa INFN Sezione di Ferraraa; Dipartimento di Fisica e Scienze della
Terra, Università di Ferrarab, I-44122 Ferrara, Italy  A. Calcaterra  R. de
Sangro  G. Finocchiaro  S. Martellotti  P. Patteri  I. M. Peruzzi  M. Piccolo
M. Rotondo  A. Zallo  INFN Laboratori Nazionali di Frascati, I-00044 Frascati,
Italy  S. Passaggio  C. Patrignani  Now at: Università di Bologna and INFN
Sezione di Bologna, I-47921 Rimini, Italy  INFN Sezione di Genova, I-16146
Genova, Italy  B. J. Shuve  Harvey Mudd College, Claremont, California 91711,
USA  H. M. Lacker  Humboldt-Universität zu Berlin, Institut für Physik,
D-12489 Berlin, Germany  B. Bhuyan  Indian Institute of Technology Guwahati,
Guwahati, Assam, 781 039, India  U. Mallik  University of Iowa, Iowa City,
Iowa 52242, USA  C. Chen  J. Cochran  S. Prell  Iowa State University, Ames,
Iowa 50011, USA  A. V. Gritsan  Johns Hopkins University, Baltimore, Maryland
21218, USA  N. Arnaud  M. Davier  F. Le Diberder  A. M. Lutz  G. Wormser
Université Paris-Saclay, CNRS/IN2P3, IJCLab, F-91405 Orsay, France  D. J.
Lange  D. M. Wright  Lawrence Livermore National Laboratory, Livermore,
California 94550, USA  J. P. Coleman  E. Gabathuler  D. E. Hutchcroft  D. J.
Payne  C. Touramanis  University of Liverpool, Liverpool L69 7ZE, United
Kingdom  A. J. Bevan  F. Di Lodovico  Now at: King’s College, London, WC2R
2LS, UK  R. Sacco  Queen Mary, University of London, London, E1 4NS, United
Kingdom  G. Cowan  University of London, Royal Holloway and Bedford New
College, Egham, Surrey TW20 0EX, United Kingdom  Sw. Banerjee  D. N. Brown  C.
L. Davis  University of Louisville, Louisville, Kentucky 40292, USA  A. G.
Denig  W. Gradl  K. Griessinger  A. Hafner  K. R. Schubert  Johannes
Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
R. J. Barlow  Now at: University of Huddersfield, Huddersfield HD1 3DH, UK  G.
D. Lafferty  University of Manchester, Manchester M13 9PL, United Kingdom  R.
Cenci  A. Jawahery  D. A. Roberts  University of Maryland, College Park,
Maryland 20742, USA  R. Cowan  Massachusetts Institute of Technology,
Laboratory for Nuclear Science, Cambridge, Massachusetts 02139, USA  S. H.
Robertsonab R. M. Seddonb Institute of Particle Physics${}^{\,a}$; McGill
Universityb, Montréal, Québec, Canada H3A 2T8  N. Neria F. Palomboab INFN
Sezione di Milanoa; Dipartimento di Fisica, Università di Milanob, I-20133
Milano, Italy  L. Cremaldi  R. Godang  Now at: University of South Alabama,
Mobile, Alabama 36688, USA  D. J. Summers  University of Mississippi,
University, Mississippi 38677, USA  P. Taras  Université de Montréal, Physique
des Particules, Montréal, Québec, Canada H3C 3J7  G. De Nardo  C. Sciacca
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Università di
Napoli Federico II, I-80126 Napoli, Italy  G. Raven  NIKHEF, National
Institute for Nuclear Physics and High Energy Physics, NL-1009 DB Amsterdam,
The Netherlands  C. P. Jessop  J. M. LoSecco  University of Notre Dame, Notre
Dame, Indiana 46556, USA  K. Honscheid  R. Kass  Ohio State University,
Columbus, Ohio 43210, USA  A. Gaza M. Margoniab M. Posoccoa G. Simiab F.
Simonettoab R. Stroiliab INFN Sezione di Padovaa; Dipartimento di Fisica,
Università di Padovab, I-35131 Padova, Italy  S. Akar  E. Ben-Haim  M. Bomben
G. R. Bonneaud  G. Calderini  J. Chauveau  G. Marchiori  J. Ocariz
Laboratoire de Physique Nucléaire et de Hautes Energies, Sorbonne Université,
Paris Diderot Sorbonne Paris Cité, CNRS/IN2P3, F-75252 Paris, France  M.
Biasiniab E. Manonia A. Rossia INFN Sezione di Perugiaa; Dipartimento di
Fisica, Università di Perugiab, I-06123 Perugia, Italy  G. Batignaniab S.
Bettariniab M. Carpinelliab Also at: Università di Sassari, I-07100 Sassari,
Italy  G. Casarosaab M. Chrzaszcza F. Fortiab M. A. Giorgiab A. Lusianiac B.
Oberhofab E. Paoloniab M. Ramaa G. Rizzoab J. J. Walsha L. Zaniab INFN Sezione
di Pisaa; Dipartimento di Fisica, Università di Pisab; Scuola Normale
Superiore di Pisac, I-56127 Pisa, Italy  A. J. S. Smith  Princeton University,
Princeton, New Jersey 08544, USA  F. Anullia R. Facciniab F. Ferrarottoa F.
Ferronia Also at: Gran Sasso Science Institute, I-67100 L’Aquila, Italy  A.
Pilloniab G. Pireddaa INFN Sezione di Romaa; Dipartimento di Fisica,
Università di Roma La Sapienzab, I-00185 Roma, Italy  C. Bünger  S. Dittrich
O. Grünberg  M. He$s\overline{s}$ T. Leddig  C. Vo$s\overline{s}$ R. Waldi
Universität Rostock, D-18051 Rostock, Germany  T. Adye  F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United
Kingdom  S. Emery  G. Vasseur  IRFU, CEA, Université Paris-Saclay, F-91191
Gif-sur-Yvette, France  D. Aston  C. Cartaro  M. R. Convery  J. Dorfan  W.
Dunwoodie  M. Ebert  R. C. Field  B. G. Fulsom  M. T. Graham  C. Hast  W. R.
Innes  P. Kim  D. W. G. S. Leith  S. Luitz  D. B. MacFarlane  D. R. Muller  H.
Neal  B. N. Ratcliff  A. Roodman  M. K. Sullivan  J. Va’vra  W. J. Wisniewski
SLAC National Accelerator Laboratory, Stanford, California 94309 USA  M. V.
Purohit  J. R. Wilson  University of South Carolina, Columbia, South Carolina
29208, USA  A. Randle-Conde  S. J. Sekula  Southern Methodist University,
Dallas, Texas 75275, USA  H. Ahmed  St. Francis Xavier University, Antigonish,
Nova Scotia, Canada B2G 2W5  M. Bellis  P. R. Burchat  E. M. T. Puccio
Stanford University, Stanford, California 94305, USA  M. S. Alam  J. A. Ernst
State University of New York, Albany, New York 12222, USA  R. Gorodeisky  N.
Guttman  D. R. Peimer  A. Soffer  Tel Aviv University, School of Physics and
Astronomy, Tel Aviv, 69978, Israel  S. M. Spanier  University of Tennessee,
Knoxville, Tennessee 37996, USA  J. L. Ritchie  R. F. Schwitters  University
of Texas at Austin, Austin, Texas 78712, USA  J. M. Izen  X. C. Lou
University of Texas at Dallas, Richardson, Texas 75083, USA  F. Bianchiab F.
De Moriab A. Filippia D. Gambaab INFN Sezione di Torinoa; Dipartimento di
Fisica, Università di Torinob, I-10125 Torino, Italy  L. Lanceri  L. Vitale
INFN Sezione di Trieste and Dipartimento di Fisica, Università di Trieste,
I-34127 Trieste, Italy  F. Martinez-Vidal  A. Oyanguren  IFIC, Universitat de
Valencia-CSIC, E-46071 Valencia, Spain  J. Albertb A. Beaulieub F. U.
Bernlochnerb G. J. Kingb R. Kowalewskib T. Lueckb I. M. Nugentb J. M. Roneyb
R. J. Sobieab N. Tasneemb Institute of Particle Physics${}^{\,a}$; University
of Victoriab, Victoria, British Columbia, Canada V8W 3P6  T. J. Gershon  P. F.
Harrison  T. E. Latham  Department of Physics, University of Warwick, Coventry
CV4 7AL, United Kingdom  R. Prepost  S. L. Wu  University of Wisconsin,
Madison, Wisconsin 53706, USA

###### Abstract

We study the processes
$\gamma\gamma\mbox{$\rightarrow$}\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ using a data sample of 519
${\mathrm{fb}^{-1}}$ recorded with the BABAR detector operating at the SLAC
PEP-II asymmetric-energy $e^{+}e^{-}$ collider at center-of-mass energies at
and near the $\mathchar 28935\relax(nS)$ ($n=2,3,4$) resonances. This is the
first observation of the decay
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ and we
measure the branching fraction
$\Gamma(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-})/(\Gamma(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})=0.644\pm
0.039_{\rm stat}\pm 0.032_{\rm sys}$. Significant interference is observed
between
$\gamma\gamma\mbox{$\rightarrow$}\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
and the non-resonant two-photon process
$\gamma\gamma\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$. A
Dalitz plot analysis is performed of $\eta_{c}$ decays to
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$. Combined with our previous
analysis of $\eta_{c}\mbox{$\rightarrow$}K\bar{K}\pi$, we measure the
$K^{*}_{0}(1430)$ parameters and the ratio between its $\eta^{\prime}K$ and
$\pi K$ couplings. The decay
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
is dominated by the $f_{0}(2100)$ resonance, also observed in $J/\psi$
radiative decays. A new $a_{0}(1700)\mbox{$\rightarrow$}\eta\pi$ resonance is
observed in the
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
channel. We also compare $\eta_{c}$ decays to $\eta$ and $\eta^{\prime}$ final
states in association with scalar mesons as they relate to the identification
of the scalar glueball.

###### pacs:

13.25.Gv, 14.40.Pq, 14.40.Df, 14.40.Be

##  I Introduction

Scalar mesons remain a puzzle in light meson spectroscopy: they have complex
structure, and there are too many states to be accommodated within the quark
model without difficulty polosa . In particular, the structure of the isospin
I=$1\over 2$ $K\pi$ $S$-wave is still poorly understood, which limits the
precision of measurements involving a $K\pi$ system in the final state,
including recent searches for $C\\!P$ violation in $B$ meson decay cp , and
studies of new exotic resonances zs  and charmed mesons bs .

Decays of the $\eta_{c}$, the lightest pseudoscalar $c\bar{c}$ state, provide
a window on light meson states. The BABAR experiment first performed a Dalitz
plot analysis of
$\eta_{c}\mbox{$\rightarrow$}\mbox{${K^{+}}$}K^{-}\mbox{${\pi^{0}}$}$ and
$\eta_{c}\mbox{$\rightarrow$}\mbox{${K^{+}}$}K^{-}\eta$ using an isobar model
Lees:2014iua . The analysis reported the first observation of
$K^{*}_{0}(1430)\mbox{$\rightarrow$}K\eta$, and observed that $\eta_{c}$
decays into three pseudoscalars are dominated by intermediate scalar mesons.
This newly observed $K^{*}_{0}(1430)$ decay mode was expected to be small and
in fact was not observed in the study of $K^{-}p\mbox{$\rightarrow$}K^{-}\eta
p$ interactions lass_keta . More recently, the BABAR experiment performed a
measurement of the I=$1\over 2$ $K\pi$ $S$-wave amplitude from a Dalitz plot
analyses of $\eta_{c}\mbox{$\rightarrow$}K\bar{K}\pi$ Lees:2015zzr . Further
information on the properties of the $K^{*}_{0}(1430)$ resonance has been
obtained by the CLEO experiment in an analysis of the
$D^{+}\mbox{$\rightarrow$}K^{-}\mbox{${\pi^{+}}$}\mbox{${\pi^{+}}$}$ decay
Bonvicini:2008jw , and by the BESIII experiment, which observed its decay to
$K\eta^{\prime}$ using $\chi_{c1}$ decays to
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ Ablikim:2014tww .

The existence of gluonium states is still an open issue for Quantum
Chromodynamics (QCD). Lattice QCD calculations predict the lightest gluonium
states to have quantum numbers $J^{PC}=0^{++}$ and $2^{++}$ and to be in the
mass region below 2.5 ${\mathrm{GeV}}/c^{2}$ lattice . In particular, the
$J^{PC}=0^{++}$ glueball is predicted to have a mass around 1.7
${\mathrm{GeV}}/c^{2}$. Searches for these states have been performed using
many supposed “gluon rich” reactions such as radiative decays of the heavy
quarkonium states ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ kopke ; Dobbs
and $\mathchar 28935\relax{(1S)}$ Lees:2018qrk . However, despite intense
experimental searches, there has been no conclusive experimental observation
klempt ; ochs . The identification of the scalar glueball is further
complicated by possible mixing with standard $q\bar{q}$ states. The broad
$f_{0}(500)$, $f_{0}(1370)$ mink , $f_{0}(1500)$ amsler1 ; amsler2 ,
$f_{0}(1710)$ gg ; Gui:2012gx  and possibly $f_{0}(2100)$ Ablikim:2013hq  have
been suggested as scalar glueball candidates. In the BESIII partial wave
analysis of the radiative ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ decay
to $\eta\eta$ Ablikim:2013hq , the authors conclude that the production rates
of $f_{0}(1710)$ and $f_{0}(2100)$ are both about one order of magnitude
larger than that of the $f_{0}(1500)$ and no clear evidence is found for
$f_{0}(1370)$. A feature of the scalar glueball is that its $s\bar{s}$ decay
mode should be favored with respect to $u\bar{u}$ or $d\bar{d}$ chano ; chao .

In the present analysis, we consider the three-body $\eta_{c}$ decays to
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, using two-photon interactions,
$e^{+}e^{-}\mbox{$\rightarrow$}e^{+}e^{-}\gamma^{*}\gamma^{*}\mbox{$\rightarrow$}e^{+}e^{-}\eta_{c}$.
If both of the virtual photons are quasi-real, the allowed $J^{PC}$ values of
any produced resonances are $0^{\pm+}$, $2^{\pm+}$, $4^{\pm+}$… Yang . Angular
momentum conservation, parity conservation, and charge conjugation invariance
imply that these quantum numbers also apply to these final states. The
possible presence of a gluonic component of the $\eta^{\prime}$ meson, due to
the so-called gluon anomaly, has been discussed in recent years Harland-
Lang:2013ncy ; Bass:2018xmz . A comparison of the $\eta$ and $\eta^{\prime}$
content of $\eta_{c}$ decays might yield information on the possible gluonic
content of resonances decaying to ${\pi^{+}}$${\pi^{-}}$ or
$\mbox{${K^{+}}$}K^{-}$. The
$\gamma\gamma\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
process has been recently studied by the Belle experiment Xu:2018uye  but no
Dalitz plot analysis was performed.

This article is organized as follows. In Sec. II, a brief description of the
BABAR detector is given. Section III is devoted to the event reconstruction
and data selection. In Sec. IV, we describe the efficiency and resolution
studies, while in Sec. V we report the measurement of the $\eta_{c}$ branching
fraction. In Sec. VI we describe the Dalitz plot analysis methodology, and in
Secs. VII, VIII, and IX we analyze $\eta_{c}$ decays to
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, respectively. The results are
summarized in Sec. X.

##  II The BABAR detector and dataset

The results presented here are based on the full data set collected with the
BABAR detector at the PEP-II asymmetric-energy $e^{+}e^{-}$ collider located
at SLAC, and correspond to an integrated luminosity of 519
${\mathrm{fb}^{-1}}$ BaBar:2013agn  recorded at center-of-mass energies at and
near the $\mathchar 28935\relax(nS)$ ($n=2,3,4$) resonances. The BABAR
detector is described in detail in ref. BABARNIM . Charged particles are
detected, and their momenta are measured, by means of a five-layer, double-
sided microstrip detector and a 40-layer drift chamber, both operating in the
1.5 T magnetic field of a superconducting solenoid. Photons are measured and
electrons are identified in a CsI(Tl) crystal electromagnetic calorimeter.
Charged-particle identification is provided by the measurement of specific
energy loss in the tracking devices, and by an internally reflecting, ring-
imaging Cherenkov detector. The pions tracking efficiency increases from 98%
to 100% in the momentum range 0.5-3 GeV/c while the average kaon
identification efficiency is 84%. Muons and $K^{0}_{\scriptscriptstyle L}$
mesons are detected in the instrumented flux return of the magnet. Monte Carlo
(MC) simulated events geant , with reconstructed sample sizes of the order
$10^{3}$ times larger than the corresponding data samples, are used to
evaluate the signal efficiency and to determine background features. Two-
photon events are simulated using the GamGam MC generator BabarZ . In this
article, the inclusion of charge-conjugate processes is implied, unless stated
otherwise.

##  III  Event reconstruction and selection

###  III.1 Reconstruction of the $\eta^{\prime}h^{+}h^{-}$ final state

We first study the reactions

$\gamma\gamma\mbox{$\rightarrow$}\eta^{\prime}h^{+}h^{-},$ (1)

where $h^{+}h^{-}$ indicates a ${\pi^{+}}$${\pi^{-}}$ or
$\mbox{${K^{+}}$}K^{-}$ system. The selection criteria are optimized for the
$\eta_{c}$ signal, as described below. The $\eta^{\prime}$ is reconstructed in
the two decay modes $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$,
$\rho^{0}\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta\mbox{$\rightarrow$}\gamma\gamma$. To reconstruct these final states we
select events in which the $e^{+}$ and $e^{-}$ beam particles are scattered at
small angles, and hence are undetected, ensuring that both virtual photons are
quasi-real. We consider photon candidates with reconstructed energy in the
electromagnetic calorimeter greater than 100 MeV. All pairs of photon
candidates are combined, assuming they originate from the $e^{+}e^{-}$
interaction region, and pairs with invariant-mass within $\pm 20$
${\mathrm{MeV}}/c^{2}$ ($\pm 150$ ${\mathrm{MeV}}/c^{2}$) of the neutral pion
($\eta$ meson) mass are considered ${\pi^{0}}$ ($\eta$) candidates. We
consider events with exactly 4 well-measured charged-particle tracks with
transverse momentum greater than 0.1 GeV/c, and fit them to a common vertex,
which must be within the $e^{+}e^{-}$ interaction region and have a $\chi^{2}$
fit probability greater than 0.1%. Tracks are identified as either charged
kaons or pions using a high-efficiency algorithm that rejects more than half
the background with negligible signal loss. A track can be identified as both
kaon or pion (or neither) at this point. For the
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ selection, we allow the
presence of only two $\gamma$ candidates, where ${\pi^{0}}$candidates are
excluded. For the
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ we
require exactly one $\eta$ candidate, no more than three additional background
photon candidates, and no ${\pi^{0}}$ candidate in the event. These selections
are optimized on the data using as reference the $\eta_{c}$ signal.

To reconstruct $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ decays, we
consider ${\pi^{+}}$${\pi^{-}}$ pairs in the mass region
$0.620<m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})<0.875$ ${\mathrm{GeV}}/c^{2}$.
Each of these $\rho^{0}$ candidates is combined with all $\gamma$ candidates,
and any combination with invariant-mass in the range
$0.935<m(\rho^{0}\gamma)<0.975$ ${\mathrm{GeV}}/c^{2}$ is considered an
$\eta^{\prime}$ candidate. We compute the angle $\theta_{\gamma}$, defined as
the angle between the ${\pi^{+}}$ and the $\gamma$ in the
${\pi^{+}}$${\pi^{-}}$ rest frame. The distribution of $\theta_{\gamma}$ is
expected to be proportional to $\sin^{2}\theta_{\gamma}$ Rosner . We thus scan
the $\rho^{0}\gamma$ mass spectrum with varying selection on
$|\cos\theta_{\gamma}|$ and obtain a small reduction of the combinatorial
background by requiring $|\cos\theta_{\gamma}|<0.85$. The above selection
reduces the $\eta^{\prime}$ signal and background yields by 3% and 17%,
respectively.

To improve the mass experimental resolution, the $\eta^{\prime}$ four-momentum
is constructed by adding the momenta of the ${\pi^{+}}$, ${\pi^{-}}$, and
$\gamma$, and computing the $\eta^{\prime}$ energy by assigning the Particle
Data Group (PDG) PDG  nominal mass. This method, tested on MC simulations,
improves the resolution by $\approx 20\%$.

Figure 1: Distributions of the transverse momenta of the (a,b)
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ and (c,d)
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ systems for events satisfying all other
selection criteria, in which the $\eta^{\prime}$ is reconstructed in the (a,c)
$\rho^{0}\gamma$ and (b,d) $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ decay
modes. The data are represented by points with error bars, and the $\eta_{c}$
MC simulation by solid (red) histograms with arbitrary normalization. The
(blue) dashed lines indicate the selection used to isolate two-photon event
candidates.

To reconstruct
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
decays, we perform a kinematic fit to the $\eta$ candidate, and require the
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ mass to be within $\pm 2\sigma$ of
the fitted $\eta^{\prime}$ mass ($956.8~{}\pm~{}0.5$) ${\mathrm{MeV}}/c^{2}$,
where $\sigma=2.9$ ${\mathrm{MeV}}/c^{2}$ is the width of the resolution
function describing the $\eta^{\prime}$ signal. Similarly, to improve the
experimental resolution, the $\eta^{\prime}$ four momentum is constructed by
adding the momenta of the ${\pi^{+}}$, ${\pi^{-}}$, and $\eta$, and computing
the $\eta^{\prime}$ energy by assigning the PDG mass.

Background arises mainly from random combinations of particles from
$e^{+}e^{-}$ annihilation, from other two-photon processes, and from events
with initial-state photon radiation (ISR). The ISR background is dominated by
events with a single high-energy photon recoiling against the reconstructed
hadronic system, which in the mass region of interest is typically a
$J^{PC}=1^{--}$ resonance isr . We discriminate against ISR events by
requiring the recoil mass $M^{2}_{\mathrm{\rm
rec}}\equiv(p_{e^{+}e^{-}}-p_{\rm rec})^{2}>2$ GeV2/$c^{4}$, where
$p_{e^{+}e^{-}}$ is the four-momentum of the initial state $e^{+}e^{-}$ and
$p_{\rm rec}$ is the reconstructed four-momentum of the candidate
$\eta^{\prime}(\eta)h^{+}h^{-}$ system.

We define $p_{T}$ as the magnitude of the transverse momentum of the
$\eta^{\prime}h^{+}h^{-}$ system, in the $e^{+}e^{-}$ rest frame, with respect
to the beam axis. Well reconstructed two-photon events with quasi-real photons
are expected to have low values of $p_{T}$. Substantial background arises from
$\gamma\gamma\mbox{$\rightarrow$}2h^{+}2h^{-}$ events, combined with a
background photon candidate. These are removed by requiring
$\mbox{$p_{T}$}(2h^{+}2h^{-})>0.1$ ${\mathrm{GeV}}/c$.

We retain events with $p_{T}$ below a maximum value that is optimized with
respect to the $\eta_{c}$ signal for each decay mode. We produce
$\eta^{\prime}h^{+}h^{-}$ invariant-mass spectra with different maximum
$p_{T}$ values, and fit them to extract the number of $\eta_{c}$ signal events
($N_{s}$) (defined as the 2.93-3.03 ${\mathrm{GeV}}/c^{2}$ interval) and the
number of background events underneath the $\eta_{c}$ signal ($N_{b}$). We
then compute the purity, defined as $P=N_{s}/(N_{s}+N_{b})$, the figure of
merit $S=N_{s}/\sqrt{N_{s}+N_{b}}$, and their product, $PS$.

####  III.1.1 Reconstruction of the
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ final state

For the final selection of the
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ final state, we require
all four charged tracks to be positively identified as pions, using an
algorithm based on multivariate analysis BDT  that is more than 98% efficient
for the tracks in the sample, while suppressing kaons by a factor of at least
seven.

Figures 1(a) and 1(b) show the $p_{T}$ distributions for selected events in
the charmonium region. This region is defined as reconstructed invariant-mass
$m(\eta^{\prime}(\eta)h^{+}h^{-})>2.7$ ${\mathrm{GeV}}/c^{2}$. In the case of
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ an upper mass requirement
$m(\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})<3.5$
${\mathrm{GeV}}/c^{2}$ is applied because of the large number of combinations
produced by the presence of the $\gamma$. The data are compared with
expectations from $\eta_{c}$ signal MC simulations; a signal from two-photon
production is observed in the data in both cases, and is particularly clean
for
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.
In a scan of the $S$, $P$, and $PS$ variables as functions of the maximum
$p_{T}$ value, we observe a broad maximum of $S$ starting at 0.05
${\mathrm{GeV}}/c$ for the $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$
decay candidates, and a maximum of $PS$ at 0.15 ${\mathrm{GeV}}/c$ for the
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates. We require $\mbox{$p_{T}$}<0.05$ ${\mathrm{GeV}}/c$ and
$\mbox{$p_{T}$}<0.15$ ${\mathrm{GeV}}/c$, respectively, as indicated by the
dashed lines in the figures.

Figure 2: Invariant-mass distributions of (a) $\rho^{0}\gamma$ and (b)
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ for
$\gamma\gamma\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates satisfying all other selection criteria. Corresponding (c)
$\rho^{0}\gamma$ and (d) $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-
mass distributions for
$\gamma\gamma\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$
candidates. The shaded areas indicate the $\eta^{\prime}$ selections.

Figures 2(a) and 2(b) show the $\rho^{0}\gamma$ and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-mass distributions,
respectively, for events satisfying all selection criteria except that on
these masses. Clear $\eta^{\prime}$ signals are visible, and the shaded
regions indicate the selection windows, (0.935-0.975) ${\mathrm{GeV}}/c^{2}$
for $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ and (0.948-0.966)
${\mathrm{GeV}}/c^{2}$ for
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.
Figures 3(a,b) show the $\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
invariant-mass spectra for the selected events in the data. Prominent
$\eta_{c}$ signals are observed, and there is some activity in the
$\eta_{c}(2S)$ mass region.

Figure 3: The $\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-
mass spectra for selected events with (a)
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ and (b)
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.
The $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ invariant-mass spectra for selected
events with (c) $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ and (d)
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.

Figure 4: Distributions of the transverse momentum
$p_{T}(\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ for selected
$\gamma\gamma\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates with (a) $\eta\mbox{$\rightarrow$}\gamma\gamma$ and (b)
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$,
in the charmonium mass region. The data are represented by the points with
error bars, and the $\eta_{c}$ MC simulation as solid (red) histograms with
arbitrary normalizations. The dashed (blue) lines indicate the selection used
to isolate two-photon event candidates.

If there are multiple candidates in the same event, we retain them all. The
fraction of events having two combinations in the $\eta_{c}$ mass region is 3%
(and 3.4% in $\eta_{c}$ signal MC simulations) for
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
with $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$. No multiple candidates
are found for
$\eta^{\prime}\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$
or any of the other final states discussed below.

####  III.1.2 Reconstruction of the $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ final
state

For the $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ final state, we require the two
charged tracks assigned to the $\eta^{\prime}$ decay to be positively
identified as pions and the other two to be positively identified as kaons.
The algorithm is more than 92% efficient for kaon identification, while
suppressing pions by a factor of at least five. The $p_{T}$ distributions for
events in the charmonium region, compared with MC $\eta_{c}$ signal
simulations, are shown in Figs. 1(c)-(d), where signals of the two-photon
reaction can be seen. To minimize systematic uncertainties in the measurements
of the branching fractions, the same $p_{T}$ requirements as for the
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ final state are used,
indicated by the dashed lines in the figures.

The corresponding $\eta^{\prime}$ signals for this final state are shown in
Figs. 2(c)-(d), and the $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ invariant-mass
spectra are shown in Figs. 3(c)-(d). Prominent $\eta_{c}$ signals with low
background are present in both invariant-mass spectra with possible weak
activity in the $\eta_{c}(2S)$ mass region. The decay
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ is observed
here for the first time.

###  III.2 Reconstruction of the $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
final state

We study the reaction

$\gamma\gamma\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$},$
(2)

where $\eta\mbox{$\rightarrow$}\gamma\gamma$ and
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$.

####  III.2.1 $\eta\mbox{$\rightarrow$}\gamma\gamma$

For reaction (2), where $\eta\mbox{$\rightarrow$}\gamma\gamma$, we again
consider well-measured charged-particle tracks with transverse momenta greater
than 0.1 ${\mathrm{GeV}}/c$ and photons with energy greater than 0.1
${\mathrm{GeV}}$, and each pair of $\gamma$’s is kinematically fitted to the
${\pi^{0}}$ and $\eta$ hypotheses. We require exactly two selected tracks, fit
them to a common vertex, and require the fitted vertex to be within the
interaction region and the $\chi^{2}$ probability of the fit to be greater
than 0.1%. We retain events having exactly one $\eta$ candidate, no
${\pi^{0}}$ candidates, and no more than three background $\gamma$’s.

The two charged tracks are required to be loosely identified as pions. Most
ISR events are removed by requiring $M^{2}_{\mathrm{\rm
rec}}\equiv(p_{e^{+}e^{-}}-p_{\rm rec})^{2}>2$ GeV2/$c^{4}$. Further
background is due to the presence of ISR events from
$\psi(2S)\mbox{$\rightarrow$}\eta
J/\psi\mbox{$\rightarrow$}\eta\mu^{+}\mu^{-}$, where the two muons are
misidentified as pions. This background is efficiently removed by vetoing
events having two loosely identified muons. Background from the process
$\gamma\gamma\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ is
removed by requiring
$\mbox{$p_{T}$}(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})>0.05$
${\mathrm{GeV}}/c$.

The $p_{T}$ distribution for such events in the charmonium mass region is
compared with $\eta_{c}$ signal MC simulation in Fig. 4(a), where a clear
signal of the two-photon reaction is observed. Optimizing the $\eta_{c}$
figure of merit ($S$) and purity ($P$), we require $\mbox{$p_{T}$}<0.1$
${\mathrm{GeV}}/c$. The resulting $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
invariant-mass spectrum is shown in Fig. 6(a), where the $\eta_{c}$ signal can
be observed together with some weak activity in the $\eta_{c}(2S)$ mass
region.

####  III.2.2
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$

For reaction (2), where
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$,
we require exactly four well-measured charged-particle tracks with the vertex
$\chi^{2}$ fit probability greater than 0.1%. In order to have sensitivity to
low momentum ${\pi^{0}}$ mesons, we consider photons with energy greater than
30 ${\mathrm{MeV}}/c^{2}$. We allow no more than two kinematically fitted
${\pi^{0}}$ candidates and no more than five background $\gamma$’s. Candidate
$\gamma\gamma\mbox{$\rightarrow$}2\mbox{${\pi^{+}}$}2\mbox{${\pi^{-}}$}$
events are removed by requiring
$\mbox{$p_{T}$}(2\mbox{${\pi^{+}}$}2\mbox{${\pi^{-}}$})>0.05$
${\mathrm{GeV}}/c$. Background ISR events are removed by requiring
$M^{2}_{\mathrm{\rm rec}}\equiv(p_{e^{+}e^{-}}-p_{\rm rec})^{2}>2$
GeV2/$c^{4}$. All four charged tracks are required to be loosely identified as
pions.

The $\eta$ candidates are reconstructed by combining every pair of oppositely
charged tracks with each of the ${\pi^{0}}$ candidates in the event. The
resulting ${\pi^{+}}$${\pi^{-}}$${\pi^{0}}$ invariant-mass spectrum is shown
in Fig. 5. A clean $\eta$ signal can be seen; we select candidates in the mass
region
$538<m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$})<557~{}\mbox{${\mathrm{MeV}}/c^{2}$}$.
The $\eta$ is then reconstructed by adding the momentum three-vectors of the
three pions and computing the $\eta$ energy using its nominal PDG mass.

The $p_{T}$ distribution for such events in the charmonium mass region is
compared with $\eta_{c}$ signal MC simulation in Fig. 4(b), where a clear
signal of the two-photon reaction is observed. In this case, a maximum of the
$PS$ figure of merit leads to the requirement $\mbox{$p_{T}$}<0.1$
${\mathrm{GeV}}/c$. The resulting $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
invariant-mass spectrum is shown in Fig. 6(b), where the $\eta_{c}$ signal can
be observed together with some weak activity in the $\eta_{c}(2S)$ mass
region.

Figure 5: Distribution of the reconstructed ${\pi^{+}}$${\pi^{-}}$${\pi^{0}}$
mass for selected
$\gamma\gamma\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidate events. The shaded area indicates the $\eta$ selection region.

Figure 6: The $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-mass
spectra for selected events with (a) $\eta\mbox{$\rightarrow$}\gamma\gamma$
and (b)
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$.

##  IV  Efficiency and $\eta_{c}$ invariant-mass resolution

To compute the reconstruction and selection efficiency, MC signal events are
generated using a detailed detector simulation geant ; BabarZ  in which the
$\eta_{c}$ mesons decay uniformly in phase space. These simulated events are
reconstructed and analyzed in the same manner as data. We define the helicity
angle $\theta_{H}$ as the angle formed by the $h^{+}$ (where $h=\pi,K$), in
the $h^{+}h^{-}$ rest frame, and the $\eta^{\prime}$ ($\eta$) direction in the
$h^{+}h^{-}\eta^{\prime}$ ($h^{+}h^{-}\eta$) rest frame. For each final state,
we compute the raw efficiency in 50$\times$50 intervals of the invariant-mass,
$m(h^{+}h^{-})$, and $\cos\theta_{H}$, as the ratio of reconstructed to
generated events in that interval.

To smoothen statistical fluctuations, the efficiency maps are parameterized as
follows. We first fit the efficiency as a function of $\cos\theta_{H}$ in each
of the 100 ${\mathrm{MeV}}/c^{2}$ wide intervals of $m(h^{+}h^{-})$, using
Legendre polynomials up to $L=12$:

$\epsilon(\cos\theta_{H})=\sum_{L=0}^{12}a_{L}(m)Y^{0}_{L}(\cos\theta_{H}),$
(3)

where $m$ denotes the $h^{+}h^{-}$ invariant-mass. For a given value of
$m(h^{+}h^{-})$, the efficiency is interpolated linearly between adjacent mass
intervals.

Figure 7 shows the resulting efficiency maps $\epsilon(m,\cos\theta_{H})$ for
the four $\eta^{\prime}h^{+}h^{-}$ final states, and Fig. 8 shows the maps for
the two $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ final states. The small
regions of very low efficiency near $|\cos\theta_{H}|\sim 1$ are the result of
the difficulty of reconstructing $K^{\pm}$ mesons with laboratory momentum
less than $\approx$ 200 ${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, and
${\pi^{\pm}}$ mesons with laboratory momentum less than $\approx$100
${\mathrm{\,Me\kern-1.00006ptV\\!/}c}$, due to energy loss in the beam pipe
and inner-detector material.

Figure 7: Parametrized detection efficiencies in the $\cos\theta_{H}\ vs.\
m(h^{+}h^{-})$ plane for simulated (a)
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$, (b)
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
(c) $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$, and (d)
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
events. The average value of the efficiency is shown in each interval.

The mass resolution is determined from the distribution of the difference
($\Delta m$) between the generated and reconstructed $\eta^{\prime}h^{+}h^{-}$
or $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-mass values. The
$\Delta m$ distributions are parameterized by the sum of a Crystal Ball cb
and a Gaussian function, which describe well the distributions, and have root-
mean-squared values of: 11.5 ${\mathrm{MeV}}/c^{2}$ for
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$; 13.9 ${\mathrm{MeV}}/c^{2}$
for $\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$;
8.2 ${\mathrm{MeV}}/c^{2}$ for $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$; 12.2 ${\mathrm{MeV}}/c^{2}$
for $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$;
15.9 ${\mathrm{MeV}}/c^{2}$ for $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta\mbox{$\rightarrow$}\gamma\gamma$; and 13.8 ${\mathrm{MeV}}/c^{2}$ for
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$.

##  V Yields and branching fractions

In this section, we fit the invariant-mass distributions to obtain the numbers
of selected $\eta_{c}$ events, $N_{\eta^{\prime}K^{+}K^{-}}$,
$N_{\eta^{\prime}\pi^{+}\pi^{-}}$, and $N_{\eta\pi^{+}\pi^{-}}$, for each
$\eta^{\prime}$ or $\eta$ decay mode. We then use the
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ and
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ yields to compute the
ratio of branching fractions for $\eta_{c}$ to the
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ and
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ final states. This ratio
is computed as

$\begin{split}{\cal R}=&\frac{{\cal
B}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-})}{{\cal
B}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})}\\\
=&\frac{N_{\eta^{\prime}K^{+}K^{-}}}{N_{\eta^{\prime}\pi^{+}\pi^{-}}}\frac{\epsilon_{\eta^{\prime}\pi^{+}\pi^{-}}}{\epsilon_{\eta^{\prime}K^{+}K^{-}}}\end{split}$
(4)

for each $\eta^{\prime}$ decay mode, where
$\epsilon_{\eta^{\prime}K^{+}K^{-}}$ and
$\epsilon_{\eta^{\prime}\pi^{+}\pi^{-}}$ are the corresponding weighted
efficiencies described in the following Sec. V.2.

###  V.1 Fits to the invariant-mass spectra

We determine $N_{K^{+}K^{-}\eta^{\prime}}$ and
$N_{\pi^{+}\pi^{-}\eta^{\prime}}$ from $\eta_{c}$ decays by performing binned
$\chi^{2}$ fits to the $\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ and
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-mass spectra, in
the 2.7-3.3 ${\mathrm{GeV}}/c^{2}$ mass region, separately for the two
$\eta^{\prime}$ decay modes. In these fits, the $\eta_{c}$ signal contribution
is described by a simple Breit-Wigner (BW) function convolved with a fixed
resolution function described above, with $\eta_{c}$ parameters fixed to PDG
values PDG . An additional BW function is used to describe the residual
background from ISR $J/\psi$ events, and the remaining background is
parameterized by a $2^{nd}$ order polynomial. The fitted
$\eta^{\prime}h^{+}h^{-}$ invariant-mass spectra are shown in Fig. 9. The fits
generally describe the data well, although the fit to the
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ invariant-mass spectrum for
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
(Fig. 9(d)), which has low statistics, appears to the eye to have a somewhat
distorted lineshape. For this fit, we add two additional parameters by leaving
free the parameters of the Gaussian component of the resolution function. To
minimize the dependence of the $N$’s on the fit quality, the $\eta_{c}$ signal
yields are obtained by integrating the data over the $\eta_{c}$ signal region
after subtracting the fitted backgrounds.

Statistical errors on the $\eta_{c}$ yields are evaluated by generating, from
each invariant-mass spectrum, 500 new spectra by random Poisson fluctuations
of the content of each bin. The generated mass spectra are fitted using the
same model as for the original one and the resulting distributions of the
$\eta_{c}$ subtracted yields are fitted using a Gaussian function, whose
$\sigma$ is taken as the statistical uncertainty. The resulting yields and
$\chi^{2}$ per degree of freedom for the fits, $\chi^{2}/{\rm ndf}$ are
reported in Table 1.

Figure 8: Parametrized detection efficiencies in the $\cos\theta_{H}\ vs.\
m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ plane for simulated
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ events
with (a) $\eta\mbox{$\rightarrow$}\gamma\gamma$ and (b)
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$.
The average value of the efficiency is shown in each interval.

Figure 9: Invariant-mass distributions of selected (left)
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ and (right)
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ candidates for (top)
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ and (bottom)
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.
The lines are the results from the fits described in the text.

We test the fitting procedure by leaving free the $\eta_{c}$ parameters and
find agreement, within the errors, with world averages. For the decay
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
however, the fits without interference do not describe the data well for
either $\eta$ decay mode. Leaving free the $\eta_{c}$ parameters, the fits
return masses shifted down by $\approx 10$ ${\mathrm{MeV}}/c^{2}$ with respect
to PDG averages. We test the possibility of interference effects of the
$\eta_{c}$ with each non-resonant two-photon process Zhang:2012tj , modifying
the fitting function by defining

$f(m)=|A_{\rm nres}|^{2}+|A_{\eta_{c}}|^{2}+c\cdot 2Re(A_{\rm
nres}A^{*}_{\eta_{c}}),$ (5)

where $A_{\rm nres}$ is the non-resonant amplitude with $|A_{\rm nres}|^{2}$
described by a $2^{nd}$ order polynomial; the coherence factor $c$ is the
fraction of the non-resonant events that are true two-photon production of the
same final state; the resonant contribution is $A_{\eta_{c}}=\alpha\cdot
BW(m)\cdot\exp(i\phi)$, where $BW(m)$ is a simple Breit-Wigner with parameters
fixed to PDG values; and $\alpha$, $\phi$, and $c$ are free parameters. The
sum of $f(m)$ and the ${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$
contribution is convolved with the experimental resolution.

Fits with interference and fixed PDG parameters give values of $\chi^{2}/{\rm
ndf}=77/54$ ($p$-value=2.2%) and $\chi^{2}/{\rm ndf}=46/54$ ($p$-value=77%)
for $\eta\mbox{$\rightarrow$}\gamma\gamma$ and
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$
decay modes, respectively. The fitted relative phases are $\phi=1.41\pm
0.02_{\rm stat}\pm 0.02_{\rm sys}$ rad and $\phi=1.26\pm 0.03_{\rm stat}\pm
0.02_{\rm sys}$ rad. Systematic uncertainties are related to the use of
$\eta_{c}$ fixed parameters and on errors in the background shape. The fits,
on the other hand, show little sensitivity to the $c$ parameter. The fitted
invariant-mass spectra are shown in Fig. 10, where reasonable descriptions of
the data are evident. As a comparison we also fit the two mass spectra with no
interference and fixed $\eta_{c}$ parameters and obtain the dotted lines
distributions shown in Fig. 10 with corresponding $\chi^{2}/{\rm ndf}=160/55$
and $\chi^{2}/{\rm ndf}=139/55$, respectively.

We find that the interference model does not produce significant improvements
in the description of the data for final states that include an
$\eta^{\prime}$. As a cross check, we reanalyze the data reported in Ref.
Lees:2014iua , and find no evidence for such interference effects also for the
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${K^{+}}$}K^{-}$ decay mode.

Systematic uncertainties on the yields due to the fitting procedure are
estimated by varying the $\eta_{c}$ parameters according to the PDG
uncertainties. An additional uncertainty of 4% is assigned to the yield for
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ with
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
due to the variation of the resolution function. We also take the integral of
each full function used to describe the $\eta_{c}$ as an estimate of the
yield, and take the difference as the systematic uncertainty. The quadratic
sums of these uncertainties are given in Table 1.

Figure 10:  Invariant-mass spectra for selected
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ candidate events with (a)
$\eta\mbox{$\rightarrow$}\gamma\gamma$ and (b)
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$.
The solid (red) lines represent the fits including interference described in
the text. The dashed (blue) line represents the fitted non-resonant
components. The dotted lines represent the fits without interference.

###  V.2 Branching fractions

We estimate $\epsilon_{\eta^{\prime}K^{+}K^{-}}$ and
$\epsilon_{\eta^{\prime}\pi^{+}\pi^{-}}$ for the $\eta_{c}$ signals using the
2-D raw efficiency functions described in Sec. IV. Each event is first
weighted by $1/\epsilon(m,\cos\theta_{H})$. Since the backgrounds below the
$\eta_{c}$ signals have different distributions in the Dalitz plot, we perform
a sideband subtraction by assigning an additional weight of $+1$ to events in
the $\eta_{c}$ signal region, defined as the (2.93-3.03)
${\mathrm{GeV}}/c^{2}$ mass region, and a weight $-1$ to events in the
sideband regions, (2.77-2.87) ${\mathrm{GeV}}/c^{2}$ and (3.09-3.19)
${\mathrm{GeV}}/c^{2}$. The weights in the sideband regions are scaled by a
small amount to match the fitted $\eta_{c}$ signal/background ratio, and added
to those in the signal region, to produce the weighted yields shown in Table
1.

Systematic uncertainties on the efficiencies have been evaluated as follows.
The uncertainty due to the limited MC statistics is computed by generating 500
new efficiency tables, obtained from the original tables by random variation,
according to a Poisson distribution, of the generated and reconstructed MC
yields in each cell. The distributions of the resulting weights are fitted
using a Gaussian function whose $\sigma$ values are taken as systematic
uncertainties and are listed in Table 1. To estimate an uncertainty on the
method of sideband subtraction, we use the average weights in the signal
region, and take the difference as an uncertainty. The quadratic sums of these
uncertainties are given in Table 1.

Table 1: Information for the evaluation of the branching fractions. The reported yields are obtained from the integration of the $\eta_{c}$ signal after background subtraction in the $\eta_{c}$ signal region. The first error is statistical, the second systematic.  Final state | yield | weight | weighted yields | $\chi^{2}/{\rm ndf}$
---|---|---|---|---
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ ($\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$) | $1160\pm 57\pm 47$ | $17.37\pm 0.28$ | $20149\pm 990\pm 878$ | 51/55
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ ($\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$) | $473\pm 29\pm 3$ | $26.79\pm 0.35$ | $12672\pm 777\pm 184$ | 58/55
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ ($\eta^{\prime}\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\eta$) | $619\pm 35\pm 11$ | $18.42\pm 0.18$ | $11401\pm 645\pm 231$ | 72/55
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ ($\eta^{\prime}\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\eta$) | $249\pm 20\pm 11$ | $30.77\pm 0.40$ | $7662\pm 615\pm 353$ | 90/53

We label with ${\cal R}_{1}(\rho^{0}\gamma)$ and ${\cal
R}_{2}(\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ the measurements of the
branching fraction for the two $\eta^{\prime}$ decay modes. In each case, the
numerator and denominator involve the same number of charged tracks and
$\gamma$’s, so the systematic uncertainties on their reconstruction
efficiencies cancel in the ratio. The only difference is the presence of two
kaons in the numerator and two pions in the denominator. The uncertainties in
the particle identification efficiencies are correlated; we assign a
systematic uncertainty of 1% to the identification of each kaon and 0.5% to
each pion. Table 2 summarizes the largest systematic uncertainties on the
branching fraction, which arise from MC statistics, the use of the full
fitting function in extracting the yield (labelled full-BW), the sideband
subtraction in the efficiencies (labelled no-sideband), and the kaon/pion
identification (labelled PID).

Table 2: Summary of the systematic uncertainties on the branching fraction.  ${\cal R}$ | MC stat. | full-BW | no-sideband | PID | Total
---|---|---|---|---|---
${\cal R}_{1}(\rho^{0}\gamma)$ | 0.029 | 0.014 | 0.003 | 0.014 | 0.035
${\cal R}_{2}(\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ | 0.034 | 0.066 | 0.019 | 0.015 | 0.078

Adding the systematic uncertainties in quadrature, we obtain the following
values of the branching ratios:

${\cal R}_{1}(\rho^{0}\gamma)=0.629\pm 0.049_{\rm stat}\pm 0.035_{\rm sys},$
(6) ${\cal R}_{2}(\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})=0.672\pm
0.066_{\rm stat}\pm 0.078_{\rm sys},$ (7)

and an average value of

$\frac{\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-})}{\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})}=0.644\pm
0.039_{\rm stat}\pm 0.032_{\rm sys}.$ (8)

##  VI Dalitz plot analyses

We perform Dalitz plot analyses of the
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$, and
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ systems in the $\eta_{c}$ mass
region using unbinned maximum likelihood fits. The likelihood function is
written as

$\displaystyle\mathcal{L}=$ $\displaystyle\prod_{n=1}^{N}$
$\displaystyle\bigg{[}$ $\displaystyle f_{\rm
sig}\cdot\epsilon(x^{\prime}_{n},y^{\prime}_{n})\frac{\sum_{i,j}c_{i}c_{j}^{*}A_{i}(x_{n},y_{n})A_{j}^{*}(x_{n},y_{n})}{\sum_{i,j}c_{i}c_{j}^{*}I_{A_{i}A_{j}^{*}}}$
(9) $\displaystyle+(1-f_{\rm
sig})\frac{\sum_{i}k_{i}B_{i}(x_{n},y_{n})}{\sum_{i}k_{i}I_{B_{i}}}\bigg{]},$

where:

  * •

$N$ is the number of events in the signal region;

  * •

$f_{\rm sig}$ is the fraction of those events attributed to $\eta_{c}$ decays;

  * •

for the $n$-th event, $x_{n}=m^{2}(\eta/\eta^{\prime}h^{+})$,
$y_{n}=m^{2}(\eta/\eta^{\prime}h^{-})$, and

  * •

$\epsilon(x^{\prime}_{n},y^{\prime}_{n})$ is the efficiency, parameterized as
a function of $x^{\prime}_{n}=m(h^{+}h^{-})$ and
$y^{\prime}_{n}=\cos\theta_{H}$ (see Sec. IV);

  * •

$c_{i}$ is the complex amplitude of the $i-$th signal component; the $c_{i}$
are free parameters of the fit;

  * •

for the $n$-th event, $A_{i}(x_{n},y_{n})$ describe the $i-th$ complex signal-
amplitude contribution;

  * •

$k_{i}$ is the magnitude of the $i-$th background component; the $k_{i}$
parameters are obtained by fitting the sideband regions;

  * •

for the $n$-th event, $B_{i}(x_{n},y_{n})$ is the probability-density function
of the $i$-th background contribution; we assume that interference between
signal and background amplitudes can be ignored;

  * •

$I_{A_{i}A_{j}^{*}}=\int
A_{i}(x,y)A_{j}^{*}(x,y)\epsilon(m(h^{+}h^{-}),\cos\theta_{H})\ {\rm d}x{\rm
d}y$ and $I_{B_{i}}~{}=~{}\int B_{i}(x,y){\rm d}x{\rm d}y$ are normalization
integrals; numerical integration is performed on phase-space generated events.

Amplitudes are parameterized as described in Refs. Asner:2003gh  and ds . They
include a relativistic Breit-Wigner function having a variable width modulated
by the Blatt-Weisskopf blatt  spin form factors and the relevant spin-angular
information. Note that these factors are both one for scalar resonances.

The efficiency-corrected fractional contribution $f_{i}$ due to resonant or
non-resonant contribution (NR) is defined as follows:

$f_{i}=\frac{|c_{i}|^{2}\int|A_{i}(x_{n},y_{n})|^{2}{\rm d}x{\rm
d}y}{\int|\sum_{j}c_{j}A_{j}(x,y)|^{2}{\rm d}x{\rm d}y}.$ (10)

The $f_{i}$ do not necessarily sum to 100% because of interference effects.
The uncertainty for each $f_{i}$ is evaluated by propagating the full
covariance matrix obtained from the fit.

The search for the amplitudes contributing to the signal or background is
performed by starting with the largest resonance observed in the mass
projections, which is taken as the reference amplitude with $c_{1}=1$ and
phase zero. We then add, one by one, possible processes that could contribute
to the decay, testing for an increase in the likelihood value. Amplitudes are
discarded if no significant improvement in the likelihood ($\Delta(-2\log{\cal
L})>2$) is obtained. Each excluded resonance is reiterated many times in
combination with other possible resonant contributions. Where possible,
resonance parameters are left free, for comparison with existing values;
otherwise, they are fixed to PDG values.

Table 3 summarizes the information on the structure of the samples used in the
Dalitz analyses. Yields and purities are computed in the $\eta_{c}$ signal
region, defined as the mass ranges (2.93-3.03) ${\mathrm{GeV}}/c^{2}$ for
$\eta^{\prime}h^{+}h^{-}$ and (2.92-3.02) ${\mathrm{GeV}}/c^{2}$ for
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.

Table 3: Information for the Dalitz analysis.  Final state | Decay mode | Yield | Fraction | Purity (%)
---|---|---|---|---
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ | $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ | 656 | 0.705 | $69.7\pm 1.7$
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ | $\eta^{\prime}\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\eta$ | 274 | 0.295 | $85.7\pm 2.0$
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ | $\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ | 2239 | 0.717 | $51.8\pm 1.1$
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ | $\eta^{\prime}\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\eta$ | 883 | 0.283 | $69.0\pm 1.6$
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ | $\eta\mbox{$\rightarrow$}\gamma\gamma$ | 6512 | 0.700 | $58.0\pm 0.6$
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ | $\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$ | 2791 | 0.300 | $52.7\pm 1.0$

The widths of the resonances contributing to the $\eta_{c}$ decays are much
larger than the experimental resolution, and therefore resolution effects are
ignored. The only exception is the $\phi(1020)$ resonance, which contributes
to the background to
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$. We obtain an
enhanced $\phi(1020)$ signal by relaxing the selection criteria and in
particular the $p_{T}$ selection. The resulting $\mbox{${K^{+}}$}K^{-}$ mass
distribution shows a prominent $\phi(1020)$ signal, which is fitted with a
$P$-wave relativistic BW function yielding a width $6.1\pm 0.3$
${\mathrm{MeV}}/c^{2}$. The fitted BW function is used to describe this
contribution to the background.

Each Dalitz plot analysis deals with two sets of data contributing to the
given $\eta_{c}$ final state, with different efficiencies and purities:
$\eta^{\prime}\mbox{$\rightarrow$}\rho^{0}\gamma$ and
$\eta^{\prime}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
for $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}h^{+}h^{-}$,
$\eta\mbox{$\rightarrow$}\gamma\gamma$ and
$\eta\mbox{$\rightarrow$}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}\mbox{${\pi^{0}}$}$
for $\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.
Therefore we use the sum of two different likelihood functions, which share
the free parameters and fitting model. Due to the lack of statistics we do not
separate the contributing backgrounds for the two sets of data.

##  VII Dalitz plot analysis of
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$

Figure 11 shows the Dalitz plot for the selected
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ candidates in
the data, for the two $\eta^{\prime}$ decay modes combined. Figure 12(a)-(b)
shows the two squared mass projections.

Figure 11: Dalitz plot for selected
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ candidates in
the $\eta_{c}$ signal region, summed over the two $\eta^{\prime}$ decay modes.

Figure 12: Squared-mass projections (a) $m^{2}(\mbox{${K^{+}}$}K^{-})$ and (b)
$m^{2}(\eta^{\prime}K^{\pm})$ of the measured
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ Dalitz plot.
The shaded (gray) histograms are the background interpolated from fits to the
two $\eta_{c}$ sidebands. Linear-scale mass projections (c)
$m(\mbox{${K^{+}}$}K^{-})$ and (d) $m(\eta^{\prime}K^{\pm})$, after
subtraction of the background. The solid (red) histograms represent the
results of the fit described in the text (solution (A)). The other histograms
display the contributions from each of the listed components. The
$\eta^{\prime}K^{\pm}$ mass projections have two entries per event.

We observe that this $\eta_{c}$ decay mode is dominated by a diagonal band on
the low mass side of the Dalitz plot. The $m^{2}(\mbox{${K^{+}}$}K^{-})$
spectrum shows a large structure in the region of the $f_{0}(1710)$ resonance.
The combined $m^{2}(\eta^{\prime}K^{\pm})$ invariant-mass spectrum shows a
structure at threshold due to the $K^{*}_{0}(1430)$ accompanied by weaker
resonant structures.

We first fit the two $\eta_{c}$ sidebands separately, using an incoherent sum
of amplitudes, which includes contributions from the $\phi(1020)$,
$\phi(1680)$, $f_{2}^{\prime}(1525)$, $K^{*}_{0}(1430)$, and $K^{*}_{0}(1950)$
resonances. To model the background composition in the $\eta_{c}$ signal
region, we take a weighted average of the two fitted fractional contributions,
and normalize using the results from the fit to the
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ invariant-mass spectrum. The estimated
background contributions are indicated by the shaded regions in Figs.
12(a)-(b), and we show the corresponding background-subtracted invariant-mass
spectra in Figs. 12(c)-(d).

The $K^{*}_{0}(1430)$ is a relatively broad resonance decaying to $K\pi$,
$K\eta$, and $K\eta^{\prime}$. The measured $K\eta$ relative branching
fraction is $\frac{{\cal B}(K^{*}_{0}(1430)\mbox{$\rightarrow$}K\eta)}{{\cal
B}(K^{*}_{0}(1430)\mbox{$\rightarrow$}K\pi)}=0.092\pm 0.025^{+0.010}_{-0.025}$
Lees:2014iua , while the $K\eta^{\prime}$ has only been observed in Ref.
Ablikim:2014tww . To describe the $K^{*}_{0}(1430)$ lineshape in the
$K\eta^{\prime}$ projection, we model it using a simplified coupled-channel
Breit-Wigner function, which ignores the small $K\eta$ contribution. We
parameterize the $K^{*}_{0}(1430)$ signal as

$BW(m)=\frac{1}{m_{0}^{2}-m^{2}-i(\rho_{1}(m)g^{2}_{K\pi}+\rho_{2}(m)g^{2}_{K\eta^{\prime}})},$
(11)

where $m_{0}$ is the resonance mass, $g_{K\pi}$ and $g_{K\eta^{\prime}}$ are
the couplings to the $K\pi$ and $K\eta^{\prime}$ final states, and
$\rho_{j}(m)=2P/m$ are the respective Lorentz-invariant phase-space factors,
with $P$ the decay particle momentum in the $K^{*}_{0}(1430)$ rest frame. The
$\rho_{2}(m)$ function becomes imaginary below the $K\eta^{\prime}$ threshold.
The values of $m_{0}$ and the $g_{Kj}$ couplings cannot be derived from the
$K\eta^{\prime}$ system only, and therefore we make use of the $K\pi$ $S$-wave
measurement from BABAR Lees:2015zzr . We average the reported quasi model-
independent (QMI) measurements of the $K\pi$ $S$-wave from
$\eta_{c}\mbox{$\rightarrow$}K^{0}_{\scriptscriptstyle S}K\pi$ and
$\eta_{c}\mbox{$\rightarrow$}\mbox{${K^{+}}$}K^{-}\mbox{${\pi^{0}}$}$ decays,
and obtain the modulus squared of the amplitude and the phase shown in Fig.
13.

Figure 13: The (a) squared modulus and (b) phase of the $K\pi$ $S$-wave
averaged over the $\eta_{c}\mbox{$\rightarrow$}K^{0}_{\scriptscriptstyle
S}K\pi$ and
$\eta_{c}\mbox{$\rightarrow$}\mbox{${K^{+}}$}K^{-}\mbox{${\pi^{0}}$}$ from the
BABAR Lees:2015zzr  QMI analysis. Statistical uncertainties only are shown.
The full (red) lines represent the result from the fit with free
$g^{2}_{K\eta^{\prime}}$ and $g^{2}_{K\pi}$ parameters. The dashed (blue)
lines represent the result from the fit with a fixed
$g^{2}_{K\eta^{\prime}}/g^{2}_{K\pi}$ ratio. The dotted (black) line in (a)
represents the empirical background contribution.

We perform a simultaneous binned $\chi^{2}$ fit to the $K\pi$ $S$-wave
amplitude and phase from threshold up to 1.72 ${\mathrm{GeV}}/c^{2}$. Above
this mass, other resonant contributions are present, which make the amplitude
and phase more complicated. We model the $K\pi$ $S$-wave in this region as:

$S{\text{-}\rm wave}(m)=B(m)+c\cdot BW_{K\pi}(m)e^{i\phi},$ (12)

where $BW_{K\pi}(m)$ is given by Eq.( 11), $B(m)$ is an empirical background
term, parameterized as

$B(m)=\rho_{1}(m)e^{-\alpha m},$ (13)

and $c$, $\phi$, and $\alpha$ are free parameters. The results of the fit are
shown in Fig. 13 as the solid (red) lines. We obtain a $\chi^{2}/{\rm
ndf}=55/31$ ($\chi^{2}/{\rm ndf}=25/31$ with included systematic
uncertainties) and the $K^{*}_{0}(1430)$ parameters listed in Table 4.

Table 4: Resonance parameters from the Dalitz plot analyses of $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$, $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$, and $\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$. In the case of the $K^{*}_{0}(1430)$, the first two rows report results from fits to the $K\pi$ $S$-wave with free $K^{*}_{0}(1430)$ parameters and fixed $\frac{g^{2}_{\eta^{\prime}K}}{g^{2}_{\pi K}}$ ratio, respectively. When two errors are listed the first is statistical, the second systematic. The calculated significances do not include systematic uncertainties.  Resonance | Mass (${\mathrm{MeV}}/c^{2}$) |  $g_{K\pi}^{2}$ (${\mathrm{GeV^{2}}}/c^{4}$) |  $g_{K\eta^{\prime}}^{2}$ (${\mathrm{GeV^{2}}}/c^{4}$) |
---|---|---|---|---
|  | $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ |  |
|  |  |  |
$K^{*}_{0}(1430)$ |  |  |  |
$\eta_{c}\mbox{$\rightarrow$}K\bar{K}\pi$ | $1447\pm 8$ | $0.414\pm 0.026$ | $0.197\pm 0.105$ |
fixed $\frac{g^{2}_{\eta^{\prime}K}}{g^{2}_{\pi K}}$ | $1453\pm 22$ | $0.462\pm 0.036$ |  |
Resonance | Mass (${\mathrm{MeV}}/c^{2}$) |  $\Gamma$ (${\mathrm{MeV}}$) |  | significance (n$\sigma)$
$f_{0}(1710)$ | $1757\pm 24\pm 9$ | $175\pm 23\pm 4$ |  | 11.4
$K^{*}_{0}(1950)$ | $1942\pm 22\pm 5$ | $80\pm 32\pm 20$ |  | 3.3
|  |  |  |
|  | $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ |  |
|  |  |  |
$f_{0}(500)$ | $953\pm 90$ | $335\pm 81$ |  |
$f_{2}(1430)$ | $1440\pm 11\pm 3$ | $46\pm 15\pm 5$ |  | 4.4
$f_{0}(2100)$ | $2116\pm 27\pm 17$ | $289\pm 34\pm 15$ |  | 10
|  |  |  |
|  | $\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ |  |
|  |  |  |
$a_{0}(1700)$ | $1704\pm 5\pm 2$ | $110\pm 15\pm 11$ |  | 8
|  |  |  |

We note a large statistical error on $g^{2}_{K\eta^{\prime}}$ that is expected
because of the weak sensitivity of the $K\pi$ $S$-wave to the opening of the
$K\eta^{\prime}$ threshold. We also note the presence of a very small
background term. We attempt to replace the background term with a BW function
with parameters fixed to the PDG averages for the $\kappa/K^{*}_{0}(700)$
resonance, but obtain a poor description of the data. For comparison, the
$K^{*}_{0}(1430)$ parameters used by BESIII in the Dalitz plot analysis of
$\chi_{c1}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$
Ablikim:2014tww  are those measured by the CLEO
$D^{+}\mbox{$\rightarrow$}K^{-}\mbox{${\pi^{+}}$}\mbox{${\pi^{+}}$}$ Dalitz
plot analysis Bonvicini:2008jw , $m=1471.2\ \mbox{${\mathrm{MeV}}/c^{2}$}$,
$g^{2}_{K\pi}=0.299$ ${\mathrm{GeV^{2}}}/c^{4}$, and
$g^{2}_{K\eta^{\prime}}=0.0529$ ${\mathrm{GeV^{2}}}/c^{4}$.

We perform a Dalitz plot analysis of the
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ decay channel
by using the $\eta^{\prime}f_{0}(1710)$ intermediate state as the reference
amplitude. If there are regions of the phase space not well described by the
fit, we add postulated $\mbox{${K^{+}}$}K^{*-}_{0}$, $\eta^{\prime}f_{0,2}$,
or $\eta^{\prime}a_{0}$ intermediate states, and accept them if
$\Delta(-2\log{\cal L})>2$. At each stage, we test for the presence of a non-
resonant contribution.

We describe the $K^{*}_{0}(1430)$ according to Eq. (11) first with $m_{0}$ and
$g^{2}_{K\pi}$ parameters fixed to the values from the fit to the $K\pi$
$S$-wave and $g^{2}_{K\eta^{\prime}}$ free. We observe little sensitivity to
the $g^{2}_{K\eta^{\prime}}$ parameter, expressed by the large error, and
therefore we also fix the value of this parameter to that from the fit to the
$K\pi$ $S$-wave.

The projections of the fit result are shown in Fig. 12, along with the largest
signal components. To test the fit quality, we generate a large number of
phase-space MC-simulated events, which are weighted by the likelihood function
obtained by the fit. These MC-simulated events are then normalized to the
observed yield and are superimposed to the data. To test the fit quality we
also project the fit on the ($m(\mbox{${K^{+}}$}K^{-}),\cos\theta_{H})$ plane
and compare data and simulation in each cell of the plane. Labelling with
${\rm ndf}=N_{\rm cells}-N_{\rm par}$, where $N_{\rm cells}$ is the number of
cells having at least two expected events and $N_{\rm par}$ the number of free
parameters in the Dalitz analysis, we obtain $\chi^{2}/{\rm ndf}=285/264=1.1$
corresponding to a $p$-value of 18%.

The intermediate states retained by this procedure are listed in the left half
of Table 5, together with their fitted fractions and relative phases. We label
this fit as solution (A). The non-resonant contribution is consistent with
zero.

Table 5: Fractions and relative phases from the Dalitz plot analysis of $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$. The first errors are statistical, the second systematic.  Intermediate state | fraction (%) | phase (rad) | fraction (%) | phase (rad)
---|---|---|---|---
| Solution (A) |  | Solution (B) |
$f_{0}(1710)\eta^{\prime}$ | $29.5\pm 4.7\pm 1.6$ | 0. | $29.4\pm 4.5\pm 1.6$ | 0.
$K^{*}_{0}(1430)^{+}K^{-}$ | $53.9\pm 7.2\pm 2.0$ | $0.61\pm 0.13\pm 0.45$ | $61.4\pm 8.1\pm 2.6$ | $0.79\pm 0.12\pm 0.59$
$K^{*}_{0}(1950)^{+}K^{-}$ | $2.4\pm 1.2\pm 0.4$ | $0.46\pm 0.29\pm 0.50$ | $2.6\pm 1.2\pm 0.5$ | $0.21\pm 0.28\pm 1.10$
$f_{0}(1500)\eta^{\prime}$ | $0.8\pm 1.0\pm 0.3$ | $0.32\pm 0.54\pm 0.10$ | $0.9\pm 1.0\pm 0.3$ | $0.24\pm 0.52\pm 0.10$
$f_{0}(980)\eta^{\prime}$ | $4.7\pm 2.7\pm 0.4$ | $-0.74\pm 0.55\pm 0.05$ | $5.8\pm 3.0\pm 0.5$ | $-1.01\pm 0.46\pm 0.05$
$f_{2}(1270)\eta^{\prime}$ | $2.9\pm 1.5\pm 0.1$ | $2.9\pm 0.38\pm 0.09$ | $2.6\pm 1.6\pm 0.2$ | $2.73\pm 0.39\pm 0.09$
sum | $94.3\pm 9.3\pm 2.6$ |  | $102.6\pm 10.0\pm 3.2$ |
$\chi^{2}/{\rm ndf}$ | 285/264=1.1 |  | 281/260=1.1 |
$p$-value | 18% |  | 18% |

We measure the $f_{0}(1710)$ parameters, listed in Table 4. In addition to the
strong $f_{0}(1710)\eta^{\prime}$ and $K^{*}_{0}(1430)^{+}K^{-}$ contributions
there is evidence for a signal of the $K^{*}_{0}(1950)^{+}K^{-}$ decay mode.
We measure the parameters of the $K^{*}_{0}(1950)$ (see Table 4) for which
there is only one previous measurement from the LASS collaboration
Aston:1987ir . There are smaller contributions from $f_{0}(980)\eta^{\prime}$,
$f_{2}(1270)\eta^{\prime}$, and $f_{0}(1510)\eta^{\prime}$. The latter is
indistinguishable from an $f_{2}^{\prime}(1525)\eta^{\prime}$ contribution,
but for simplicity, we report only the $f_{0}(1510)\eta^{\prime}$, which gives
a slighly larger likelihood improvement.

Statistical significances of resonances contributing to the decay are
evaluated using the Wilks’ theorem wilks  from the difference in log
likelihood between fits with and without the specific signal component, taking
into account the difference of two free parameters. For
$f_{0}(1710)\eta^{\prime}$ and $K^{*}_{0}(1950)^{+}K^{-}$ we obtain
$\Delta(-2\log{\cal L})=135.9$ and $\Delta(-2\log{\cal L})=15.3$,
respectively. The corresponding significances are listed in Table 4.

We evaluate systematic uncertainties on the fitted fractions, phases, and
resonance parameters. For resonances having parameters fixed to PDG values, we
vary these parameters according to their PDG uncertainties. We modify the
purity of the $\eta_{c}$ signal according to its statistical uncertainty. We
replace the fitted efficiency with the raw efficiency, defined in Sec. IV. The
Blatt-Weisskopf blatt  form factor present in the relativistic BW functions,
nominally fixed at 1.5 GeV-1, is varied between 0 and 3.0 GeV-1. The
background description is modified by varying each resonant fraction by its
statistical uncertainties in the fits to the sidebands. All the contributions
are added in quadrature.

An inspection of Fig. 12(b)-(d) suggests an additional enhancement in the
$m^{(2)}(\eta^{\prime}K^{\pm})$ around a mass of $\approx 2100$
${\mathrm{MeV}}/c^{2}$. We explore this possibility adding, in the Dalitz plot
analysis, an additional scalar resonance in this mass region with free
parameters. The presence of this additional resonance also affects the
parameters of the $K^{*}_{0}(1950)$ which are also left free in the fit. The
fit returns the following values of the parameters of these resonances

$\begin{split}m(K^{*}_{0}(1950))&=1979\pm 26_{\rm stat}\pm 3_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(K^{*}_{0}(1950))&=144\pm 44_{\rm
stat}\pm 21_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$},\end{split}$

and

$\begin{split}m(K^{*}_{0}(2130))&=2128\pm 31_{\rm stat}\pm 9_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(K^{*}_{0}(2130))&=95\pm 42_{\rm
stat}\pm 76_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$}.\end{split}$

A comparison between the two fits on the $m(\eta^{\prime}K^{\pm})$ projection
is shown in Fig. 14.

Figure 14:  Linear-scale mass projection $m(\eta^{\prime}K^{\pm})$, after
subtraction of the background. The solid (red) histogram represent the results
of the fit described in the text (solution (A)). The dashed (blue) histogram
represent results of the fit (solution (A)) allowing the presence of an
additional $K^{*}_{0}(2130)$ resonance. The $\eta^{\prime}K^{\pm}$ mass
projection has two entries per event.

This new hypothesis gives an overall improvement of the likelihood by a factor
$\Delta(-2\log{\cal L})=8.3$. However, an application of the Wilks theorem for
the individual significances of the $K^{*}_{0}(1950$ and $K^{*}_{0}(2130)$ in
this new fit, obtain values of 4.3$\sigma$ and 2.7$\sigma$, respectively.
Since the local significance of the $K_{0}^{*}(2130)$ is less than $3\sigma$,
we do not consider anymore in the following the presence of this contribution.

###  VII.1 Measurement of the relative
$K^{*}_{0}(1430)\mbox{$\rightarrow$}K\eta^{\prime}$ coupling

We make use of previous measurements of $\eta_{c}$ decays, combined with the
results of the present analysis, to obtain a measurement of the
$K^{*}_{0}(1430)$ couplings to the $K\eta^{\prime}$ and $K\pi$ final states.
The product of the $\eta_{c}$ two-photon width and its branching fraction to
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
$\Gamma_{\gamma\gamma}\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})=65.4\pm
2.6_{\rm stat}\pm 7.8_{\rm sys}$ eV, has been measured by the Belle experiment
Xu:2018uye , while $\Gamma_{\gamma\gamma}\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}K\bar{K}\pi)=386\pm 0.008_{\rm stat}\pm
0.021_{\rm sys}$ eV has been measured by the BABAR experiment
delAmoSanchez:2011bt . The isospin decomposition of the $\eta_{c}$ decay to
$K\bar{K}\pi$ includes decays to
$\bar{K}^{0}\mbox{${K^{+}}$}\mbox{${\pi^{-}}$}$,
$K^{0}K^{-}\mbox{${\pi^{+}}$}$, $K^{0}\bar{K}^{0}\mbox{${\pi^{0}}$}$, and
$\mbox{${K^{+}}$}K^{-}\mbox{${\pi^{0}}$}$, where the latter contributes with a
factor 1/6. Dividing the BABAR result by a factor 6 to obtain the
$\eta_{c}\mbox{$\rightarrow$}\mbox{${\pi^{0}}$}\mbox{${K^{+}}$}K^{-}$
component, we have

$\frac{\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})}{\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\mbox{${\pi^{0}}$}\mbox{${K^{+}}$}K^{-})}=1.016\pm
0.040_{\rm stat}\pm 0.121_{\rm sys}.$ (14)

Combined with the $\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-})/\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$,
given above, Eq. (8), this gives

$\frac{\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-})}{\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}\mbox{${\pi^{0}}$}\mbox{${K^{+}}$}K^{-})}=0.655\pm
0.047_{\rm stat}\pm 0.085_{\rm sys}.$ (15)

The BABAR Dalitz plot analysis of
$\eta_{c}\mbox{$\rightarrow$}\mbox{${\pi^{0}}$}\mbox{${K^{+}}$}K^{-}$ measured
the fraction $\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}K^{-}K^{*}_{0}(1430)^{+}(\mbox{$\rightarrow$}\mbox{${K^{+}}$}\mbox{${\pi^{0}}$}))=(33.8\pm
1.9_{\rm stat}\pm 0.4_{\rm sys})\%$ Lees:2014iua . The present analysis
measures $\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}K^{-}K^{*}_{0}(1430)^{+}(\mbox{$\rightarrow$}\mbox{${K^{+}}$}\eta^{\prime}))=(53.9\pm
7.2_{\rm stat}\pm 2.0_{\rm sys})\%$ (left section of Table 5). Combining
these, and applying a factor 3 due to the isospin related unseen decay modes,
we obtain the ratio

$\mbox{${\cal B}$}=\frac{\mbox{${\cal
B}$}(K^{*}_{0}(1430)\mbox{$\rightarrow$}K\eta^{\prime})}{\mbox{${\cal
B}$}(K^{*}_{0}(1430)\mbox{$\rightarrow$}K\pi)}=0.348\pm 0.056_{\rm stat}\pm
0.047_{\rm sys}.$ (16)

This ratio can be written as

$\mbox{${\cal
B}$}=\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}\frac{I_{K\eta^{\prime}}}{I_{K\pi}},$
(17)

where $I_{K\eta^{\prime}}$ and $I_{K\pi}$ are the integrals over the
$\eta_{c}$ phase space of the coupled-channel Breit-Wigner function describing
the $K^{*}_{0}(1430)$ in the
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ and
$\eta_{c}\mbox{$\rightarrow$}\mbox{${\pi^{0}}$}\mbox{${K^{+}}$}K^{-}$ decay
modes (Eq. (11)). Using Eq. (17), we obtain the ratio of the couplings
$\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}=1.43\pm 0.23_{\rm stat}\pm
0.22_{\rm sys}$, to be compared with the results from the fit to the $K\pi$
$S$-wave, (from the first row in Table 4), of
$\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}=0.476\pm 0.254$.

To resolve this discrepancy (of the order of $2.3\sigma$), we perform several
fits to the $K\pi$ $S$-wave with $\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}$
varying from 0.476 to 1.75, observing a steady increase in $\chi^{2}$ from 55
to 80. Using each set of fitted $K^{*}_{0}(1430)$ resonance parameters, we
repeat the Dalitz plot analysis to obtain new values of the fractional
contributions, and recalculate the ratio
$\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}$ according to Eq. (11). This
ratio depends weakly on the resonance parameters, varying between 1.40 to
1.67. Therefore, we fix $\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}=1.43$ in
the fit to the $K\pi$ $S$-wave, and show the result as the dashed (blue) lines
in Fig. 13. This fit has a $\chi^{2}/{\rm ndf}=70/32$ ($\chi^{2}/{\rm
ndf}=32/32$ when systematic uncertainties are included). The fitted
$K^{*}_{0}(1430)$ parameters are then used in a new Dalitz plot analysis,
which we denote solution (B), the results of which are listed in the right
half of Table 5. The fitted $K^{*}_{0}(1430)^{+}K^{-}$ contribution increases
to $\mbox{${\cal
B}$}(\eta_{c}\mbox{$\rightarrow$}K^{-}K^{*}_{0}(1430)^{+}(\mbox{$\rightarrow$}\mbox{${K^{+}}$}\eta^{\prime}))=(61.4\pm
8.1_{\rm stat}\pm 2.6_{\rm sys})\%$ which gives the ratio

$\mbox{${\cal B}$}=\frac{\mbox{${\cal
B}$}(K^{*}_{0}(1430)\mbox{$\rightarrow$}K\eta^{\prime})}{\mbox{${\cal
B}$}(K^{*}_{0}(1430)\mbox{$\rightarrow$}K\pi)}=0.397\pm 0.064_{\rm stat}\pm
0.054_{\rm sys}$ (18)

and

$\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}=1.50\pm 0.24_{\rm stat}\pm
0.24_{\rm sys},$ (19)

where we have included the change from solution (A) in the systematic
uncertainty, as an estimate of the model uncertainty. Similarly, we use the
estimates of the $K^{*}_{0}(1430)$ mass and $g^{2}_{K\pi}$ from solution (B),
along with the differences from solution (A) (see Table 4), to obtain

$\begin{split}m(K^{*}_{0}(1430))=&1449\pm 17_{\rm stat}\pm 2_{\rm
sys}\mbox{${\mathrm{MeV}}/c^{2}$},\\\ g^{2}_{K\pi}=&0.458\pm 0.032_{\rm
stat}\pm 0.044_{\rm sys}\ \mbox{${\mathrm{GeV^{2}}}/c^{4}$}.\end{split}$ (20)

The inconsistency between the $\frac{g^{2}_{K\eta^{\prime}}}{g^{2}_{K\pi}}$
values may be associated with an imperfect model describing the
$K^{*}_{0}(1430)$ shape. The Dalitz plot fit quality of the solution (B) is
similar to that of solution (A) with $\Delta(-2\log{\cal L})=4.8$ and
$\chi^{2}/N_{\rm cells}=281/260=1.1$.

##  VIII Dalitz plot analysis of
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$

Figure 15 shows the Dalitz plot for the selected
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates in the data, in the $\eta_{c}$ signal region, for the two
$\eta^{\prime}$ decay modes combined, and Figs. 16(a)-(b) show two squared-
mass projections. We observe several diagonal bands in the Dalitz plot, in
particular at the lower-left edge. There are corresponding structures in the
$m^{2}(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ spectrum, including peaks
attributable to the $f_{0}(980)$ and $f_{2}(1270)$ resonances, and a large
structure at high ${\pi^{+}}$${\pi^{-}}$ mass. In the
$m^{2}(\eta^{\prime}\mbox{${\pi^{\pm}}$})$ spectrum, a large structure is
present; there is no known resonance decaying to $\eta^{\prime}\pi$ in this
mass region, but this could be a reflection of the structure in the high
$m^{2}(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ region.

Figure 15: Dalitz plot for selected
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates in the $\eta_{c}$ signal region, summed over the two
$\eta^{\prime}$ decay modes.

Figure 16: Squared-mass projections (a)
$m^{2}(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ and (b)
$m^{2}(\eta^{\prime}\mbox{${\pi^{\pm}}$})$ of the measured
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
Dalitz plot. The shaded (gray) histograms are the background interpolated from
fits to the two $\eta_{c}$ sidebands. Linear-scale mass projections (c)
$m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ and (d)
$m(\eta^{\prime}\mbox{${\pi^{\pm}}$})$, after subtraction of the background.
The solid (red) histograms represent the results of the fit described in the
text, and the other histograms display the contributions from each of the
listed components. The $\eta^{\prime}\mbox{${\pi^{\pm}}$}$ projections have
two entries per event.

We fit the two $\eta_{c}$ sidebands using an incoherent sum of amplitudes,
which includes contributions from the $\rho^{0}(770)$, $f_{2}(1270)$,
$f_{0}(1370)$, and $f_{0}(2100)$ resonances. To model the background in the
$\eta_{c}$ signal region we take a weighted average of the fitted fractional
contributions, and normalize using the results from the fit to the
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-mass spectrum.
The estimated background contributions are indicated by the shaded regions in
Figs. 16(a)-(b), and we show the corresponding background-subtracted
invariant-mass spectra in Figs. 16(c)-(d).

A candidate for the large structure in the high ${\pi^{+}}$${\pi^{-}}$ mass
region is the $f_{0}(2100)$ resonance, observed in radiative $J/\psi$ decay to
$\gamma\eta\eta$ Ablikim:2013hq . We take $f_{0}(2100)\eta^{\prime}$ as the
reference contribution, and perform a Dalitz plot analysis as described in
Sec. VI. Again, no non-resonant contribution is needed, and the list of the
resonances contributing to this $\eta_{c}$ decay mode is given in Table 6,
together with their fitted fractions and relative phases.

Table 6: Fractions and relative phases from the Dalitz plot analysis of $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$. The first errors are statistical, the second systematic.  Intermediate state | fraction (%) | phase (rad)
---|---|---
$f_{0}(2100)\eta^{\prime}$ | $74.9\pm 7.5\pm 3.6$ | 0.
$f_{0}(500)\eta^{\prime}$ | $4.3\pm 2.3\pm 0.7$ | $-5.89\pm 0.24\pm 0.10$
$f_{0}(980)\eta^{\prime}$ | $16.1\pm 2.4\pm 0.5$ | $-5.31\pm 0.16\pm 0.04$
$f_{2}(1270)\eta^{\prime}$ | $22.1\pm 2.9\pm 2.4$ | $-3.60\pm 0.16\pm 0.03$
$f_{2}(1430)\eta^{\prime}$ | $1.9\pm 0.7\pm 0.1$ | $-2.45\pm 0.32\pm 0.11$
$a_{2}(1710)\pi$ | $3.2\pm 1.9\pm 0.5$ | $-0.75\pm 0.27\pm 0.11$
$a_{0}(1950)\pi$ | $2.5\pm 1.1\pm 0.1$ | $-0.02\pm 0.32\pm 0.06$
$f_{2}(1800)\eta^{\prime}$ | $5.3\pm 2.2\pm 1.4$ | $0.67\pm 0.24\pm 0.08$
sum | $130.5\pm 9.5\pm 4.7$ |
$\chi^{2}/{\rm ndf}$=409/386=1.1 |  |
$p$-value | 20% |

The $f_{0}(2100)$ parameters are first left free in the fit, and we obtain the
values listed in Table 4, which are in agreement with BESIII measurement
($m=2081\pm 13^{+24}_{-36}$ ${\mathrm{MeV}}/c^{2}$,
$\Gamma=273^{+27+70}_{-24-23}$) ${\mathrm{MeV}}$ Ablikim:2013hq . We then fix
them to the values listed in the PDG. We also leave free the $f_{0}(500)$
parameters and obtain the values listed in Table 4 which give a good
description of the data. Given the low statistics, we do not assign systematic
uncertainties to the fitted $f_{0}(500)$ resonance parameters, which are
within the range of other measurements PDG . The $f_{0}(980)$ is parameterized
by a coupled-channel Breit-Wigner function with parameters fixed to the
measurement from Ref. Armstrong . To describe the small enhancement around
1.43 ${\mathrm{GeV}}/c^{2}$, we test both spin-2 and spin-0 hypotheses with
free resonance parameters; we obtain $\Delta(-2\log{\cal L})=2.4$ in favor of
the spin-2 hypothesis, so we attribute this signal to the $f_{2}(1430)$
resonance, and report the fitted parameter values in Table 4. We test the
significance of this signal by removing it from the list of the resonances,
obtaining $\Delta(-2\log{\cal L})=23.8$ and a significance of $4.4\sigma$.
Replacing the $f_{2}(1430)$ resonance with $f_{0}(1500)$ or $f_{0}(1370)$, we
obtain poor fits with fractions from these possible contributions consistent
with zero. The $f_{0}(2100)$ statistical significance is 10$\sigma$.

The projections of the fit result are compared with the data in Fig. 16. To
test the fit quality, we generate a large number of phase-space MC-simulated
events, which are weighted by the likelihood function obtained from the fit.
These MC-simulated events are then normalized to the observed yield and
superimposed to the data. We also project the fit on the
($m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}),\cos\theta_{H})$ plane and compare
data and simulation in each cell, obtaining $\chi^{2}/{\rm ndf}=409/386=1.1$.
The systematic uncertainties on the fitted fractions, phases and resonance
parameters are evaluated as in the previous section.

##  IX Dalitz plot analysis of
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$

Figure 17 shows the Dalitz plot for the selected
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates in the data, in the $\eta_{c}$ signal region, for the two $\eta$
decay modes combined, and Figs. 18(a)-(b) show two squared-mass projections.
We observe that the Dalitz plot is dominated by horizontal and vertical bands
due to the $a_{0}(980)$ and diagonal bands due to resonances in the
${\pi^{+}}$${\pi^{-}}$ final state. The squared-mass projections show signals
of $f_{0}(500)$, $f_{0}(980)$, and $f_{2}(1270)$.

Figure 17: Dalitz plot for selected
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
candidates in the $\eta_{c}$ signal region, summed over the two $\eta$ decay
modes.

Figure 18: Squared-mass projections (a)
$m^{2}(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ and (b)
$m^{2}(\eta\mbox{${\pi^{\pm}}$})$ of the measured
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ Dalitz
plot. The shaded (gray) histograms represent the background interpolated from
fits to the two $\eta_{c}$ sidebands. Linear-scale mass projections (c)
$m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})$ and (d)
$m(\eta\mbox{${\pi^{\pm}}$})$ after subtraction of the background. The solid
(red) histograms represent the results of the fit described in the text, and
the other histograms display the contributions from each of the listed
components. The $\eta\pi^{\pm}$ projections have two entries per event.

The $\eta_{c}$ sidebands are also rich in resonant structure, and are fitted
using an incoherent sum of amplitudes, including contributions from the
$a_{0}(980)$, $f_{2}(1270)$, $a_{2}(1310)$, and $f_{2}(1950)$ resonances. We
take a weighted average of the fitted fractions in the two sidebands,
normalized using the results from the fit to the
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ invariant-mass spectrum, to
estimate the background in the signal region, shown as the shaded regions in
Figs. 18(a)-(b).

We take $a_{0}(980)^{+}\mbox{${\pi^{-}}$}$ as the reference contribution, and
perform a Dalitz plot analysis as described above. The resulting list of
contributions to this $\eta_{c}$ decay mode is given in Table 7, together with
fitted fractions and relative phases.

Table 7: Fractions and relative phases from the Dalitz plot analysis of $\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$. The first errors are statistical, the second systematic.  Intermediate state | fraction (%) | phase (rad)
---|---|---
$a_{0}(980)^{+}\mbox{${\pi^{-}}$}$ | $12.3\pm 1.2\pm 2.8$ | 0.
$a_{2}(1310)^{+}\mbox{${\pi^{-}}$}$ | $2.5\pm 0.7\pm 0.9$ | $-1.04\pm 0.13\pm 0.20$
$f_{0}(500)\eta$ | $4.3\pm 1.3\pm 1.1$ | $0.54\pm 0.14\pm 0.24$
$f_{2}(1270)\eta$ | $4.6\pm 0.9\pm 0.8$ | $-1.15\pm 0.11\pm 0.05$
$f_{0}(980)\eta$ | $5.7\pm 1.3\pm 1.5$ | $-2.41\pm 0.09\pm 0.07$
$f_{0}(1500)\eta$ | $4.2\pm 0.7\pm 0.9$ | $2.32\pm 0.13\pm 0.17$
$a_{0}(1450)^{+}\mbox{${\pi^{-}}$}$ | $15.0\pm 2.4\pm 3.2$ | $2.60\pm 0.09\pm 0.11$
$a_{0}(1700)^{+}\mbox{${\pi^{-}}$}$ | $3.5\pm 0.8\pm 0.8$ | $1.39\pm 0.15\pm 0.20$
$f_{2}(1950)\eta$ | $4.2\pm 1.0\pm 1.0$ | $-1.59\pm 0.15\pm 0.21$
resonant sum | $56.3\pm 3.7\pm 10.0$ |
$NR$ | $172.7\pm 8.0\pm 10.0$ | $1.67\pm 0.07\pm 0.06$
sum | $229.0\pm 8.8\pm 14.1$ |
$\chi^{2}/{\rm ndf}$ | 419/382=1.1 |
$p$-value | 9.3% |

We find little sensitivity to the parameters of the $f_{0}(500)$ resonance,
and therefore we use the parameters from the
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
Dalitz plot analysis, listed in Table 4. A new $a_{0}(1700)$ resonance is
observed in the $\eta\mbox{${\pi^{\pm}}$}$ invariant-mass spectrum, with
fitted parameters listed in Table 4. The likelihood change obtained when the
resonance is excluded from the fit is $\Delta(-2\log{\cal L})=72.3$,
corresponding to a significance greater than $8\sigma$. Possible contributions
from the $a_{2}(1710)$ and $f_{0}(2100)$ resonances have been tested, but both
are found to be consistent with zero.

We note the presence of a very large non-resonant scalar contribution, and in
Table 7, we list both the sum of resonant contributions and the sum including
the non-resonant contribution. A similar effect has been observed in charmless
$B$ decays Aubert:2006nu . This effect could be correlated with the
interference of the $\eta_{c}$ with the two-photon continuum described in Sec.
V.

We test the fit quality as described above, with the comparison in the
$(m(\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}),\cos\theta_{H})$ plane giving
$\chi^{2}/{\rm ndf}=419/382=1.1$. We evaluate systematic uncertainties as
described above but adding an additional uncertainty due to the possible
interference between intermediate resonances from the $\eta_{c}$ decay and
those present in the background. To obtain the order of magnitude of the
effect we compare the fits to the $\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
mass spectra described in Sec. VI.A with and without the interference and
obtain an average difference in the $\eta_{c}$ yield of the order of 26%.
Multiplying this factor by the sum of all the resonant fractions given in
Table 7, we obtain an estimate of the uncertainty of the order of 15% which is
added in quadrature to the other sources of systematic uncertanties. We also
vary the $\eta_{c}$ signal region width from 100 ${\mathrm{MeV}}/c^{2}$ to 60
${\mathrm{MeV}}/c^{2}$ and add in quadrature the resulting differences in
amplitudes fractions and phases as an additional source of systematic
uncertainties.

##  X Summary

We study the processes
$\gamma\gamma\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
$\gamma\gamma\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$,
and $\gamma\gamma\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
using a data sample of 519 ${\mathrm{fb}^{-1}}$ recorded with the BABAR
detector operating at the SLAC PEP-II asymmetric-energy $e^{+}e^{-}$ collider
at center-of-mass energies at and near the $\mathchar 28935\relax(nS)$
($n=2,3,4$) resonances. We observe $\eta_{c}$ decays to all the above final
states and perform Dalitz plot analyses to measure intermediate resonant
fractions and relative phases. Significant interference effects of the
$\eta_{c}$ with the two-photon background are observed only for the decay
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$.

The decay $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ is
observed for the first time and we measure the branching fraction relative to
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$

$\frac{{\cal
B}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-})}{{\cal
B}(\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$})}=0.644\pm
0.039_{\rm stat}\pm 0.032_{\rm sys}.$

Using published information from the BABAR and Belle experiments, and this
analysis of $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$,
we obtain measurements of the $K^{*}_{0}(1430)$ resonance parameters:

$\begin{split}m(K^{*}_{0}(1430))&=1449\pm 17_{\rm stat}\pm 2_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ g^{2}_{K\pi}&=0.458\pm 0.032_{\rm stat}\pm
0.044_{\rm sys}\ \mbox{${\mathrm{GeV^{2}}}/c^{4}$};\\\
\frac{g^{2}_{\eta^{\prime}K}}{g^{2}_{\pi K}}&=1.50\pm 0.24_{\rm stat}\pm
0.24_{\rm sys}.\end{split}$

We also measure the ratio of couplings of the $K^{*}_{0}(1430)$ resonance to
$\eta^{\prime}K$ and $\pi K$,

$\frac{\mbox{${\cal
B}$}(K^{*}_{0}(1430)^{+}\mbox{$\rightarrow$}\eta^{\prime}K)}{\mbox{${\cal
B}$}(K^{*}_{0}(1430)^{+}\mbox{$\rightarrow$}\pi K)}=0.450\pm 0.072_{\rm
stat}\pm 0.061_{\rm sys}.$

The $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ decay
contains a significant contribution from
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}f_{0}(1710)$, and we measure the
$f_{0}(1710)$ resonance parameters:

$\begin{split}m(f_{0}(1710)&=1757\pm 24_{\rm stat}\pm 9_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(f_{0}(1710))&=175\pm 23_{\rm stat}\pm
4_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$}.\end{split}$

Evidence is also found for the $K^{*}_{0}(1950)$, whose parameters are
measured as:

$\begin{split}m(K^{*}_{0}(1950))&=1942\pm 22_{\rm stat}\pm 21_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(K^{*}_{0}(1950))&=80\pm 32_{\rm
stat}\pm 20_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$}.\end{split}$

We find no evidence for the $\kappa/K^{*}_{0}(700)$ in $\eta_{c}$ decays.

The
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
decay is found to be dominated by the $f_{0}(2100)$ resonance, also observed
in radiative $J/\psi$ decays, and we measure the resonance parameters:

$\begin{split}m(f_{0}(2100))&=2116\pm 27_{\rm stat}\pm 17_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(f_{0}(2100))&=289\pm 34_{\rm stat}\pm
15_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$}.\end{split}$

Evidence is also found for the $f_{2}(1430)$, and we measure the resonance
parameters:

$\begin{split}m(f_{2}(1430))&=1440\pm 11_{\rm stat}\pm 3_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(f_{2}(1430))&=46\pm 15_{\rm stat}\pm
5_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$}.\end{split}$

The Dalitz plot analysis of the
$\eta_{c}\mbox{$\rightarrow$}\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ decay
shows the presence of a new $a_{0}(1700)\mbox{$\rightarrow$}\eta\pi$
resonance, for which we measure the following parameters:

$\begin{split}m(a_{0}(1700))&=1704\pm 5_{\rm stat}\pm 2_{\rm sys}\
\mbox{${\mathrm{MeV}}/c^{2}$};\\\ \Gamma(a_{0}(1700))&=110\pm 15_{\rm stat}\pm
11_{\rm sys}\ \mbox{${\mathrm{MeV}}/c^{2}$}.\end{split}$

In the framework of the identification of scalar gluonium states, it is
interesting to compare the rates of $\eta_{c}$ decays into a gluonium
candidate state and an $\eta$ or an $\eta^{\prime}$ meson. Table 8 summarizes
relevant results from this and our previous analysis.

Table 8: Fractional contributions to $\eta_{c}\mbox{$\rightarrow$}\eta h^{+}h^{-}$ and $\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}h^{+}h^{-}$ decays of selected scalar mesons, uncorrected for unseen decay modes. The first errors are statistical, the second systematic.  Final state |  $f_{0}(1500)$(%) |  $f_{0}(1710)$(%) |  $f_{0}(2100)$(%)
---|---|---|---
$\eta\mbox{${K^{+}}$}K^{-}$ | $23.7\pm 7.0\pm 1.8$ | $8.9\pm 0.2\pm 0.4$ |
$\eta\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ | $4.2\pm 0.7\pm 0.9$ |  | 0.
$\eta^{\prime}\mbox{${K^{+}}$}K^{-}$ | $0.8\pm 1.0\pm 0.3$ | $29.5\pm 4.7\pm 1.6$ |
$\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$ | $0.3\pm 0.2$ |  | $74.9\pm 7.5\pm 3.5$

We observe an enhanced contribution of $f_{0}(1710)$ in $\eta_{c}$ decays to
$\eta^{\prime}$ and an enhanced contribution of $f_{0}(1500)$ in $\eta_{c}$
decays to $\eta$. This effect may point to an enhanced gluonium content in the
$f_{0}(1710)$ meson. A similar conclusion is drawn in the study of $J/\psi$
radiative decays Ablikim:2013hq . In particular, Ref. Gui:2012gx  finds that
the production rate of the pure gauge scalar glueball in
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ radiative decays predicted by
lattice QCD is compatible with the production rate of
${J\mskip-3.0mu/\mskip-2.0mu\psi\mskip 2.0mu}$ radiative decays to
$f_{0}(1710)$ and this suggests that $f_{0}(1710)$ has a larger overlap with
the glueball compared to other glueball candidates (e.g., $f_{0}(1500)$). The
observation of $f_{0}(2100)$ in both $J/\psi$ radiative decays and in
$\eta_{c}\mbox{$\rightarrow$}\eta^{\prime}\mbox{${\pi^{+}}$}\mbox{${\pi^{-}}$}$
allows to add this state in the list of the candidates for the scalar
glueball.

##  XI Acknowledgments

We are grateful for the extraordinary contributions of our PEP-II colleagues
in achieving the excellent luminosity and machine conditions that have made
this work possible. The success of this project also relies critically on the
expertise and dedication of the computing organizations that support BABAR.
The collaborating institutions wish to thank SLAC for its support and the kind
hospitality extended to them. This work is supported by the US Department of
Energy and National Science Foundation, the Natural Sciences and Engineering
Research Council (Canada), the Commissariat à l’Energie Atomique and Institut
National de Physique Nucléaire et de Physique des Particules (France), the
Bundesministerium für Bildung und Forschung and Deutsche
Forschungsgemeinschaft (Germany), the Istituto Nazionale di Fisica Nucleare
(Italy), the Foundation for Fundamental Research on Matter (The Netherlands),
the Research Council of Norway, the Ministry of Education and Science of the
Russian Federation, Ministerio de Economia y Competitividad (Spain), and the
Science and Technology Facilities Council (United Kingdom). Individuals have
received support from the Marie-Curie IEF program (European Union), the A. P.
Sloan Foundation (USA) and the Binational Science Foundation (USA-Israel).

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