of 16
Dalitz plot analysis of
η
c
K
þ
K
η
and
η
c
K
þ
K
π
0
in
two-photon interactions
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano,
3a,3b
G. Eigen,
4
B. Stugu,
4
D. N. Brown,
5
L. T. Kerth,
5
Yu. G. Kolomensky,
5
M. J. Lee,
5
G. Lynch,
5
H. Koch,
6
T. Schroeder,
6
C. Hearty,
7
T. S. Mattison,
7
J. A. McKenna,
7
R. Y. So,
7
A. Khan,
8
V. E. Blinov,
9a,9c
A. R. Buzykaev,
9a
V. P. Druzhinin,
9a,9b
V. B. Golubev,
9a,9b
E. A. Kravchenko,
9a,9b
A. P. Onuchin,
9a,9c
S. I. Serednyakov,
9a,9b
Yu. I. Skovpen,
9a,9b
E. P. Solodov,
9a,9b
K. Yu. Todyshev,
9a,9b
A. J. Lankford,
10
M. Mandelkern,
10
B. Dey,
11
J. W. Gary,
11
O. Long,
11
C. Campagnari,
12
M. Franco Sevilla,
12
T. M. Hong,
12
D. Kovalskyi,
12
J. D. Richman,
12
C. A. West,
12
A. M. Eisner,
13
W. S. Lockman,
13
W. Panduro Vazquez,
13
B. A. Schumm,
13
A. Seiden,
13
D. S. Chao,
14
C. H. Cheng,
14
B. Echenard,
14
K. T. Flood,
14
D. G. Hitlin,
14
T. S. Miyashita,
14
P. Ongmongkolkul,
14
F. C. Porter,
14
R. Andreassen,
15
Z. Huard,
15
B. T. Meadows,
15
B. G. Pushpawela,
15
M. D. Sokoloff,
15
L. Sun,
15
P. C. Bloom,
16
W. T. Ford,
16
A. Gaz,
16
J. G. Smith,
16
S. R. Wagner,
16
R. Ayad,
17
,
W. H. Toki,
17
B. Spaan,
18
D. Bernard,
19
M. Verderi,
19
S. Playfer,
20
D. Bettoni,
21a
C. Bozzi,
21a
R. Calabrese,
21a,21b
G. Cibinetto,
21a,21b
E. Fioravanti,
21a,21b
I. Garzia,
21a,21b
E. Luppi,
21a,21b
L. Piemontese,
21a
V. Santoro,
21a
A. Calcaterra,
22
R. de Sangro,
22
G. Finocchiaro,
22
S. Martellotti,
22
P. Patteri,
22
I. M. Peruzzi,
22
M. Piccolo,
22
M. Rama,
22
A. Zallo,
22
R. Contri,
23a,23b
M. Lo Vetere,
23a,23b
M. R. Monge,
23a,23b
S. Passaggio,
23a
C. Patrignani,
23a,23b
E. Robutti,
23a
B. Bhuyan,
24
V. Prasad,
24
M. Morii,
25
A. Adametz,
26
U. Uwer,
26
H. M. Lacker,
27
P. D. Dauncey,
28
U. Mallik,
29
C. Chen,
30
J. Cochran,
30
S. Prell,
30
H. Ahmed,
31
A. V. Gritsan,
32
N. Arnaud,
33
M. Davier,
33
D. Derkach,
33
G. Grosdidier,
33
F. Le Diberder,
33
A. M. Lutz,
33
B. Malaescu,
33
,
P. Roudeau,
33
A. Stocchi,
33
G. Wormser,
33
D. J. Lange,
34
D. M. Wright,
34
J. P. Coleman,
35
J. R. Fry,
35
E. Gabathuler,
35
D. E. Hutchcroft,
35
D. J. Payne,
35
C. Touramanis,
35
A. J. Bevan,
36
F. Di Lodovico,
36
R. Sacco,
36
G. Cowan,
37
J. Bougher,
38
D. N. Brown,
38
C. L. Davis,
38
A. G. Denig,
39
M. Fritsch,
39
W. Gradl,
39
K. Griessinger,
39
A. Hafner,
39
E. Prencipe,
39
K. R. Schubert,
39
R. J. Barlow,
40
G. D. Lafferty,
40
R. Cenci,
41
B. Hamilton,
41
A. Jawahery,
41
D. A. Roberts,
41
R. Cowan,
42
G. Sciolla,
42
R. Cheaib,
43
P. M. Patel,
43
,*
S. H. Robertson,
43
N. Neri,
44a
F. Palombo,
44a,44b
L. Cremaldi,
45
R. Godang,
45
,**
P. Sonnek,
45
D. J. Summers,
45
M. Simard,
46
P. Taras,
46
G. De Nardo,
47a,47b
G. Onorato,
47a,47b
C. Sciacca,
47a,47b
M. Martinelli,
48
G. Raven,
48
C. P. Jessop,
49
J. M. LoSecco,
49
K. Honscheid,
50
R. Kass,
50
E. Feltresi,
51a,51b
M. Margoni,
51a,51b
M. Morandin,
51a
M. Posocco,
51a
M. Rotondo,
51a
G. Simi,
51a,51b
F. Simonetto,
51a,51b
R. Stroili,
51a,51b
S. Akar,
52
E. Ben-Haim,
52
M. Bomben,
52
G. R. Bonneaud,
52
H. Briand,
52
G. Calderini,
52
J. Chauveau,
52
Ph. Leruste,
52
G. Marchiori,
52
J. Ocariz,
52
M. Biasini,
53a,53b
E. Manoni,
53a
S. Pacetti,
53a,53b
A. Rossi,
53a
C. Angelini,
54a,54b
G. Batignani,
54a,54b
S. Bettarini,
54a,54b
M. Carpinelli,
54a,54b
,
††
G. Casarosa,
54a,54b
A. Cervelli,
54a,54b
M. Chrzaszcz,
54a,54b
F. Forti,
54a,54b
M. A. Giorgi,
54a,54b
A. Lusiani,
54a,54c
B. Oberhof,
54a,54b
E. Paoloni,
54a,54b
A. Perez,
54a
G. Rizzo,
54a,54b
J. J. Walsh,
54a
D. Lopes Pegna,
55
J. Olsen,
55
A. J. S. Smith,
55
R. Faccini,
56a,56b
F. Ferrarotto,
56a
F. Ferroni,
56a,56b
M. Gaspero,
56a,56b
L. Li Gioi,
56a
G. Piredda,
56a
C. Bünger,
57
S. Dittrich,
57
O. Grünberg,
57
T. Hartmann,
57
M. Hess,
57
T. Leddig,
57
C. Voß,
57
R. Waldi,
57
T. Adye,
58
E. O. Olaiya,
58
F. F. Wilson,
58
S. Emery,
59
G. Vasseur,
59
F. Anulli,
60
,
‡‡
D. Aston,
60
D. J. Bard,
60
C. Cartaro,
60
M. R. Convery,
60
J. Dorfan,
60
G. P. Dubois-Felsmann,
60
W. Dunwoodie,
60
M. Ebert,
60
R. C. Field,
60
B. G. Fulsom,
60
M. T. Graham,
60
C. Hast,
60
W. R. Innes,
60
P. Kim,
60
D. W. G. S. Leith,
60
P. Lewis,
60
D. Lindemann,
60
S. Luitz,
60
V. Luth,
60
H. L. Lynch,
60
D. B. MacFarlane,
60
D. R. Muller,
60
H. Neal,
60
M. Perl,
60
T. Pulliam,
60
B. N. Ratcliff,
60
A. Roodman,
60
A. A. Salnikov,
60
R. H. Schindler,
60
A. Snyder,
60
D. Su,
60
M. K. Sullivan,
60
J. Va
vra,
60
A. P. Wagner,
60
W. F. Wang,
60
W. J. Wisniewski,
60
H. W. Wulsin,
60
M. V. Purohit,
61
R. M. White,
61
,§§
J. R. Wilson,
61
A. Randle-Conde,
62
S. J. Sekula,
62
M. Bellis,
63
P. R. Burchat,
63
E. M. T. Puccio,
63
M. S. Alam,
64
J. A. Ernst,
64
R. Gorodeisky,
65
N. Guttman,
65
D. R. Peimer,
65
A. Soffer,
65
S. M. Spanier,
66
J. L. Ritchie,
67
A. M. Ruland,
67
R. F. Schwitters,
67
B. C. Wray,
67
J. M. Izen,
68
X. C. Lou,
68
F. Bianchi,
69a,69b
F. De Mori,
69a,69b
A. Filippi,
69a
D. Gamba,
69a,69b
L. Lanceri,
70a,70b
L. Vitale,
70a,70b
F. Martinez-Vidal,
71
A. Oyanguren,
71
P. Villanueva-Perez,
71
J. Albert,
72
Sw. Banerjee,
72
A. Beaulieu,
72
F. U. Bernlochner,
72
H. H. F. Choi,
72
G. J. King,
72
R. Kowalewski,
72
M. J. Lewczuk,
72
T. Lueck,
72
I. M. Nugent,
72
J. M. Roney,
72
R. J. Sobie,
72
N. Tasneem,
72
T. J. Gershon,
73
P. F. Harrison,
73
T. E. Latham,
73
H. R. Band,
74
S. Dasu,
74
Y. Pan,
74
R. Prepost,
74
and S. L. Wu
74
(The
BABAR
Collaboration)
1
Laboratoire d
Annecy-le-Vieux de Physique des Particules (LAPP), Université de Savoie, CNRS/IN2P3,
F-74941 Annecy-Le-Vieux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
3a
INFN Sezione di Bari, I-70126 Bari, Italy
3b
Dipartimento di Fisica, Università di Bari, I-70126 Bari, Italy
4
University of Bergen, Institute of Physics, N-5007 Bergen, Norway
5
Lawrence Berkeley National Laboratory and University of California, Berkeley, California 94720, USA
6
Ruhr Universität Bochum, Institut für Experimentalphysik 1, D-44780 Bochum, Germany
7
University of British Columbia, Vancouver, British Columbia, Canada V6T 1Z1
PHYSICAL REVIEW D
89,
112004 (2014)
1550-7998
=
2014
=
89(11)
=
112004(16)
112004-1
© 2014 American Physical Society
8
Brunel University, Uxbridge, Middlesex UB8 3PH, United Kingdom
9a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
9b
Novosibirsk State University, Novosibirsk 630090, Russia
9c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
10
University of California at Irvine, Irvine, California 92697, USA
11
University of California at Riverside, Riverside, California 92521, USA
12
University of California at Santa Barbara, Santa Barbara, California 93106, USA
13
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
14
California Institute of Technology, Pasadena, California 91125, USA
15
University of Cincinnati, Cincinnati, Ohio 45221, USA
16
University of Colorado, Boulder, Colorado 80309, USA
17
Colorado State University, Fort Collins, Colorado 80523, USA
18
Technische Universität Dortmund, Fakultät Physik, D-44221 Dortmund, Germany
19
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
20
University of Edinburgh, Edinburgh EH9 3JZ, United Kingdom
21a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
21b
Dipartimento di Fisica e Scienze della Terra, Università di Ferrara, I-44122 Ferrara, Italy
22
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
23a
INFN Sezione di Genova, I-16146 Genova, Italy
23b
Dipartimento di Fisica, Università di Genova, I-16146 Genova, Italy
24
Indian Institute of Technology Guwahati, Guwahati, Assam 781 039, India
25
Harvard University, Cambridge, Massachusetts 02138, USA
26
Universität Heidelberg, Physikalisches Institut, D-69120 Heidelberg, Germany
27
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
28
Imperial College London, London SW7 2AZ, United Kingdom
29
University of Iowa, Iowa City, Iowa 52242, USA
30
Iowa State University, Ames, Iowa 50011-3160, USA
31
Physics Department, Jazan University, Jazan 22822, Kingdom of Saudia Arabia
32
Johns Hopkins University, Baltimore, Maryland 21218, USA
33
Laboratoire de l
Accélérateur Linéaire, IN2P3/CNRS et Université Paris-Sud 11, Centre Scientifique
d
Orsay, F-91898 Orsay Cedex, France
34
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
35
University of Liverpool, Liverpool L69 7ZE, United Kingdom
36
Queen Mary, University of London, London E1 4NS, United Kingdom
37
University of London, Royal Holloway and Bedford New College, Egham,
Surrey TW20 0EX, United Kingdom
38
University of Louisville, Louisville, Kentucky 40292, USA
39
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
40
University of Manchester, Manchester M13 9PL, United Kingdom
41
University of Maryland, College Park, Maryland 20742, USA
42
Massachusetts Institute of Technology, Laboratory for Nuclear Science,
Cambridge, Massachusetts 02139, USA
43
McGill University, Montréal, Québec, Canada H3A 2T8
44a
INFN Sezione di Milano, I-20133 Milano, Italy
44b
Dipartimento di Fisica, Università di Milano, I-20133 Milano, Italy
45
University of Mississippi, University, Mississippi 38677, USA
46
Université de Montréal, Physique des Particules, Montréal, Québec, Canada H3C 3J7
47a
INFN Sezione di Napoli, I-80126 Napoli, Italy
47b
Dipartimento di Scienze Fisiche, Università di Napoli Federico II, I-80126 Napoli, Italy
48
NIKHEF, National Institute for Nuclear Physics and High Energy Physics,
NL-1009 DB Amsterdam, The Netherlands
49
University of Notre Dame, Notre Dame, Indiana 46556, USA
50
Ohio State University, Columbus, Ohio 43210, USA
51a
INFN Sezione di Padova, I-35131 Padova, Italy
51b
Dipartimento di Fisica, Università di Padova, I-35131 Padova, Italy
52
Laboratoire de Physique Nucléaire et de Hautes Energies, IN2P3/CNRS, Université Pierre et Marie
Curie-Paris6, Université Denis Diderot-Paris7, F-75252 Paris, France
53a
INFN Sezione di Perugia, I-06123 Perugia, Italy
53b
Dipartimento di Fisica, Università di Perugia, I-06123 Perugia, Italy
54a
INFN Sezione di Pisa, I-56127 Pisa, Italy
J. P. LEES
et al.
PHYSICAL REVIEW D
89,
112004 (2014)
112004-2
54b
Dipartimento di Fisica, Università di Pisa, I-56127 Pisa, Italy
54c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
55
Princeton University, Princeton, New Jersey 08544, USA
56a
INFN Sezione di Roma, I-00185 Roma, Italy
56b
Dipartimento di Fisica, Università di Roma La Sapienza, I-00185 Roma, Italy
57
Universität Rostock, D-18051 Rostock, Germany
58
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon OX11 0QX, United Kingdom
59
CEA, Irfu, SPP, Centre de Saclay, F-91191 Gif-sur-Yvette, France
60
SLAC National Accelerator Laboratory, Stanford, California 94309, USA
61
University of South Carolina, Columbia, South Carolina 29208, USA
62
Southern Methodist University, Dallas, Texas 75275, USA
63
Stanford University, Stanford, California 94305-4060, USA
64
State University of New York, Albany, New York 12222, USA
65
Tel Aviv University, School of Physics and Astronomy, Tel Aviv 69978, Israel
66
University of Tennessee, Knoxville, Tennessee 37996, USA
67
University of Texas at Austin, Austin, Texas 78712, USA
68
University of Texas at Dallas, Richardson, Texas 75083, USA
69a
INFN Sezione di Torino, I-10125 Torino, Italy
69b
Dipartimento di Fisica, Università di Torino, I-10125 Torino, Italy
70a
INFN Sezione di Trieste, I-34127 Trieste, Italy
70b
Dipartimento di Fisica, Università di Trieste, I-34127 Trieste, Italy
71
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
72
University of Victoria, Victoria, British Columbia, Canada V8W 3P6
73
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
74
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 27 March 2014; published 16 June 2014)
We study the processes
γγ
K
þ
K
η
and
γγ
K
þ
K
π
0
using a data sample of
519
fb
1
recorded with
the
BABAR
detector operating at the SLAC PEP-II asymmetric-energy
e
þ
e
collider at center-of-mass
energies at and near the
Υ
ð
nS
Þ
(
n
¼
2
;
3
;
4
) resonances. We observe
η
c
K
þ
K
π
0
and
η
c
K
þ
K
η
decays, measure their relative branching fraction, and perform a Dalitz plot analysis for each decay. We
observe the
K

0
ð
1430
Þ
K
η
decay and measure its branching fraction relative to the
K
π
decay mode to be
R
ð
K

0
ð
1430
ÞÞ ¼
B
ð
K

0
ð
1430
Þ
K
η
Þ
B
ð
K

0
ð
1430
Þ
K
π
Þ
¼
0
.
092

0
.
025
þ
0
.
010
0
.
025
. The
η
c
K
þ
K
η
and
K

0
ð
1430
Þ
K
η
results
correspond to the first observations of these channels. The data also show evidence for
η
c
ð
2
S
Þ
K
þ
K
π
0
and first evidence for
η
c
ð
2
S
Þ
K
þ
K
η
.
DOI:
10.1103/PhysRevD.89.112004
PACS numbers: 13.25.Gv, 14.40.Be, 14.40.Df, 14.40.Pq
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 region 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
consistent with the quark model. In particular, the
f
0
ð
1500
Þ
resonance, discovered in
̄
pp
annihilations, has
been interpreted as a scalar glueball
[5]
. However, no
evidence for the
f
0
ð
1500
Þ
state has been found in char-
monium decays. Another glueball candidate is the
f
0
ð
1710
Þ
discovered in radiative
J=
ψ
decays. Recently,
f
0
ð
1500
Þ
and
f
0
ð
1710
Þ
signals have been incorporated in a
Dalitz plot analysis of
B
3
K
decays
[6]
. Charmless
B
KX
decays could show enhanced gluonium production
[7]
.
Another puzzling state is the
K

0
ð
1430
Þ
resonance, never
observed as a clear peak in the
K
π
mass spectrum. In the
*
Deceased.
Present address: the University of Tabuk, Tabuk 71491, Saudi
Arabia.
Also at Università di Perugia, Dipartimento di Fisica, Perugia,
Italy.
§
Present address: Laboratoire de Physique Nucléaire et de
Hautes Energies, IN2P3/CNRS, Paris, France.
Present address: the University of Huddersfield, Huddersfield
HD1 3DH, United Kingdom.
**
Present address: University of South Alabama, Mobile,
Alabama 36688, USA
††
Also at Università di Sassari, Sassari, Italy
‡‡
Also with INFN Sezione di Roma, Roma, Italy
§§
Present address: Universidad Técnica Federico Santa Maria,
Valparaiso, Chile 2390123
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
89,
112004 (2014)
112004-3
description of the scalar amplitude in
K
π
scattering, the
K

0
ð
1430
Þ
resonance is added coherently to an effective-
range description of the low-mass
K
π
system in such a way
that the net amplitude actually decreases rapidly at the
resonance mass. The
K

0
ð
1430
Þ
parameter values were
measured by the LASS experiment in the reaction
K
p
K
π
þ
n
[8]
; the corrected
S
-wave amplitude representation
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
ð
2
S
Þ
decay modes remain unobserved,
while others have been studied with very limited statistical
precision. In particular, the branching fraction for the decay
mode
η
c
K
þ
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
þ
K
η
and
K
þ
K
π
0
systems produced in two-photon interactions. Two-photon
events in which at least one of the interacting photons is not
quasireal are strongly suppressed by the selection criteria
described below. This implies that the allowed
J
PC
values
of any produced resonances are
0
;
2
;
3
þþ
;
4
...
[11]
. Angular momentum conservation, parity conserva-
tion, and charge conjugation invariance imply that these
quantum numbers also apply to the final state except
that the
K
þ
K
η
and
K
þ
K
π
0
states cannot be in a
J
P
¼
0
þ
state.
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 study of efficiency and resolution,
while in Sec.
V
the mass spectra are presented. Section
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
ð
1430
Þ
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
BABAR
DETECTOR AND DATA SET
The results presented here are based on data collected
with the
BABAR
detector at the PEP-II asymmetric-energy
e
þ
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
ð
nS
Þ
(
n
¼
2
;
3
;
4
) resonances. The
BABAR
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 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. 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 back-
ground 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
þ
and
e
beam particles are scattered at small angles and are
undetected in the final state. We study the following
reactions
γγ
K
þ
K
η
;
ð
η
γγ
Þ
;
ð
1
Þ
γγ
K
þ
K
η
;
ð
η
π
þ
π
π
0
Þ
;
ð
2
Þ
and
γγ
K
þ
K
π
0
:
ð
3
Þ
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 originate from the
interaction region and that the
χ
2
probability 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 particular the
c
̄
c
η
c
resonance. In the
optimization procedure, we retain only selection criteria
that do not remove significant
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
hypothesis 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
J. P. LEES
et al.
PHYSICAL REVIEW D
89,
112004 (2014)
112004-4
the
π
0
candidates in the event. The
η
signal mass region
is defined as
541
<m
ð
π
þ
π
π
0
Þ
<
554
MeV
=c
2
. The
momentum three-vectors of the final-state pions are com-
bined 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
þ
K
η
mass resolution.
We check for possible background from the reaction
γγ
K
þ
K
π
þ
π
π
0
[17]
using
η
sideband regions and find it to
be consistent with zero. Background arises mainly from
random combinations of particles from
e
þ
e
annihilation,
from other two-photon processes, and from events with
initial-state photon radiation (ISR). The ISR background is
dominated by
J
PC
¼
1
−−
resonance production
[18]
.
We discriminate against
K
þ
K
η
(
K
þ
K
π
0
) events pro-
duced via ISR by requiring
M
2
rec
ð
p
e
þ
e
p
rec
Þ
2
>
10
ð
GeV
2
=c
4
, where
p
e
þ
e
is the four-momentum of the
initial state and
p
rec
is the four-momentum of the
K
þ
K
η
(
K
þ
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 consistent
with pions. For reactions (1) and (3), with only two charged
particles in the final state, we loosely identify 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
þ
e
rest frame, of
the final-state particles with respect to the beam axis.
Since well-reconstructed two-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
þ
K
where soft photon background simulates
the presence of a low momentum
π
0
. We reconstruct this
mode and reject events having a
γγ
K
þ
K
candidate
with
p
T
<
0
.
1
GeV
=c
.
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
ð
2
S
Þ
MC signal
events for the different channels are generated using a
detailed detector simulation
[14]
in which the
η
c
and
η
c
ð
2
S
Þ
mesons decay uniformly in phase space. These simulated
events are reconstructed and analyzed in the same manner
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
ð
2
S
Þ
mass regions. We
express the efficiency as a function of the
m
ð
K
þ
K
Þ
mass
and cos
θ
, where
θ
is the angle in the
K
þ
K
rest frame
between the directions of the
K
þ
and the boost from the
K
þ
K
η
or
K
þ
K
π
0
rest frame. To smooth statistical
fluctuations, this efficiency is then parametrized as follows.
First we fit the efficiency as a function of cos
θ
in
separate intervals of
m
ð
K
þ
K
Þ
, in terms of Legendre
polynomials up to
L
¼
12
:
ε
ð
cos
θ
Þ¼
X
12
L
¼
0
a
L
ð
m
Þ
Y
0
L
ð
cos
θ
Þ
;
ð
4
Þ
(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 (color online). Distributions of
p
T
for
(a)
γγ
K
þ
K
η
ð
η
γγ
Þ
, (b)
γγ
K
þ
K
η
ð
η
π
þ
π
π
0
Þ
,
and (c)
γγ
K
þ
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 vertical dashed line indicates the selection applied
to isolate two-photon events.
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
89,
112004 (2014)
112004-5
where
m
denotes
K
þ
K
invariant mass. For each value of
L
, we fit the mass dependent coefficients
a
L
ð
m
Þ
with a
seventh-order polynomial in
m
. Figure
2
shows the result-
ing fitted efficiency
ε
ð
m;
cos
θ
Þ
for each of the three
reactions. We observe a significant decrease in efficiency
for cos
θ

1
and
1
.
1
<m
ð
K
þ
K
Þ
<
1
.
5
GeV
=c
2
due to
the impossibility of reconstructing low-momentum kaons
(p
<
200
MeV
=c
in the laboratory frame) which have
experienced significant energy loss in the beampipe and
inner-detector material. The efficiency decrease at high
m
for
η
c
K
þ
K
η
(
η
π
þ
π
π
0
) [Fig.
2(b)
] results from the
loss of a low-momentum
π
0
from the
η
decay.
The mass resolution,
Δ
m
, is measured as the difference
between the generated and reconstructed
K
þ
K
η
or
K
þ
K
π
0
invariant-mass values. Figure
3
shows the
Δ
m
distribution for each of the
η
c
signal regions; these deviate
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
þ
K
η
(
η
π
þ
π
π
0
) final state to a
Crystal Ball function
[19]
, and those for the
K
þ
K
η
(
η
γγ
) and
K
þ
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
ð
2
S
Þ
mass, for the
K
þ
K
η
(
η
γγ
),
K
þ
K
η
(
η
π
þ
π
π
0
), and
K
þ
K
π
0
final states, respectively.
V. MASS SPECTRA
Figure
4(a)
shows the
K
þ
K
η
mass spectrum, summed
over the two
η
decay modes, before applying the efficiency
0
0.02
0.04
0.06
0.08
0.1
)
2
) (GeV/c
-
K
+
m(K
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
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 (color online). Fitted detection efficiency in the
cos
θ
vs: m
ð
K
þ
K
Þ
plane for (a)
η
c
K
þ
K
η
(
η
γγ
),
(b)
η
c
K
þ
K
η
(
η
π
þ
π
π
0
), and (c)
η
c
K
þ
K
π
0
. Each
bin shows the average value of the fit in that region.
)
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 (color online). MC mass resolution for (a)
η
c
K
þ
K
η
(
η
γγ
), (b)
η
c
K
þ
K
η
(
η
π
þ
π
π
0
), and (c)
η
c
K
þ
K
π
0
. The curves represent the fits described in the text.
J. P. LEES
et al.
PHYSICAL REVIEW D
89,
112004 (2014)
112004-6
correction. There are 2950 events in the 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
þ
π
π
0
decay mode. We observe a strong
η
c
signal and a small
enhancement at the position of the
η
c
ð
2
S
Þ
. The
η
c
signal-to-
background ratio for each of the
η
decay modes is
approximately the same. We perform a simultaneous fit
to the
K
þ
K
η
mass spectra for the two
η
decay modes. For
each resonance, the mass and width are constrained to take
the same fitted values in both distributions. Backgrounds
are described by second-order polynomials, and each
resonance is represented by a simple Breit-Wigner function
convolved with the corresponding resolution function. In
addition, we include a signal function 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
ð
2
S
Þ
parameter values. We have only a weak constraint
on the
η
c
ð
2
S
Þ
width and so we fix its value to 11.3 MeV
[16]
. Indicating with
ν
the number of degrees of freedom,
we obtain a good description of the data with
χ
2
=
ν
¼
194
=
204
and a
χ
2
probability of 68.1%.
The
K
þ
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
ð
2
S
Þ
on top of a sizeable
background. We perform a fit to the
K
þ
K
π
0
mass
spectrum using the background function
B
ð
m
Þ¼
e
a
1
m
þ
a
2
m
2
for
m<m
0
and
B
ð
m
Þ¼
e
b
0
þ
b
1
m
þ
b
2
m
2
for
m>m
0
, where
m
¼
m
ð
K
þ
K
π
0
Þ
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 parameters. 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
þ
K
π
0
mass spectrum for the ISR data sample obtained
requiring
j
M
2
rec
j
<
1
ð
GeV
=c
2
Þ
2
. Figure
4(b)
shows the fit
to the
K
þ
K
π
0
mass spectrum, and Table
I
summarizes the
resulting
η
c
and
η
c
ð
2
S
Þ
parameter values. We obtain a
reasonable description of the data with
χ
2
=
ν
¼
225
=
189
and a
χ
2
probability of 3.8%.
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
þ
K
η
,we
perform 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 different
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
ð
2
S
Þ
decays to the
K
þ
K
η
final state compared to the
respective branching fractions to the
K
þ
K
π
0
final state.
The ratios are computed as
R
¼
B
ð
η
c
=
η
c
ð
2
S
Þ
K
þ
K
η
Þ
B
ð
η
c
=
η
c
ð
2
S
Þ
K
þ
K
π
0
Þ
¼
N
K
þ
K
η
N
K
þ
K
π
0
ε
K
þ
K
π
0
ε
K
þ
K
η
1
B
η
:
ð
5
Þ
For each
η
decay mode,
N
K
þ
K
η
and
N
K
þ
K
π
0
represent the
fitted yields for
η
c
and
η
c
ð
2
S
Þ
in the
K
þ
K
η
and
K
þ
K
π
0
mass spectra,
ε
K
þ
K
η
and
ε
K
þ
K
π
0
are the corresponding
efficiencies, and
B
η
indicates the particular
η
branching
fraction. The PDG values of the branching fractions are
)
2
) (GeV/c
η
-
K
+
m(K
33.5
)
2
events/(10 MeV/c
0
50
100
150
200
(a)
)
2
) (GeV/c
0
π
-
K
+
m(K
33.5
)
2
events/(6 MeV/c
0
200
400
(b)
FIG. 4 (color online). (a) The
K
þ
K
η
mass spectrum summed over the two
η
decay modes. (b) The
K
þ
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.
TABLE I. Fitted
η
c
and
η
c
ð
2
S
Þ
parameter values. The first
uncertainty is statistical and the second is systematic.
Resonance
Mass (MeV
=c
2
)
Γ
(MeV)
η
c
K
þ
K
η
2984
.
1

1
.
1

2
.
134
.
8

3
.
1

4
.
0
η
c
K
þ
K
π
0
2979
.
8

0
.
8

3
.
525
.
2

2
.
6

2
.
4
η
c
ð
2
S
Þ
K
þ
K
η
3635
.
1

5
.
8

2
.
1
11.3 (fixed)
η
c
ð
2
S
Þ
K
þ
K
π
0
3637
.
0

5
.
7

3
.
4
11.3 (fixed)
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
89,
112004 (2014)
112004-7
ð
39
.
41

0
.
20
Þ
%
and
ð
22
.
92

0
.
28
Þ
%
for the
η
γγ
and
η
π
þ
π
π
0
, respectively
[16]
.
We estimate the weighted efficiencies
ε
K
þ
K
η
and
ε
K
þ
K
π
0
for the
η
c
signals by making use of the 2-D
efficiency functions described in Sec.
IV
. 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
w
¼
1
=
ε
ð
m;
cos
θ
Þ
to events in the signal region and a negative
weight
w
¼
f=
ε
ð
m;
cos
θ
Þ
to events in the sideband
regions. The weight in the sideband regions is scaled down
by the factor
f
to match the fitted
η
c
signal/background
ratio. Therefore we obtain the weighted efficiencies as
ε
K
þ
K
η
=
π
0
¼
P
N
i
¼
1
f
i
P
N
i
¼
1
f
i
=
ε
ð
m
i
;
cos
θ
i
Þ
;
ð
6
Þ
where
N
indicates the number of events in the signal
+sidebands regions. 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
ð
2
S
Þ
, we use the average
efficiency value from the simulation.
We determine
N
K
þ
K
η
and
N
K
þ
K
π
0
for the
η
c
by
performing fits to the
K
þ
K
η
and
K
þ
K
π
0
mass spectra.
The width is extracted from the simultaneous fit to the
K
þ
K
η
mass spectra, and is fixed to this value in the fit to
the
K
þ
K
π
0
mass spectrum. This procedure is adopted
because the signal-to-background ratio at the peak is much
better for the
K
þ
K
η
mode (
8
1
compared to
2
1
for
the
K
þ
K
π
0
mode) while the residual
J=
ψ
contamination
is much smaller. The
η
c
and
η
c
ð
2
S
Þ
mass values are
determined from the fits. For the
η
c
ð
2
S
Þ
, 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 contribu-
tions in quadrature.
We calculate the weighted mean of the
η
c
branching-ratio
estimates for the two
η
decay modes and obtain
R
ð
η
c
Þ¼
B
ð
η
c
K
þ
K
η
Þ
B
ð
η
c
K
þ
K
π
0
Þ
¼
0
.
571

0
.
025

0
.
051
;
ð
7
Þ
which is consistent with the BESIII measurement of
0
.
46

0
.
23
[10]
. Since the sample size for
η
c
ð
2
S
Þ
K
þ
K
η
decays with
η
π
þ
π
π
0
is small, we use only the
η
γγ
decay mode, and obtain
R
ð
η
c
ð
2
S
ÞÞ ¼
B
ð
η
c
ð
2
S
Þ
K
þ
K
η
Þ
B
ð
η
c
ð
2
S
Þ
K
þ
K
π
0
Þ
¼
0
.
82

0
.
21

0
.
27
:
ð
8
Þ
In evaluating
R
ð
η
c
Þ
for the
η
γγ
decay mode, we note
that the number of charged-particle tracks and
γ
sisthe
same in the numerator and in the denominator of the ratio,
so that several systematic uncertainties cancel. Concerning
the contribution of the
η
π
þ
π
π
0
decay, we find sys-
tematic 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 poly-
nomials or exponential functions. The uncertainty due to
the efficiency weight is evaluated by computing 1000 new
weights obtained by randomly modifying the weight in
each cell of the
ε
ð
m
ð
K
þ
K
Þ
;
cos
θ
Þ
plane according to its
statistical uncertainty. The widths of the resulting Gaussian
distributions yield the estimate of the systematic uncer-
tainty for the efficiency weighting procedure. These values
are reported as the weight uncertainties in Table
II
.
VII. DALITZ PLOT ANALYSES
We perform Dalitz plot analyses of the
K
þ
K
η
and
K
þ
K
π
0
systems in the
η
c
mass region using unbinned
TABLE II. Summary of the results from the fits to the
K
þ
K
η
and
K
þ
K
π
0
mass spectra. The table lists event yields, efficiency
correction weights, resulting branching ratios and significances. For event yields, the first uncertainty is statistical and the second is
systematic. In the evaluation of significances, systematic uncertainties are included.
Channel
Event yield
Weights
R
Significance
η
c
K
þ
K
π
0
4518

131

50
17
.
0

0
.
732
σ
η
c
K
þ
K
η
(
η
γγ
)
853

38

11
21
.
3

0
.
621
σ
B
ð
η
c
K
þ
K
η
Þ
=
B
ð
η
c
K
þ
K
π
0
Þ
0
.
602

0
.
032

0
.
065
η
c
K
þ
K
η
(
η
π
þ
π
π
0
)
292

20

731
.
2

2
.
114
σ
B
ð
η
c
K
þ
K
η
Þ
=
B
ð
η
c
K
þ
K
π
0
Þ
0
.
523

0
.
040

0
.
083
η
c
ð
2
S
Þ
K
þ
K
π
0
178

29

39
14
.
3

1
.
33
.
7
σ
η
c
ð
2
S
Þ
K
þ
K
η
47

9

317
.
4

0
.
44
.
9
σ
B
ð
η
c
ð
2
S
Þ
K
þ
K
η
Þ
=
B
ð
η
c
ð
2
S
Þ
K
þ
K
π
0
Þ
0
.
82

0
.
21

0
.
27
χ
c
2
K
þ
K
π
0
88

27

23
2
.
5
σ
χ
c
2
K
þ
K
η
2

5

20
.
0
σ
J. P. LEES
et al.
PHYSICAL REVIEW D
89,
112004 (2014)
112004-8
maximum likelihood fits. The likelihood function is
written as
L
¼
Y
N
n
¼
1

f
sig
ð
m
n
Þ
·
ε
ð
x
0
n
;y
0
n
Þ
P
i;j
c
i
c

j
A
i
ð
x
n
;y
n
Þ
A

j
ð
x
n
;y
n
Þ
P
i;j
c
i
c

j
I
A
i
A

j
þð
1
f
sig
ð
m
n
ÞÞ
P
i
k
i
B
i
ð
x
n
;y
n
Þ
P
i
k
i
I
B
i

ð
9
Þ
where
(i)
N
is the number of events in the signal region;
(ii) for the
n
th event,
m
n
is the
K
þ
K
η
or the
K
þ
K
π
0
invariant mass;
(iii) for the
n
th event,
x
n
¼
m
2
ð
K
þ
η
Þ
,
y
n
¼
m
2
ð
K
η
Þ
for
K
þ
K
η
;
x
n
¼
m
2
ð
K
þ
π
0
Þ
,
y
n
¼
m
2
ð
K
π
0
Þ
for
K
þ
K
π
0
;
(iv)
f
sig
is the mass-dependent fraction of signal
obtained from the fit to the
K
þ
K
η
or
K
þ
K
π
0
mass spectrum;
(v) for the
n
th event,
ε
ð
x
0
n
;y
0
n
Þ
is the efficiency
parametrized as a function
x
0
n
¼
m
ð
K
þ
K
Þ
and
y
0
n
¼
cos
θ
(see Sec.
IV
);
(vi) for the
n
th event, the
A
i
ð
x
n
;y
n
Þ
describe the
complex signal-amplitude contributions;
(vii)
c
i
is the complex amplitude of the
i
th signal
component; the
c
i
parameters are allowed to vary
during the fit process;
(viii) for the
n
th event, the
B
i
ð
x
n
;y
n
Þ
describe the back-
ground probability-density functions assuming
that interference between signal and background
amplitudes can be ignored;
(ix)
k
i
is the magnitude of the
i
th background compo-
nent; the
k
i
parameters are obtained by fitting the
sideband regions;
(x)
I
A
i
A

j
¼
R
A
i
ð
x; y
Þ
A

j
ð
x; y
Þ
ε
ð
m
ð
K
þ
K
Þ
;
cos
θ
Þ
d
x
d
y
and
I
B
i
¼
R
B
i
ð
x; y
Þ
d
x
d
y
are normalization
integrals; numerical integration is performed on
phase space generated events.
Amplitudes are parametrized as described in Refs.
[16]
and
[21]
. The efficiency-corrected fractional contribution
f
i
due to resonant or nonresonant contribution
i
is defined as
follows:
f
i
¼
j
c
i
j
2
R
j
A
i
ð
x; y
Þj
2
d
x
d
y
R
j
P
j
c
j
A
j
ð
x; y
Þj
2
d
x
d
y
:
ð
10
Þ
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.
A. Dalitz plot analysis of
η
c
K
þ
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
ð
76
.
1

1
.
3
Þ
%
purity, defined as
S=
ð
S
þ
B
Þ
where
S
and
B
indicate the number of signal and background
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.
We observe signals in the
K
þ
K
projections correspond-
ing to the
f
0
ð
980
Þ
,
f
0
ð
1500
Þ
,
f
0
ð
1710
Þ
, and
f
0
ð
2200
Þ
states. We also observe a broad signal in the
1
.
43
GeV
=c
2
mass region in the
K
þ
η
and
K
η
projections.
In describing the Dalitz plot, we note that amplitude
contributions to the
K
þ
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 sym-
metrized as
ð
K
K
þ
K

K
þ
Þ
=
ffiffiffi
2
p
so that the decay
conserves C-parity. For convenience, these amplitudes
are denoted by
K
K
in the following.
The
f
0
ð
980
Þ
is parametrized as in a
BABAR
Dalitz plot
analysis of
D
þ
s
K
þ
K
π
þ
decay
[21]
. For the
f
0
ð
1430
Þ
we use the BES parametrization
[22]
. For the
K

0
ð
1430
Þ
,
we use our results from the Dalitz plot analysis
(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 nonreso-
nant (
NR
) contribution is parametrized as an amplitude that
is constant in magnitude and phase over the Dalitz plot. The
f
0
ð
1500
Þ
η
amplitude is taken as 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
0
2
ð
1525
Þ
,
f
0
ð
2200
Þ
,
K

3
ð
1780
Þ
, and
K

0
ð
1950
Þ
resonances, as well as from an incoherent
FIG. 5 (color online). Dalitz plot for the
η
c
K
þ
K
η
events in
the signal region. The shaded area denotes the accessible
kinematic region.
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
89,
112004 (2014)
112004-9
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
distributions.
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
ð
1500
Þ
η
amplitude provides the
largest contribution. We also observe important contribu-
tions from the
K

0
ð
1430
Þ
þ
K
,
f
0
ð
980
Þ
η
,
f
0
ð
2200
Þ
η
, and
f
0
ð
1710
Þ
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
ð
1430
Þ
þ
K
contribution by removing it from the list of amplitudes.
We obtain a change of the negative log likelihood
Δ
ð
2
ln
L
Þ¼þ
107
and an increase of the
χ
2
on the
Dalitz plot
Δ
χ
2
¼þ
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
ð
1430
Þ

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 normalization. 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 population in each cell is at least
eight events. We compute
χ
2
¼
P
N
cells
i
¼
1
ð
N
i
obs
N
i
exp
Þ
2
=
N
i
exp
, where
N
i
obs
and
N
i
exp
are event yields from data
and simulation, respectively. Denoting by
n
ð¼
16
Þ
the
number of free parameters in the fit, we obtain
χ
2
=
ν
¼
87
=
65
(
ν
¼
N
cells
n
), which indicates that the description
of the data is adequate.
We compute the uncorrected Legendre polynomial
moments
h
Y
0
L
i
in each
K
þ
K
and
η
K

mass interval by
weighting each event by the relevant
Y
0
L
ð
cos
θ
Þ
function.
These distributions are shown in Figs.
7
and
8
. We also
compute the expected Legendre polynomial moments from
the weighted MC events and compare with the experimen-
tal distributions. We observe good agreement for all the
distributions, which indicates that the fit is able to repro-
duce the local structures apparent in the Dalitz plot.
Systematic uncertainty estimates for the fractions and
relative phases are computed in two different ways: (i) the
purity function is scaled up and down by its statistical
uncertainty, and (ii) the parameters of each resonance
contributing to the decay are modified within one standard
TABLE III. Results of the Dalitz plot analysis of the
η
c
K
þ
K
η
channel.
Final state
Fraction %
Phase (radians)
f
0
ð
1500
Þ
η
23
.
7

7
.
0

1
.
8
0.
f
0
ð
1710
Þ
η
8
.
9

3
.
2

0
.
42
.
2

0
.
3

0
.
1
K

0
ð
1430
Þ
þ
K
16
.
4

4
.
2

1
.
02
.
3

0
.
2

0
.
1
f
0
ð
2200
Þ
η
11
.
2

2
.
8

0
.
52
.
1

0
.
3

0
.
1
K

0
ð
1950
Þ
þ
K
2
.
1

1
.
3

0
.
2
0
.
2

0
.
4

0
.
1
f
0
2
ð
1525
Þ
η
7
.
3

3
.
8

0
.
41
.
0

0
.
1

0
.
1
f
0
ð
1350
Þ
η
5
.
0

3
.
7

0
.
50
.
9

0
.
2

0
.
1
f
0
ð
980
Þ
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
)
4
/c
2
) (GeV
-
K
+
(K
2
m
)
4
/c
2
events/(0.114 GeV
0
20
40
60
80
(a)
)
4
/c
2
) (GeV
η
+
(K
2
m
)
4
/c
2
events/(0.11 GeV
0
20
40
60
(b)
)
4
/c
2
) (GeV
η
-
(K
2
m
12 3 4 5 6
246
246
)
4
/c
2
events/(0.11 GeV
0
20
40
60
(c)
FIG. 6 (color online). The
η
c
K
þ
K
η
Dalitz plot projections.
The superimposed curves result from the Dalitz plot analysis
described in the text. The shaded regions show the background
estimates obtained by interpolating the results of the Dalitz plot
analyses of the sideband regions.
J. P. LEES
et al.
PHYSICAL REVIEW D
89,
112004 (2014)
112004-10
deviation of their uncertainties in the PDG average. The
two contributions are added in quadrature.
B. Dalitz plot analysis of
η
c
K
þ
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
ð
55
.
2

0
.
6
Þ
%
purity. Sideband regions are defined as the
ranges
2
.
720
-
2
.
840
GeV
=c
2
and
3
.
100
-
3
.
220
GeV
=c
2
,
respectively. Figure
9
shows the Dalitz plot for the
η
c
signal region, and Fig.
10
shows the corresponding Dalitz
plot projections. The Dalitz plot and the mass projections
are very similar to the distributions in Ref.
[23]
for the
decay
η
c
K
0
s
K

π
.
)
2
) (GeV/c
-
K
+
m(K
1
1.5
2
2.5
)
2
weight sum/(50 MeV/c
0
5
10
15
20
25
0
0
Y
)
2
) (GeV/c
-
K
+
m(K
11.5 22.5
-10
-5
0
5
0
1
Y
)
2
) (GeV/c
-
K
+
m(K
11.5 22.5
-10
-5
0
5
0
2
Y
)
2
) (GeV/c
-
K
+
m(K
1
1.5
2
2.5
)
2
weight sum/(50 MeV/c
-5
0
0
3
Y
)
2
) (GeV/c
-
K
+
m(K
11.5 22.5
-5
0
5
0
4
Y
)
2
) (GeV/c
-
K
+
m(K
11.5 22.5
0
5
0
5
Y
FIG. 7 (color online). Legendre polynomial moments for
η
c
K
þ
K
η
as a function of
K
þ
K
mass. The superimposed curves result
from the Dalitz plot analysis described in the text.
)
2
) (GeV/c
η
±
m(K
1
1.5
2
2.5
)
2
weight sum/(50 MeV/c
0
10
20
30
40
0
0
Y
)
2
) (GeV/c
η
±
m(K
1
1.5
2
2.5
-10
0
10
20
0
1
Y
)
2
) (GeV/c
η
±
m(K
1
1.5
2
2.5
-10
-5
0
5
10
0
2
Y
)
2
) (GeV/c
η
±
m(K
1
1.5
2
2.5
)
2
weight sum/(50 MeV/c
-10
0
10
0
3
Y
)
2
) (GeV/c
η
±
m(K
1
1.5
2
2.5
-10
0
10
0
4
Y
)
2
) (GeV/c
η
±
m(K
1
1.5
2
2.5
-20
-10
0
10
0
5
Y
FIG. 8 (color online). Legendre polynomial moments for
η
c
K
þ
K
η
as a function of
K

η
mass. The superimposed curves result
from the Dalitz plot analysis described in the text. The corresponding
K
þ
η
and
K
η
distributions are combined.
DALITZ PLOT ANALYSIS OF
...
PHYSICAL REVIEW D
89,
112004 (2014)
112004-11