arXiv:1403.7051v1 [hep-ex] 27 Mar 2014
B
A
B
AR
-PUB-13/021
SLAC-PUB-15928
Dalitz plot analysis of
η
c
→
K
+
K
−
η
and
η
c
→
K
+
K
−
π
0
in two-photon interactions
J. P. Lees, V. Poireau, and V. Tisserand
Laboratoire d’Annecy-le-Vieux de Physique des Particules
(LAPP),
Universit ́e de Savoie, CNRS/IN2P3, F-74941 Annecy-Le-Vie
ux, France
E. Grauges
Universitat de Barcelona, Facultat de Fisica, Departament
ECM, E-08028 Barcelona, Spain
A. Palano
ab
INFN Sezione di Bari
a
; Dipartimento di Fisica, Universit`a di Bari
b
, I-70126 Bari, Italy
G. Eigen and B. Stugu
University of Bergen, Institute of Physics, N-5007 Bergen,
Norway
D. N. Brown, L. T. Kerth, Yu. G. Kolomensky, M. J. Lee, and G. Lyn
ch
Lawrence Berkeley National Laboratory and University of Ca
lifornia, Berkeley, California 94720, USA
H. Koch and T. Schroeder
Ruhr Universit ̈at Bochum, Institut f ̈ur Experimentalphys
ik 1, D-44780 Bochum, Germany
C. Hearty, T. S. Mattison, J. A. McKenna, and R. Y. So
University of British Columbia, Vancouver, British Columb
ia, Canada V6T 1Z1
A. Khan
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kin
gdom
V. E. Blinov
ac
, A. R. Buzykaev
a
, V. P. Druzhinin
ab
, V. B. Golubev
ab
, E. A. Kravchenko
ab
,
A. P. Onuchin
ac
, S. I. Serednyakov
ab
, Yu. I. Skovpen
ab
, E. P. Solodov
ab
, and K. Yu. Todyshev
ab
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630
090
a
,
Novosibirsk State University, Novosibirsk 630090
b
,
Novosibirsk State Technical University, Novosibirsk 6300
92
c
, Russia
A. J. Lankford and M. Mandelkern
University of California at Irvine, Irvine, California 926
97, USA
B. Dey, J. W. Gary, and O. Long
University of California at Riverside, Riverside, Califor
nia 92521, USA
C. Campagnari, M. Franco Sevilla, T. M. Hong, D. Kovalskyi, J. D. Rich
man, and C. A. West
University of California at Santa Barbara, Santa Barbara, C
alifornia 93106, USA
A. M. Eisner, W. S. Lockman, W. Panduro Vazquez, B. A. Schumm, a
nd A. Seiden
University of California at Santa Cruz, Institute for Parti
cle Physics, Santa Cruz, California 95064, USA
D. S. Chao, C. H. Cheng, B. Echenard, K. T. Flood, D. G. Hitlin, T. S.
Miyashita, P. Ongmongkolkul, and F. C. Porter
California Institute of Technology, Pasadena, California
91125, USA
R. Andreassen, Z. Huard, B. T. Meadows, B. G. Pushpawela, M. D.
Sokoloff, and L. Sun
University of Cincinnati, Cincinnati, Ohio 45221, USA
P. C. Bloom, W. T. Ford, A. Gaz, J. G. Smith, and S. R. Wagner
University of Colorado, Boulder, Colorado 80309, USA
R. Ayad
∗
and W. H. Toki
2
Colorado State University, Fort Collins, Colorado 80523, U
SA
B. Spaan
Technische Universit ̈at Dortmund, Fakult ̈at Physik, D-44
221 Dortmund, Germany
D. Bernard and M. Verderi
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS
/IN2P3, F-91128 Palaiseau, France
S. Playfer
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
D. Bettoni
a
, C. Bozzi
a
, R. Calabrese
ab
, G. Cibinetto
ab
, E. Fioravanti
ab
,
I. Garzia
ab
, E. Luppi
ab
, L. Piemontese
a
, and V. Santoro
a
INFN Sezione di Ferrara
a
; Dipartimento di Fisica e Scienze della Terra, Universit`a
di Ferrara
b
, I-44122 Ferrara, Italy
A. Calcaterra, R. de Sangro, G. Finocchiaro, S. Martellotti,
P. Patteri, I. M. Peruzzi,
†
M. Piccolo, M. Rama, and A. Zallo
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, I
taly
R. Contri
ab
, M. Lo Vetere
ab
, M. R. Monge
ab
, S. Passaggio
a
, C. Patrignani
ab
, and E. Robutti
a
INFN Sezione di Genova
a
; Dipartimento di Fisica, Universit`a di Genova
b
, I-16146 Genova, Italy
B. Bhuyan and V. Prasad
Indian Institute of Technology Guwahati, Guwahati, Assam,
781 039, India
M. Morii
Harvard University, Cambridge, Massachusetts 02138, USA
A. Adametz and U. Uwer
Universit ̈at Heidelberg, Physikalisches Institut, D-691
20 Heidelberg, Germany
H. M. Lacker
Humboldt-Universit ̈at zu Berlin, Institut f ̈ur Physik, D-
12489 Berlin, Germany
P. D. Dauncey
Imperial College London, London, SW7 2AZ, United Kingdom
U. Mallik
University of Iowa, Iowa City, Iowa 52242, USA
C. Chen, J. Cochran, and S. Prell
Iowa State University, Ames, Iowa 50011-3160, USA
H. Ahmed
Physics Department, Jazan University, Jazan 22822, Kingdo
m of Saudia Arabia
A. V. Gritsan
Johns Hopkins University, Baltimore, Maryland 21218, USA
N. Arnaud, M. Davier, D. Derkach, G. Grosdidier, F. Le Diberder,
A. M. Lutz, B. Malaescu,
‡
P. Roudeau, A. Stocchi, and G. Wormser
Laboratoire de l’Acc ́el ́erateur Lin ́eaire, IN2P3/CNRS et
Universit ́e Paris-Sud 11,
Centre Scientifique d’Orsay, F-91898 Orsay Cedex, France
D. J. Lange and D. M. Wright
Lawrence Livermore National Laboratory, Livermore, Calif
ornia 94550, USA
J. P. Coleman, J. R. Fry, E. Gabathuler, D. E. Hutchcroft, D. J. P
ayne, and C. Touramanis
3
University of Liverpool, Liverpool L69 7ZE, United Kingdom
A. J. Bevan, F. Di Lodovico, and R. Sacco
Queen Mary, University of London, London, E1 4NS, United Kin
gdom
G. Cowan
University of London, Royal Holloway and Bedford New Colleg
e, Egham, Surrey TW20 0EX, United Kingdom
J. Bougher, D. N. Brown, and C. L. Davis
University of Louisville, Louisville, Kentucky 40292, USA
A. G. Denig, M. Fritsch, W. Gradl, K. Griessinger, A. Hafner, E. Pre
ncipe, and K. R. Schubert
Johannes Gutenberg-Universit ̈at Mainz, Institut f ̈ur Ker
nphysik, D-55099 Mainz, Germany
R. J. Barlow
§
and G. D. Lafferty
University of Manchester, Manchester M13 9PL, United Kingd
om
R. Cenci, B. Hamilton, A. Jawahery, and D. A. Roberts
University of Maryland, College Park, Maryland 20742, USA
R. Cowan and G. Sciolla
Massachusetts Institute of Technology, Laboratory for Nuc
lear Science, Cambridge, Massachusetts 02139, USA
R. Cheaib, P. M. Patel,
¶
and S. H. Robertson
McGill University, Montr ́eal, Qu ́ebec, Canada H3A 2T8
N. Neri
a
and F. Palombo
ab
INFN Sezione di Milano
a
; Dipartimento di Fisica, Universit`a di Milano
b
, I-20133 Milano, Italy
L. Cremaldi, R. Godang,
∗∗
P. Sonnek, and D. J. Summers
University of Mississippi, University, Mississippi 38677
, USA
M. Simard and P. Taras
Universit ́e de Montr ́eal, Physique des Particules, Montr ́
eal, Qu ́ebec, Canada H3C 3J7
G. De Nardo
ab
, G. Onorato
ab
, and C. Sciacca
ab
INFN Sezione di Napoli
a
; Dipartimento di Scienze Fisiche,
Universit`a di Napoli Federico II
b
, I-80126 Napoli, Italy
M. Martinelli and G. Raven
NIKHEF, National Institute for Nuclear Physics and High Ene
rgy Physics, NL-1009 DB Amsterdam, The Netherlands
C. P. Jessop and J. M. LoSecco
University of Notre Dame, Notre Dame, Indiana 46556, USA
K. Honscheid and R. Kass
Ohio State University, Columbus, Ohio 43210, USA
E. Feltresi
ab
, M. Margoni
ab
, M. Morandin
a
, M. Posocco
a
, M. Rotondo
a
, G. Simi
ab
, F. Simonetto
ab
, and R. Stroili
ab
INFN Sezione di Padova
a
; Dipartimento di Fisica, Universit`a di Padova
b
, I-35131 Padova, Italy
S. Akar, E. Ben-Haim, M. Bomben, G. R. Bonneaud, H. Briand,
G. Calderini, J. Chauveau, Ph. Leruste, G. Marchiori, and J. Ocariz
Laboratoire de Physique Nucl ́eaire et de Hautes Energies,
IN2P3/CNRS, Universit ́e Pierre et Marie Curie-Paris6,
Universit ́e Denis Diderot-Paris7, F-75252 Paris, France
M. Biasini
ab
, E. Manoni
a
, S. Pacetti
ab
, and A. Rossi
a
4
INFN Sezione di Perugia
a
; Dipartimento di Fisica, Universit`a di Perugia
b
, I-06123 Perugia, Italy
C. Angelini
ab
, G. Batignani
ab
, S. Bettarini
ab
, M. Carpinelli
ab
,
††
G. Casarosa
ab
, A. Cervelli
ab
, M. Chrzaszcz
ab
,
F. Forti
ab
, M. A. Giorgi
ab
, A. Lusiani
ac
, B. Oberhof
ab
, E. Paoloni
ab
, A. Perez
a
, G. Rizzo
ab
, and J. J. Walsh
a
INFN Sezione di Pisa
a
; Dipartimento di Fisica, Universit`a di Pisa
b
; Scuola Normale Superiore di Pisa
c
, I-56127 Pisa, Italy
D. Lopes Pegna, J. Olsen, and A. J. S. Smith
Princeton University, Princeton, New Jersey 08544, USA
R. Faccini
ab
, F. Ferrarotto
a
, F. Ferroni
ab
, M. Gaspero
ab
, L. Li Gioi
a
, and G. Piredda
a
INFN Sezione di Roma
a
; Dipartimento di Fisica,
Universit`a di Roma La Sapienza
b
, I-00185 Roma, Italy
C. B ̈unger, S. Dittrich, O. Gr ̈unberg, T. Hartmann, M. Hess, T.
Leddig, C. Voß, and R. Waldi
Universit ̈at Rostock, D-18051 Rostock, Germany
T. Adye, E. O. Olaiya, and F. F. Wilson
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX
11 0QX, United Kingdom
S. Emery and G. Vasseur
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, F
rance
F. Anulli,
‡‡
D. Aston, D. J. Bard, C. Cartaro, M. R. Convery, J. Dorfan, G. P
. Dubois-Felsmann,
W. Dunwoodie, M. Ebert, R. C. Field, B. G. Fulsom, M. T. Graham, C. H
ast, W. R. Innes, P. Kim,
D. W. G. S. Leith, P. Lewis, D. Lindemann, S. Luitz, V. Luth, H. L. Ly
nch, D. B. MacFarlane, D. R. Muller,
H. Neal, M. Perl, T. Pulliam, B. N. Ratcliff, A. Roodman, A. A. Salnikov, R
. H. Schindler, A. Snyder,
D. Su, M. K. Sullivan, J. Va’vra, A. P. Wagner, W. F. Wang, W. J. Wisnie
wski, and H. W. Wulsin
SLAC National Accelerator Laboratory, Stanford, Californ
ia 94309 USA
M. V. Purohit, R. M. White,
§§
and J. R. Wilson
University of South Carolina, Columbia, South Carolina 292
08, USA
A. Randle-Conde and S. J. Sekula
Southern Methodist University, Dallas, Texas 75275, USA
M. Bellis, P. R. Burchat, and E. M. T. Puccio
Stanford University, Stanford, California 94305-4060, US
A
M. S. Alam and J. A. Ernst
State University of New York, Albany, New York 12222, USA
R. Gorodeisky, N. Guttman, D. R. Peimer, and A. Soffer
Tel Aviv University, School of Physics and Astronomy, Tel Av
iv, 69978, Israel
S. M. Spanier
University of Tennessee, Knoxville, Tennessee 37996, USA
J. L. Ritchie, A. M. Ruland, R. F. Schwitters, and B. C. Wray
University of Texas at Austin, Austin, Texas 78712, USA
J. M. Izen and X. C. Lou
University of Texas at Dallas, Richardson, Texas 75083, USA
F. Bianchi
ab
, F. De Mori
ab
, A. Filippi
a
, and D. Gamba
ab
INFN Sezione di Torino
a
; Dipartimento di Fisica, Universit`a di Torino
b
, I-10125 Torino, Italy
L. Lanceri
ab
and L. Vitale
ab
INFN Sezione di Trieste
a
; Dipartimento di Fisica, Universit`a di Trieste
b
, I-34127 Trieste, Italy
5
F. Martinez-Vidal, A. Oyanguren, and P. Villanueva-Perez
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spa
in
J. Albert, Sw. Banerjee, A. Beaulieu, F. U. Bernlochner, H. H. F. C
hoi, G. J. King, R. Kowalewski,
M. J. Lewczuk, T. Lueck, I. M. Nugent, J. M. Roney, R. J. Sobie, a
nd N. Tasneem
University of Victoria, Victoria, British Columbia, Canad
a V8W 3P6
T. J. Gershon, P. F. Harrison, and T. E. Latham
Department of Physics, University of Warwick, Coventry CV4
7AL, United Kingdom
H. R. Band, S. Dasu, Y. Pan, R. Prepost, and S. L. Wu
University of Wisconsin, Madison, Wisconsin 53706, USA
We study the processes
γγ
→
K
+
K
−
η
and
γγ
→
K
+
K
−
π
0
using a data sample of 519 fb
−
1
recorded
with the
B
A
B
AR
detector operating at the SLAC PEP-II asymmetric-energy
e
+
e
−
collider at center-
of-mass energies at and near the
Υ
(
nS
) (
n
= 2
,
3
,
4) resonances. We observe
η
c
→
K
+
K
−
π
0
and
η
c
→
K
+
K
−
η
decays, measure their relative branching fraction, and per
form a Dalitz plot analysis
for each decay. We observe the
K
∗
0
(1430)
→
Kη
decay and measure its branching fraction relative to
the
Kπ
decay mode to be
R
(
K
∗
0
(1430)) =
B
(
K
∗
0
(1430)
→
Kη
)
B
(
K
∗
0
(1430)
→
Kπ
)
= 0
.
092
±
0
.
025
+0
.
010
−
0
.
025
. The
η
c
→
K
+
K
−
η
and
K
∗
0
(1430)
→
Kη
results correspond to the first observations of these channe
ls. The data also
show evidence for
η
c
(2
S
)
→
K
+
K
−
π
0
and first evidence for
η
c
(2
S
)
→
K
+
K
−
η
.
PACS numbers: 13.25.Gv, 14.40.Pq, 14.40.Df, 14.40.Be
I. INTRODUCTION
Charmonium decays, in particular
J/ψ
radiative and
hadronic decays, have been studied extensively [1, 2].
One of the motivations for these studies is the search
for non-
q
̄
q
mesons such as glueballs or molecular states
that are predicted by QCD to populate the low mass re-
gion of the hadron mass spectrum [3]. Recently, a search
for exotic resonances was performed through Dalitz plot
analyses of
χ
c
1
states [4].
Scalar mesons are still a puzzle in light-meson spec-
troscopy: there are too many states and they are not con-
sistent with the quark model. In particular, the
f
0
(1500)
resonance, discovered in ̄
pp
annihilations, has been in-
terpreted as a scalar glueball [5]. However, no evidence
for the
f
0
(1500) state has been found in charmonium
decays. Another glueball candidate is the
f
0
(1710) dis-
covered in radiative
J/ψ
decays. Recently,
f
0
(1500) and
f
0
(1710) signals have been incorporated in a Dalitz plot
∗
Now at the University of Tabuk, Tabuk 71491, Saudi Arabia
†
Also with Universit`a di Perugia, Dipartimento di Fisica, P
erugia,
Italy
‡
Now at Laboratoire de Physique Nucl ́eaire et de Hautes Energ
ies,
IN2P3/CNRS, Paris, France
§
Now at the University of Huddersfield, Huddersfield HD1 3DH,
UK
¶
Deceased
∗∗
Now at University of South Alabama, Mobile, Alabama 36688,
USA
††
Also with Universit`a di Sassari, Sassari, Italy
‡‡
Also with INFN Sezione di Roma, Roma, Italy
§§
Now at Universidad T ́ecnica Federico Santa Maria, Valparai
so,
Chile 2390123
analysis of
B
→
3
K
decays [6]. Charmless
B
→
KX
decays
could show enhanced gluonium production [7]. Another
puzzling state is the
K
∗
0
(1430) resonance, never observed
as a clear peak in the
Kπ
mass spectrum. In the de-
scription of the scalar amplitude in
Kπ
scattering, the
K
∗
0
(1430) resonance is added coherently to an effective-
range description of the low-mass
Kπ
system in such a
way that the net amplitude actually decreases rapidly
at the resonance mass. The
K
∗
0
(1430) parameter values
were measured by the LASS experiment in the reaction
K
−
p
→
K
−
π
+
n
[8]; the corrected
S
-wave amplitude rep-
resentation is given explicitly in Ref. [9]. In the present
analysis, we study three-body
η
c
decays to pseudoscalar
mesons and obtain results that are relevant to several
issues in light-meson spectroscopy.
Many
η
c
and
η
c
(2
S
) decay modes remain unobserved,
while others have been studied with very limited statis-
tical precision. In particular, the branching fraction for
the decay mode
η
c
→
K
+
K
−
η
has been measured by the
BESIII experiment based on a fitted yield of only 6
.
7
±
3
.
2
events [10]. No Dalitz plot analysis has been performed
on
η
c
three-body decays.
We describe herein a study of the
K
+
K
−
η
and
K
+
K
−
π
0
systems produced in two-photon interactions.
Two-photon events in which at least one of the interact-
ing photons is not quasi-real are strongly suppressed by
the selection criteria described below. This implies that
the allowed
J
P C
values of any produced resonances are
0
±
+
, 2
±
+
, 3
++
, 4
±
+
... [11]. Angular momentum conser-
vation, parity conservation, and charge conjugation in-
variance imply that these quantum numbers also apply
to the final state except that the
K
+
K
−
η
and
K
+
K
−
π
0
states cannot be in a
J
P
= 0
+
state.
This article is organized as follows. In Sec. II, a brief
6
description of the
B
A
B
AR
detector is given. Section III
is devoted to the event reconstruction and data selection.
In Sec. IV, we describe the study of efficiency and resolu-
tion, while in Sec. V the mass spectra are presented. Sec-
tion VI is devoted to the measurement of the branching
ratios, while Sec. VII describes the Dalitz plot analyses.
In Sec. VIII, we report the measurement of the
K
∗
0
(1430)
branching ratio, in Sec. IX we discuss its implications for
the pseudoscalar meson mixing angle, and in Sec. X we
summarize the results.
II. THE
B
A
B
AR
DETECTOR AND DATASET
The results presented here are based on data collected
with the
B
A
B
AR
detector at the PEP-II asymmetric-
energy
e
+
e
−
collider located at SLAC and correspond
to an integrated luminosity of 519 fb
−
1
[12] recorded
at center-of-mass energies at and near the
Υ
(
nS
) (
n
=
2
,
3
,
4) resonances. The
B
A
B
AR
detector is described in
detail elsewhere [13]. 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 electro-
magnetic 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. Muons and
K
0
L
mesons are
detected in the instrumented flux return of the magnet.
Monte Carlo (MC) simulated events [14], with sample
sizes more than 10 times larger than the corresponding
data samples, are used to evaluate signal efficiency and
to determine background features. Two-photon events
are simulated using the GamGam MC generator [15].
III. EVENT RECONSTRUCTION AND DATA
SELECTION
In this analysis, we select events in which the
e
+
and
e
−
beam particles are scattered at small angles and are
undetected in the final state. We study the following
reactions
γγ
→
K
+
K
−
η,
(
η
→
γγ
)
,
(1)
γγ
→
K
+
K
−
η,
(
η
→
π
+
π
−
π
0
)
,
(2)
and
γγ
→
K
+
K
−
π
0
.
(3)
For reactions (1) and (3), we consider only events
for which the number of well-measured charged-particle
tracks with transverse momenta greater than 0.1 GeV
/c
is exactly equal to two. For reaction (2), we require the
number of well-measured charged-particle tracks to be
exactly equal to four. The charged-particle tracks are fit
to a common vertex with the requirements that they orig-
inate from the interaction region and that the
χ
2
proba-
bility of the vertex fit be greater than 0.1%. We observe
prominent
η
c
signals in all three reactions and improve
the signal-to-background ratio using the data, in partic-
ular the
c
̄
c η
c
resonance. In the optimization procedure,
we retain only selection criteria that do not remove signif-
icant
η
c
signal. For the reconstruction of
π
0
→
γγ
decays,
we require the energy of the less-energetic photon to be
greater than 30 MeV for reaction (2) and 50 MeV for
reaction (3). For
η
→
γγ
decay, we require the energy of
the less energetic photon to be greater than 100 MeV.
Each pair of
γ
’s is kinematically fit to a
π
0
or
η
hypoth-
esis requiring it to emanate from the primary vertex of
the event, and with the diphoton mass constrained to
the nominal
π
0
or
η
mass, respectively [16]. Due to the
presence of soft-photon background, we do not impose
a veto on the presence of additional photons in the final
state. For reaction (1), we require the presence of exactly
one
η
candidate in each event and discard events having
additional
π
0
’s decaying to
γ
’s with energy greater than
70 MeV. For reaction (3), we accept no more than two
π
0
candidates in the event.
In reaction (2), the
η
is reconstructed by combining two
oppositely charged tracks identified as pions with each of
the
π
0
candidates in the event. The
η
signal mass re-
gion is defined as 541
< m
(
π
+
π
−
π
0
)
<
554 MeV
/c
2
.
The momentum three-vectors of the the final-state pi-
ons are combined and the energy of the
η
candidate
is computed using the nominal
η
mass. According to
tests with simulated events, this method improves the
K
+
K
−
η
mass resolution. We check for possible back-
ground from the reaction
γγ
→
K
+
K
−
π
+
π
−
π
0
[17] us-
ing
η
sideband regions and find it to be consistent with
zero. Background arises mainly from random combina-
tions of particles from
e
+
e
−
annihilation, from other two-
photon processes, and from events with initial-state pho-
ton radiation (ISR). The ISR background is dominated
by
J
P C
= 1
−−
resonance production [18]. We discrim-
inate against
K
+
K
−
η
(
K
+
K
−
π
0
) events produced via
ISR by requiring
M
2
rec
≡
(
p
e
+
e
−
−
p
rec
)
2
>
10 (GeV
2
/
c
4
,
where
p
e
+
e
−
is the four-momentum of the initial state and
p
rec
is the four-momentum of the
K
+
K
−
η
(
K
+
K
−
π
0
)
system. This requirement also removes a large fraction
of a residual
J/ψ
contribution.
Particle identification is used in two different ways. For
reaction (2), with four charged particles in the final state,
we require two oppositely charged particles to be loosely
identified as kaons and the other two tracks to be con-
sistent with pions. For reactions (1) and (3), with only
two charged particles in the final state, we loosely iden-
tify one kaon and require that neither track be a well-
identified pion, electron, or muon. We define
p
T
as the
magnitude of the vector sum of the transverse momenta,
in the
e
+
e
−
rest frame, of the final-state particles with
respect to the beam axis. Since well-reconstructed two-
7
photon events are expected to have low values of
p
T
,
we require
p
T
<
0
.
05 GeV
/c
. Reaction (3) is affected
by background from the reaction
γγ
→
K
+
K
−
where soft
photon background simulates the presence of a low mo-
mentum
π
0
. We reconstruct this mode and reject events
having a
γγ
→
K
+
K
−
candidate with
p
T
<
0
.
1 GeV
/c
.
(GeV/c)
T
p
0
0.1
0.2
0.3
0.4
0.5
events/(5 MeV/c)
0
200
400
600
800
(a)
(GeV/c)
T
p
0
0.1
0.2
0.3
0.4
0.5
events/(5 MeV/c)
0
100
200
300
(b)
(GeV/c)
T
p
0
0.1
0.2
0.3
0.4
0.5
events/(5 MeV/c)
0
5000
10000
15000
20000
(c)
FIG. 1: Distributions of
p
T
for (a)
γγ
→
K
+
K
−
η
(
η
→
γγ
),
(b)
γγ
→
K
+
K
−
η
(
η
→
π
+
π
−
π
0
), and (c)
γγ
→
K
+
K
−
π
0
. In
each figure the data are shown as points with error bars,
and the MC simulation is shown as a histogram; the verti-
cal dashed line indicates the selection applied to isolate t
wo-
photon events.
Figure 1 shows the measured
p
T
distribution for each
of the three reactions in comparison to the corresponding
p
T
distribution obtained from simulation. A peak at low
p
T
is observed in all three distributions indicating the
presence of the two-photon process. The shape of the
peak agrees well with that seen in the MC simulation.
IV. EFFICIENCY AND RESOLUTION
To compute the efficiency,
η
c
and
η
c
(2
S
) MC signal
events for the different channels are generated using a de-
tailed detector simulation [14] in which the
η
c
and
η
c
(2
S
)
mesons decay uniformly in phase space. These simulated
events are reconstructed and analyzed in the same man-
ner as data. The efficiency is computed as the ratio of
reconstructed to generated events. Due to the presence
of long tails in the Breit-Wigner (BW) representation
of the resonances, we apply selection criteria to restrict
the generated events to the
η
c
and
η
c
(2
S
) mass regions.
We express the efficiency as a function of the
m
(
K
+
K
−
)
mass and cos
θ
, where
θ
is the angle in the
K
+
K
−
rest
frame between the directions of the
K
+
and the boost
from the
K
+
K
−
η
or
K
+
K
−
π
0
rest frame. To smooth
statistical fluctuations, this efficiency is then parameter-
ized as follows.
First we fit the efficiency as a function of cos
θ
in sep-
arate intervals of
m
(
K
+
K
−
), in terms of Legendre poly-
nomials up to
L
= 12:
ǫ
(cos
θ
) =
12
∑
L
=0
a
L
(
m
)
Y
0
L
(cos
θ
)
,
(4)
where
m
denotes
K
+
K
−
invariant mass. For each value
of
L
, we fit the mass dependent coefficients
a
L
(
m
) with
a seventh-order polynomial in
m
. Figure 2 shows the re-
sulting fitted efficiency
ǫ
(
m,
cos
θ
) for each of the three re-
actions. We observe a significant decrease in efficiency for
cos
θ
∼ ±
1 and 1
.
1
< m
(
K
+
K
−
)
<
1
.
5 GeV
/c
2
due to
the impossibility of reconstructing low-momentum kaons
(p
<
200 MeV
/c
in the laboratory frame) which have expe-
rienced significant energy loss in the beampipe and inner-
detector material. The efficiency decrease at high
m
for
η
c
→
K
+
K
−
η
(
η
→
π
+
π
−
π
0
) (Fig. 2(b)) results from the
loss of a low-momentum
π
0
from the
η
decay.
The mass resolution, ∆
m
, is measured as the difference
between the generated and reconstructed
K
+
K
−
η
or
K
+
K
−
π
0
invariant-mass values. Figure 3 shows the ∆
m
distribution for each of the
η
c
signal regions; these de-
viate from Gaussian line shapes due to a low-energy tail
caused by the response of the CsI calorimeter to photons.
We fit the distribution for the
K
+
K
−
η
(
η
→
π
+
π
−
π
0
) fi-
nal state to a Crystal Ball function [19], and those for
the
K
+
K
−
η
(
η
→
γγ
) and
K
+
K
−
π
0
final states to a sum
of a Crystal Ball function and a Gaussian function. The
root-mean-squared values are 15, 14, and 21 MeV
/c
2
at
the
η
c
mass, and 18, 15, and 24 MeV
/c
2
at the
η
c
(2
S
)
mass, for the
K
+
K
−
η
(
η
→
γγ
),
K
+
K
−
η
(
η
→
π
+
π
−
π
0
),
and
K
+
K
−
π
0
final states, respectively.
V. MASS SPECTRA
Figure 4(a) shows the
K
+
K
−
η
mass spectrum,
summed over the two
η
decay modes, before applying
the efficiency correction. There are 2950 events in the
8
0
0.02
0.04
0.06
0.08
0.1
)
2
) (GeV/c
-
K
+
m(K
1
1.5
2
θ
cos
-1
-0.5
0
0.5
1
(a)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
)
2
) (GeV/c
-
K
+
m(K
1
1.5
2
θ
cos
-1
-0.5
0
0.5
1
(b)
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
)
2
) (GeV/c
-
K
+
m(K
1
1.5
2
2.5
θ
cos
-1
-0.5
0
0.5
1
(c)
FIG. 2: Fitted detection efficiency in the cos
θ vs. m
(
K
+
K
−
)
plane for (a)
η
c
→
K
+
K
−
η
(
η
→
γγ
), (b)
η
c
→
K
+
K
−
η
(
η
→
π
+
π
−
π
0
), and (c)
η
c
→
K
+
K
−
π
0
. Each bin shows the
average value of the fit in that region.
mass region between 2.7 and 3.8 GeV
/c
2
, of which 73%
are from the
η
→
γγ
decay mode and 27% are from the
η
→
π
+
π
−
π
0
decay mode. We observe a strong
η
c
signal
and a small enhancement at the position of the
η
c
(2
S
).
The
η
c
signal-to-background ratio for each of the
η
decay
modes is approximately the same. We perform a simulta-
neous fit to the
K
+
K
−
η
mass spectra for the two
η
decay
modes. For each resonance, the mass and width are con-
)
2
m (GeV/c
∆
-0.1
-0.05
0
0.05
0.1
)
2
events/(2 MeV/c
0
200
400
600
800
(a)
)
2
m (GeV/c
∆
-0.1
-0.05
0
0.05
0.1
)
2
events/(2 MeV/c
0
200
400
600
800
(b)
)
2
m (GeV/c
∆
-0.1
0
0.1
)
2
events/(2.5 MeV/c
0
500
1000
(c)
FIG. 3: MC mass resolution for (a)
η
c
→
K
+
K
−
η
(
η
→
γγ
),
(b)
η
c
→
K
+
K
−
η
(
η
→
π
+
π
−
π
0
), and (c)
η
c
→
K
+
K
−
π
0
. The
curves represent the fits described in the text.
strained to take the same fitted values in both distribu-
tions. Backgrounds are described by second-order poly-
nomials, and each resonance is represented by a simple
Breit-Wigner function convolved with the corresponding
resolution function. In addition, we include a signal func-
tion for the
χ
c
2
resonance with parameters fixed to their
PDG values [16]. Figure 4(a) shows the fit result, and
Table I summarizes the
η
c
and
η
c
(2
S
) parameter values.
We have only a weak constraint on the
η
c
(2
S
) width and
so fix its value to 11.3 MeV [16].
The
K
+
K
−
π
0
mass spectrum is shown in Fig. 4(b).
There are 23 720 events in the mass region between 2.7
and 3.9 GeV
/c
2
. We observe a strong
η
c
signal and a
small signal at the position of the
η
c
(2
S
) on top of a
sizeable background. We perform a fit to the
K
+
K
−
π
0
9
)
2
) (GeV/c
η
-
K
+
m(K
3
3.5
)
2
events/(10 MeV/c
0
50
100
150
200
(a)
)
2
) (GeV/c
0
π
-
K
+
m(K
3
3.5
)
2
events/(6 MeV/c
0
200
400
(b)
FIG. 4: (a) The
K
+
K
−
η
mass spectrum summed over the two
η
decay modes. (b) The
K
+
K
−
π
0
mass spectrum. In each
figure, the solid curve shows the total fitted function and the
dashed curve shows the fitted background contribution.
mass spectrum using the background function
B
(
m
) =
e
a
1
m
+
a
2
m
2
for
m < m
0
and
B
(
m
) =
e
b
0
+
b
1
m
+
b
2
m
2
for
m > m
0
, where
m
=
m
(
K
+
K
−
π
0
) and
a
i
,
b
i
, and
m
0
are free parameters [20]. The two functions and their
first derivatives are required to be continuous at
m
0
, so
that the resulting function has only four independent pa-
rameters. In addition, we allow for the presence of a
residual
J/ψ
contribution modeled as a simple Gaussian
function. Its parameter values are fixed to those from a
fit to the
K
+
K
−
π
0
mass spectrum for the ISR data sam-
ple obtained requiring
|
M
2
rec
|
<
1 (GeV
/c
2
)
2
. Figure 4(b)
shows the fit to the
K
+
K
−
π
0
mass spectrum, and Ta-
ble I summarizes the resulting
η
c
and
η
c
(2
S
) parameter
values.
TABLE I: Fitted
η
c
and
η
c
(2
S
) parameter values. The first
uncertainty is statistical and the second is systematic.
Resonance
Mass (MeV
/c
2
)
Γ (MeV)
η
c
→
K
+
K
−
η
2984
.
1
±
1
.
1
±
2
.
1 34
.
8
±
3
.
1
±
4
.
0
η
c
→
K
+
K
−
π
0
2979
.
8
±
0
.
8
±
3
.
5 25
.
2
±
2
.
6
±
2
.
4
η
c
(2
S
)
→
K
+
K
−
η
3635
.
1
±
5
.
8
±
2
.
1 11.3 (fixed)
η
c
(2
S
)
→
K
+
K
−
π
0
3637
.
0
±
5
.
7
±
3
.
4 11.3 (fixed)
The following systematic uncertainties are considered.
The background uncertainty contribution is estimated by
replacing each function by a third-order polynomial. The
mass scale uncertainty is estimated from fits to the
J/ψ
signal in ISR events. In the case of
η
c
→
K
+
K
−
η
, we per-
form independent fits to the mass spectra obtained for
the two
η
decay modes, and consider the mass difference
as a measurement of systematic uncertainty. The differ-
ent contributions are added in quadrature to obtain the
values quoted in Table I.
VI. BRANCHING RATIOS
We compute the ratios of the branching fractions for
η
c
and
η
c
(2
S
) decays to the
K
+
K
−
η
final state compared to
the respective branching fractions to the
K
+
K
−
π
0
final
state. The ratios are computed as
R
=
B
(
η
c
/η
c
(2
S
)
→
K
+
K
−
η
)
B
(
η
c
/η
c
(2
S
)
→
K
+
K
−
π
0
)
=
N
K
+
K
−
η
N
K
+
K
−
π
0
ǫ
K
+
K
−
π
0
ǫ
K
+
K
−
η
1
B
η
.
(5)
For each
η
decay mode,
N
K
+
K
−
η
and
N
K
+
K
−
π
0
repre-
sent the fitted yields for
η
c
and
η
c
(2
S
) in the
K
+
K
−
η
and
K
+
K
−
π
0
mass spectra,
ǫ
K
+
K
−
η
and
ǫ
K
+
K
−
π
0
are the
corresponding efficiencies, and
B
η
indicates the particu-
lar
η
branching fraction. The PDG values of the branch-
ing fractions are (39
.
41
±
0
.
20)% and (22
.
92
±
0
.
28)% for
the
η
→
γγ
and
η
→
π
+
π
−
π
0
, respectively [16]. We esti-
mate
ǫ
K
+
K
−
η
and
ǫ
K
+
K
−
π
0
for the
η
c
signals by making
use of the 2-D efficiency functions described in Sec. IV
and weighting each event by 1
/ǫ
(
m,
cos
θ
). Due to the
presence of non-negligible backgrounds in the
η
c
signals,
which have different distributions in the Dalitz plot, we
perform a sideband subtraction by assigning a weight +1
to events in the signal region and a negative weight to
events in the sideband regions. The weight in the side-
band regions is scaled down to match the fitted
η
c
sig-
10
nal/background ratio. To remove the dependence of the
fit quality on the efficiency functions we make use of the
unfitted efficiency distributions. Due to the presence of
a sizeable background for the
η
c
(2
S
), we use the average
efficiency value from the simulation.
We determine
N
K
+
K
−
η
and
N
K
+
K
−
π
0
for the
η
c
by
performing fits to the
K
+
K
−
η
and
K
+
K
−
π
0
mass spec-
tra. The width is extracted from the simultaneous fit to
the
K
+
K
−
η
mass spectra, and is fixed to this value in
the fit to the
K
+
K
−
π
0
mass spectrum. This procedure
is adopted because the signal-to-background ratio at the
peak is much better for the
K
+
K
−
η
mode (
∼
8:1 com-
pared to
∼
2:1 for the
K
+
K
−
π
0
mode) while the residual
J/ψ
contamination is much smaller. The
η
c
and
η
c
(2
S
)
mass values are determined from the fits. For the
η
c
(2
S
),
we fix the width to 11.3 MeV [16]. The resulting yields,
efficiencies, measured branching ratios, and significances
are reported in Table II. The significances are evaluated
as
N
s
/σ
T
where
N
s
is the signal event yield and
σ
T
is
the total uncertainty obtained by adding the statistical
and systematic contributions in quadrature.
We calculate the weighted mean of the
η
c
branching-
ratio estimates for the two
η
decay modes and obtain
R
(
η
c
) =
B
(
η
c
→
K
+
K
−
η
)
B
(
η
c
→
K
+
K
−
π
0
)
= 0
.
571
±
0
.
025
±
0
.
051
,
(6)
which is consistent with the BESIII measurement
of 0
.
46
±
0
.
23 [10]. Since the sample size for
η
c
(2
S
)
→
K
+
K
−
η
decays with
η
→
π
+
π
−
π
0
is small, we
use only the
η
→
γγ
decay mode, and obtain
R
(
η
c
(2
S
)) =
B
(
η
c
(2
S
)
→
K
+
K
−
η
)
B
(
η
c
(2
S
)
→
K
+
K
−
π
0
)
= 0
.
82
±
0
.
21
±
0
.
27
.
(7)
In evaluating
R
(
η
c
) for the
η
→
γγ
decay mode, we note
that the number of charged-particle tracks and
γ
’s is the
same in the numerator and in the denominator of the ra-
tio, so that several systematic uncertainties cancel. Con-
cerning the contribution of the
η
→
π
+
π
−
π
0
decay, we find
systematic uncertainties related to the difference in the
number of charged-particle tracks to be negligible. We
consider the following sources of systematic uncertainty.
We modify the
η
c
width by fixing its value to the PDG
value [16]. We modify the background model by using
fourth-order polynomials or exponential functions. The
uncertainty due to the efficiency weight is evaluated by
computing 1000 new weights obtained by randomly mod-
ifying the weight in each cell of the
ǫ
(
m
(
K
+
K
−
)
,
cos
θ
)
plane according to its statistical uncertainty. The widths
of the resulting Gaussian distributions yield the estimate
of the systematic uncertainty for the efficiency weight-
ing procedure. These values are reported as the weight
uncertainties in Table II.
VII. DALITZ PLOT ANALYSES
We perform Dalitz plot analyses of the
K
+
K
−
η
and
K
+
K
−
π
0
systems in the
η
c
mass region using unbinned
maximum likelihood fits. The likelihood function is writ-
ten as
L
=
N
∏
n
=1
[
f
sig
(
m
n
)
·
ǫ
(
x
′
n
,y
′
n
)
∑
i,j
c
i
c
∗
j
A
i
(
x
n
,y
n
)
A
∗
j
(
x
n
,y
n
)
∑
i,j
c
i
c
∗
j
I
A
i
A
∗
j
+(1
−
f
sig
(
m
n
))
∑
i
k
i
B
i
(
x
n
,y
n
)
∑
i
k
i
I
B
i
]
(8)
where
•
N
is the number of events in the signal region;
•
for the
n
-th event,
m
n
is the
K
+
K
−
η
or the
K
+
K
−
π
0
invariant mass;
•
for the
n
-th event,
x
n
=
m
2
(
K
+
η
),
y
n
=
m
2
(
K
−
η
)
for
K
+
K
−
η
;
x
n
=
m
2
(
K
+
π
0
),
y
n
=
m
2
(
K
−
π
0
) for
K
+
K
−
π
0
;
•
f
sig
is the mass-dependent fraction of signal ob-
tained from the fit to the
K
+
K
−
η
or
K
+
K
−
π
0
mass spectrum;
•
for the
n
-th event,
ǫ
(
x
′
n
,y
′
n
) is the efficiency pa-
rameterized as a function
x
′
n
=
m
(
K
+
K
−
) and
y
′
n
= cos
θ
(see Sec. IV);
•
for the
n
-th event, the
A
i
(
x
n
,y
n
) describe the com-
plex signal-amplitude contributions;
•
c
i
is the complex amplitude of the
i
−
th signal com-
ponent; the
c
i
parameters are allowed to vary dur-
ing the fit process;
•
for the
n
-th event, the
B
i
(
x
n
,y
n
) describe the back-
ground probability-density functions assuming that
interference between signal and background ampli-
tudes can be ignored;
•
k
i
is the magnitude of the
i
−
th background com-
ponent; the
k
i
parameters are obtained by fitting
the sideband regions;
•
I
A
i
A
∗
j
=
∫
A
i
(
x,y
)
A
∗
j
(
x,y
)
ǫ
(
m
(
K
+
K
−
)
,
cos
θ
) d
x
d
y
and
I
B
i
=
∫
B
i
(
x,y
)d
x
d
y
are normalization
integrals; numerical integration is performed on
phase space generated events.
Amplitudes are parameterized as described in Refs. [21]
and [22]. The efficiency-corrected fractional contribution
f
i
due to resonant or nonresonant contribution
i
is de-
fined as follows:
f
i
=
|
c
i
|
2
∫
|
A
i
(
x
n
,y
n
)
|
2
d
x
d
y
∫
|
∑
j
c
j
A
j
(
x,y
)
|
2
d
x
d
y
.
(9)
The
f
i
do not necessarily sum to 100% because of inter-
ference effects. The uncertainty for each
f
i
is evaluated
by propagating the full covariance matrix obtained from
the fit.
11
TABLE II: Summary of the results from the fits to the
K
+
K
−
η
and
K
+
K
−
π
0
mass spectra. The table lists event yields,
efficiency correction weights, resulting branching ratios a
nd significances. For event yields, the first uncertainty is s
tatistical
and the second is systematic. In the evaluation of significan
ces, systematic uncertainties are included.
Channel
Event yield
Weights
R
Significance
η
c
→
K
+
K
−
π
0
4518
±
131
±
50 17.0
±
0.7
32
σ
η
c
→
K
+
K
−
η
(
η
→
γγ
)
853
±
38
±
11 21.3
±
0.6
21
σ
B
(
η
c
→
K
+
K
−
η
)
/
B
(
η
c
→
K
+
K
−
π
0
)
0
.
602
±
0
.
032
±
0
.
065
η
c
→
K
+
K
−
η
(
η
→
π
+
π
−
π
0
)
292
±
20
±
7 31.2
±
2.1
14
σ
B
(
η
c
→
K
+
K
−
η
)
/
B
(
η
c
→
K
+
K
−
π
0
)
0
.
523
±
0
.
040
±
0
.
083
η
c
(2
S
)
→
K
+
K
−
π
0
178
±
29
±
39 14.3
±
1.3
3.7
σ
η
c
(2
S
)
→
K
+
K
−
η
47
±
9
±
3 17.4
±
0.4
4.9
σ
B
(
η
c
(2
S
)
→
K
+
K
−
η
)
/
B
(
η
c
(2
S
)
→
K
+
K
−
π
0
)
0
.
82
±
0
.
21
±
0
.
27
χ
c
2
→
K
+
K
−
π
0
88
±
27
±
23
2.5
σ
χ
c
2
→
K
+
K
−
η
2
±
5
±
2
0.0
σ
A. Dalitz plot analysis of
η
c
→
K
+
K
−
η
We define the
η
c
signal region as the range 2.922-
3.036 GeV
/c
2
. This region contains 1161 events with
(76.1
±
1.3)% purity, defined as
S/
(
S
+
B
) where
S
and
B
indicate the number of signal and back-
ground events, respectively, as determined from the fit
(Fig. 4(a)). Sideband regions are defined as the ranges
2.730-2.844 GeV
/c
2
and 3.114-3.228 GeV
/c
2
, respec-
tively. Figure 5 shows the Dalitz plot for the
η
c
signal
region and Fig. 6 shows the Dalitz plot projections.
FIG. 5: Dalitz plot for the
η
c
→
K
+
K
−
η
events in the sig-
nal region. The shaded area denotes the accessible kinemati
c
region.
We observe signals in the
K
+
K
−
projections cor-
responding to the
f
0
(980),
f
0
(1500),
f
0
(1710), and
f
0
(2200) states. We also observe a broad signal in the
1.43 GeV
/c
2
mass region in the
K
+
η
and
K
−
η
projec-
tions.
In describing the Dalitz plot, we note that amplitude
contributions to the
K
+
K
−
system must have isospin
zero in order to satisfy overall isospin conservation in
η
c
decay. In addition, amplitudes of the form
K
∗
̄
K
must
be symmetrized as (
K
∗
+
K
−
+
K
∗−
K
+
)
/
√
2 so that the
decay conserves C-parity. For convenience, these ampli-
tudes are denoted by
K
∗
+
K
−
in the following.
TABLE III: Results of the Dalitz plot analysis of the
η
c
→
K
+
K
−
η
channel.
Final state
Fraction %
Phase (radians)
f
0
(1500)
η
23.7
±
7.0
±
1.8
0.
f
0
(1710)
η
8.9
±
3.2
±
0.4 2.2
±
0.3
±
0.1
K
∗
0
(1430)
+
K
−
16.4
±
4.2
±
1.0 2.3
±
0.2
±
0.1
f
0
(2200)
η
11.2
±
2.8
±
0.5 2.1
±
0.3
±
0.1
K
∗
0
(1950)
+
K
−
2.1
±
1.3
±
0.2 -0.2
±
0.4
±
0.1
f
′
2
(1525)
η
7.3
±
3.8
±
0.4 1.0
±
0.1
±
0.1
f
0
(1350)
η
5.0
±
3.7
±
0.5 0.9
±
0.2
±
0.1
f
0
(980)
η
10.4
±
3.0
±
0.5 -0.3
±
0.3
±
0.1
NR
15.5
±
6.9
±
1.0 -1.2
±
0.4
±
0.1
Sum
100.0
±
11.2
±
2.5
χ
2
/ν
87/65
The
f
0
(980) is parameterized as in a
B
A
B
AR
Dalitz
plot analysis of
D
+
s
→
K
+
K
−
π
+
decay [22]. For the
f
0
(1430) we use the BES parameterization [23]. For the
K
∗
0
(1430), we use our results from the Dalitz plot anal-
ysis (see Sec. VII.C), since the individual measurements
of the mass and width considered for the PDG average
values [16] show a large spread for each parameter. The
non-resonant (
NR
) contribution is parameterized as an
amplitude that is constant in magnitude and phase over
the Dalitz plot. The
f
0
(1500)
η
amplitude is taken as
12
)
4
/c
2
) (GeV
-
K
+
(K
2
m
1
2
3
4
5
6
)
4
/c
2
events/(0.114 GeV
0
20
40
60
80
(a)
)
4
/c
2
) (GeV
η
+
(K
2
m
2
4
6
)
4
/c
2
events/(0.11 GeV
0
20
40
60
(b)
)
4
/c
2
) (GeV
η
-
(K
2
m
2
4
6
)
4
/c
2
events/(0.11 GeV
0
20
40
60
(c)
FIG. 6: The
η
c
→
K
+
K
−
η
Dalitz plot projections. The super-
imposed curves result from the Dalitz plot analysis describ
ed
in the text. The shaded regions show the background esti-
mates obtained by interpolating the results of the Dalitz pl
ot
analyses of the sideband regions.
the reference amplitude, and so its phase is set to zero.
The test of the fit quality is performed by computing a
two-dimensional (2-D)
χ
2
over the Dalitz plot.
We first perform separate fits to the
η
c
sidebands using
a list of incoherent sum of amplitudes. We find significant
contributions from the
f
′
2
(1525),
f
0
(2200),
K
∗
3
(1780),
and
K
∗
0
(1950) resonances, as well as from an incoherent
uniform background. The resulting amplitude fractions
are interpolated into the
η
c
signal region and normalized
to yield the fitted purity. Figure 6 shows the projections
of the estimated background contributions as shaded dis-
tributions.
For the description of the
η
c
signal, amplitudes are
added one by one to ascertain the associated increase of
the likelihood value and decrease of the 2-D
χ
2
. Table III
summarizes the fit results for the amplitude fractions and
phases. We note that the
f
0
(1500)
η
amplitude provides
the largest contribution. We also observe important con-
tributions from the
K
∗
0
(1430)
+
K
−
,
f
0
(980)
η
,
f
0
(2200)
η
,
and
f
0
(1710)
η
channels. In addition, the fit requires a
sizeable
NR
contribution. The sum of the fractions for
this
η
c
decay mode is consistent with 100%.
We test the statistical significance of the
K
∗
0
(1430)
+
K
−
contribution by removing it from
the list of amplitudes. We obtain a change of the
negative log likelihood ∆(
−
2ln
L
)=+107 and an increase
of the
χ
2
on the Dalitz plot ∆
χ
2
=+76 for the reduction
by 2 parameters. This corresponds to a statistical
significance of 10.3 standard deviations. We obtain the
first observation of the
K
∗
0
(1430)
±
→
K
±
η
decay mode.
We test the quality of the fit by examining a large
sample of MC events at the generator level weighted
by the likelihood fitting function and by the efficiency.
These events are used to compare the fit result to the
Dalitz plot and its projections with proper normaliza-
tion. The latter comparison is shown in Fig. 6, and good
agreement is obtained for all projections. We make use
of these weighted events to compute a 2-D
χ
2
over the
Dalitz plot. For this purpose, we divide the Dalitz plot
into a number of cells such that the expected popula-
tion in each cell is at least eight events. We compute
χ
2
=
∑
N
cells
i
=1
(
N
i
obs
−
N
i
exp
)
2
/N
i
exp
, where
N
i
obs
and
N
i
exp
are event yields from data and simulation, respectively.
Denoting by
n
(= 16) the number of free parameters in
the fit, we obtain
χ
2
/ν
= 87
/
65 (
ν
=
N
cells
−
n
), which
indicates that the description of the data is adequate.
We compute the uncorrected Legendre polynomial mo-
ments
h
Y
0
L
i
in each
K
+
K
−
and
ηK
±
mass interval by
weighting each event by the relevant
Y
0
L
(cos
θ
) func-
tion. These distributions are shown in Figs. 7 and 8.
We also compute the expected Legendre polynomial mo-
ments from the weighted MC events and compare with
the experimental distributions. We observe good agree-
ment for all the distributions, which indicates that the
fit is able to reproduce the local structures apparent in
the Dalitz plot.
Systematic uncertainty estimates for the fractions and
relative phases are computed in two different ways: 1)
the purity function is scaled up and down by its statisti-
cal uncertainty, and 2) the parameters of each resonance
contributing to the decay are modified within one stan-
dard deviation of their uncertainties in the PDG average.
The two contributions are added in quadrature.
B. Dalitz plot analysis of
η
c
→
K
+
K
−
π
0
We define the
η
c
signal region as the range 2.910-
3.030 GeV
/c
2
, which contains 6710 events with (55.2
±
0.6)% purity. Sideband regions are defined as the
ranges 2.720-2.840 GeV
/c
2
and 3.100-3.220 GeV
/c
2
, re-