Light meson spectroscopy from Dalitz plot analyses of
η
c
decays to
η
0
K
+
K
−
,
η
0
π
+
π
−
, and
ηπ
+
π
−
produced in two-photon interactions
J. P. Lees,
1
V. Poireau,
1
V. Tisserand,
1
E. Grauges,
2
A. Palano ,
3
G. Eigen,
4
D. N. Brown,
5
Yu. G. Kolomensky,
5
M. Fritsch,
6
H. Koch,
6
T. Schroeder,
6
R. Cheaib,
7b
C. Hearty,
7a,7b
T. S. Mattison,
7b
J. A. McKenna,
7b
R. Y. So,
7b
V. E. Blinov,
8a,8b,8c
A. R. Buzykaev,
8a
V. P. Druzhinin,
8a,8b
V. B. Golubev,
8a,8b
E. A. Kozyrev,
8a,8b
E. A. Kravchenko,
8a,8b
A. P. Onuchin,
8a,8b,8c
,*
S. I. Serednyakov,
8a,8b
Yu. I. Skovpen,
8a,8b
E. P. Solodov,
8a,8b
K. Yu. Todyshev,
8a,8b
A. J. Lankford,
9
B. Dey,
10
J. W. Gary,
10
O. Long,
10
A. M. Eisner,
11
W. S. Lockman,
11
W. Panduro Vazquez,
11
D. S. Chao,
12
C. H. Cheng,
12
B. Echenard,
12
K. T. Flood,
12
D. G. Hitlin,
12
J. Kim,
12
Y. Li,
12
D. X. Lin,
12
T. S. Miyashita,
12
P. Ongmongkolkul,
12
J. Oyang,
12
F. C. Porter,
12
M. Röhrken,
12
Z. Huard,
13
B. T. Meadows,
13
B. G. Pushpawela,
13
M. D. Sokoloff,
13
L. Sun,
13
,
†
J. G. Smith,
14
S. R. Wagner,
14
D. Bernard,
15
M. Verderi,
15
D. Bettoni,
16a
C. Bozzi,
16a
R. Calabrese,
16a,16b
G. Cibinetto,
16a,16b
E. Fioravanti,
16a,16b
I. Garzia,
16a,16b
E. Luppi,
16a,16b
V. Santoro,
16a
A. Calcaterra,
17
R. de Sangro,
17
G. Finocchiaro,
17
S. Martellotti,
17
P. Patteri,
17
I. M. Peruzzi,
17
M. Piccolo,
17
M. Rotondo,
17
A. Zallo,
17
S. Passaggio,
18
C. Patrignani,
18
,
‡
B. J. Shuve,
19
H. M. Lacker,
20
B. Bhuyan,
21
U. Mallik,
22
C. Chen,
23
J. Cochran,
23
S. Prell,
23
A. V. Gritsan,
24
N. Arnaud,
25
M. Davier,
25
F. Le Diberder,
25
A. M. Lutz,
25
G. Wormser,
25
D. J. Lange,
26
D. M. Wright,
26
J. P. Coleman,
27
E. Gabathuler,
27
,*
D. E. Hutchcroft,
27
D. J. Payne,
27
C. Touramanis,
27
A. J. Bevan,
28
F. Di Lodovico,
28
,§
R. Sacco,
28
G. Cowan,
29
Sw. Banerjee,
30
D. N. Brown,
30
C. L. Davis,
30
A. G. Denig,
31
W. Gradl,
31
K. Griessinger,
31
A. Hafner,
31
K. R. Schubert,
31
R. J. Barlow,
32
,
∥
G. D. Lafferty,
32
R. Cenci,
33
A. Jawahery,
33
D. A. Roberts,
33
R. Cowan,
34
S. H. Robertson,
35a,35b
R. M. Seddon,
35b
N. Neri,
36a
F. Palombo,
36a,36b
L. Cremaldi,
37
R. Godang,
37
,¶
D. J. Summers,
37
,*
P. Taras,
38
G. De Nardo,
39
C. Sciacca,
39
G. Raven,
40
C. P. Jessop,
41
J. M. LoSecco,
41
K. Honscheid,
42
R. Kass,
42
A. Gaz,
43a
M. Margoni,
43a,43b
M. Posocco,
43a
G. Simi,
43a,43b
F. Simonetto,
43a,43b
R. Stroili,
43a,43b
S. Akar,
44
E. Ben-Haim,
44
M. Bomben,
44
G. R. Bonneaud,
44
G. Calderini,
44
J. Chauveau,
44
G. Marchiori,
44
J. Ocariz,
44
M. Biasini,
45a,45b
E. Manoni,
45a
A. Rossi,
45a
G. Batignani,
46a,46b
S. Bettarini,
46a,46b
M. Carpinelli,
46a,46b
,**
G. Casarosa,
46a,46b
M. Chrzaszcz,
46a
F. Forti,
46a,46b
M. A. Giorgi,
46a,46b
A. Lusiani,
46a,46c
B. Oberhof,
46a,46b
E. Paoloni,
46a,46b
M. Rama,
46a
G. Rizzo,
46a,46b
J. J. Walsh,
46a
L. Zani,
46a,46b
A. J. S. Smith,
47
F. Anulli,
48a
R. Faccini,
48a,48b
F. Ferrarotto,
48a
F. Ferroni,
48a
,
††
A. Pilloni,
48a,48b
G. Piredda,
48a
,*
C. Bünger,
49
S. Dittrich,
49
O. Grünberg,
49
M. Heß,
49
T. Leddig,
49
C. Voß,
49
R. Waldi,
49
T. Adye,
50
F. F. Wilson,
50
S. Emery,
51
G. Vasseur,
51
D. Aston,
52
C. Cartaro,
52
M. R. Convery,
52
J. Dorfan,
52
W. Dunwoodie,
52
M. Ebert,
52
R. C. Field,
52
B. G. Fulsom,
52
M. T. Graham,
52
C. Hast,
52
W. R. Innes,
52
,*
P. Kim,
52
D. W. G. S. Leith,
52
,*
S. Luitz,
52
D. B. MacFarlane,
52
D. R. Muller,
52
H. Neal,
52
B. N. Ratcliff,
52
A. Roodman,
52
M. K. Sullivan,
52
J. Va
’
vra,
52
W. J. Wisniewski,
52
M. V. Purohit,
53
J. R. Wilson,
53
A. Randle-Conde,
54
S. J. Sekula,
54
H. Ahmed,
55
M. Bellis,
56
P. R. Burchat,
56
E. M. T. Puccio,
56
M. S. Alam,
57
J. A. Ernst,
57
R. Gorodeisky,
58
N. Guttman,
58
D. R. Peimer,
58
A. Soffer,
58
S. M. Spanier,
59
J. L. Ritchie,
60
R. F. Schwitters,
60
J. M. Izen,
61
X. C. Lou,
61
F. Bianchi,
62a,62b
F. De Mori,
62a,62b
A. Filippi,
62a
D. Gamba,
62a,62b
L. Lanceri,
63
L. Vitale,
63
F. Martinez-Vidal,
64
A. Oyanguren,
64
J. Albert,
65b
A. Beaulieu,
65b
F. U. Bernlochner,
65b
G. J. King,
65b
R. Kowalewski,
65b
T. Lueck,
65b
I. M. Nugent,
65b
J. M. Roney,
65b
R. J. Sobie,
65a,65b
N. Tasneem,
65b
T. J. Gershon,
66
P. F. Harrison,
66
T. E. Latham,
66
R. Prepost,
67
and S. L. Wu
67
(
BABAR
Collaboration)
1
Laboratoire d
’
Annecy-le-Vieux de Physique des Particules (LAPP), Universit ́
e de Savoie, CNRS/IN2P3,
F-74941 Annecy-Le-Vieux, France
2
Universitat de Barcelona, Facultat de Fisica, Departament ECM, E-08028 Barcelona, Spain
3
INFN Sezione 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
7a
Institute of Particle Physics, Vancouver, British Columbia V6T 1Z1, Canada
7b
University of British Columbia, Vancouver, British Columbia V6T 1Z1, Canada
8a
Budker Institute of Nuclear Physics SB RAS, Novosibirsk 630090, Russia
8b
Novosibirsk State University, Novosibirsk 630090, Russia
8c
Novosibirsk State Technical University, Novosibirsk 630092, Russia
9
University of California at Irvine, Irvine, California 92697, USA
10
University of California at Riverside, Riverside, California 92521, USA
11
University of California at Santa Cruz, Institute for Particle Physics, Santa Cruz, California 95064, USA
12
California Institute of Technology, Pasadena, California 91125, USA
PHYSICAL REVIEW D
104,
072002 (2021)
2470-0010
=
2021
=
104(7)
=
072002(23)
072002-1
Published by the American Physical Society
13
University of Cincinnati, Cincinnati, Ohio 45221, USA
14
University of Colorado, Boulder, Colorado 80309, USA
15
Laboratoire Leprince-Ringuet, Ecole Polytechnique, CNRS/IN2P3, F-91128 Palaiseau, France
16a
INFN Sezione di Ferrara, I-44122 Ferrara, Italy
16b
Dipartimento di Fisica e Scienze della Terra, Universit`
a di Ferrara, I-44122 Ferrara, Italy
17
INFN Laboratori Nazionali di Frascati, I-00044 Frascati, Italy
18
INFN Sezione di Genova, I-16146 Genova, Italy
19
Harvey Mudd College, Claremont, California 91711, USA
20
Humboldt-Universität zu Berlin, Institut für Physik, D-12489 Berlin, Germany
21
Indian Institute of Technology Guwahati, Guwahati, Assam, 781 039, India
22
University of Iowa, Iowa City, Iowa 52242, USA
23
Iowa State University, Ames, Iowa 50011, USA
24
Johns Hopkins University, Baltimore, Maryland 21218, USA
25
Universit ́
e Paris-Saclay, CNRS/IN2P3, IJCLab, F-91405 Orsay, France
26
Lawrence Livermore National Laboratory, Livermore, California 94550, USA
27
University of Liverpool, Liverpool L69 7ZE, United Kingdom
28
Queen Mary, University of London, London, E1 4NS, United Kingdom
29
University of London, Royal Holloway and Bedford New College,
Egham, Surrey TW20 0EX, United Kingdom
30
University of Louisville, Louisville, Kentucky 40292, USA
31
Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, D-55099 Mainz, Germany
32
University of Manchester, Manchester M13 9PL, United Kingdom
33
University of Maryland, College Park, Maryland 20742, USA
34
Massachusetts Institute of Technology, Laboratory for Nuclear Science,
Cambridge, Massachusetts 02139, USA
35a
Institute of Particle Physics, Montr ́
eal, Qu ́
ebec, Canada H3A 2T8
35b
McGill University, Montr ́
eal, Qu ́
ebec, Canada H3A 2T8
36a
INFN Sezione di Milano, I-20133 Milano, Italy
36b
Dipartimento di Fisica, Universit`
a di Milano, I-20133 Milano, Italy
37
University of Mississippi, University, Mississippi 38677, USA
38
Universit ́
e de Montr ́
eal, Physique des Particules, Montr ́
eal, Qu ́
ebec, Canada H3C 3J7
39
INFN Sezione di Napoli and Dipartimento di Scienze Fisiche, Universit`
a di Napoli Federico II,
I-80126 Napoli, Italy
40
NIKHEF, National Institute for Nuclear Physics and High Energy Physics,
NL-1009 DB Amsterdam, Netherlands
41
University of Notre Dame, Notre Dame, Indiana 46556, USA
42
Ohio State University, Columbus, Ohio 43210, USA
43a
INFN Sezione di Padova, I-35131 Padova, Italy
43b
Dipartimento di Fisica, Universit`
a di Padova, I-35131 Padova, Italy
44
Laboratoire de Physique Nucl ́
eaire et de Hautes Energies, Sorbonne Universit ́
e,
Paris Diderot Sorbonne Paris Cit ́
e, CNRS/IN2P3, F-75252 Paris, France
45a
INFN Sezione di Perugia, I-06123 Perugia, Italy
45b
Dipartimento di Fisica, Universit`
a di Perugia, I-06123 Perugia, Italy
46a
INFN Sezione di Pisa, I-56127 Pisa, Italy
46b
Dipartimento di Fisica, Universit`
a di Pisa, I-56127 Pisa, Italy
46c
Scuola Normale Superiore di Pisa, I-56127 Pisa, Italy
47
Princeton University, Princeton, New Jersey 08544, USA
48a
INFN Sezione di Roma, I-00185 Roma, Italy
48b
Dipartimento di Fisica, Universit`
a di Roma La Sapienza, I-00185 Roma, Italy
49
Universität Rostock, D-18051 Rostock, Germany
50
Rutherford Appleton Laboratory, Chilton, Didcot, Oxon, OX11 0QX, United Kingdom
51
IRFU, CEA, Universit ́
e Paris-Saclay, F-91191 Gif-sur-Yvette, France
52
SLAC National Accelerator Laboratory, Stanford, California 94309 USA
53
University of South Carolina, Columbia, South Carolina 29208, USA
54
Southern Methodist University, Dallas, Texas 75275, USA
55
St. Francis Xavier University, Antigonish, Nova Scotia B2G 2W5, Canada
56
Stanford University, Stanford, California 94305, USA
57
State University of New York, Albany, New York 12222, USA
58
Tel Aviv University, School of Physics and Astronomy, Tel Aviv 69978, Israel
59
University of Tennessee, Knoxville, Tennessee 37996, USA
J. P. LEES
et al.
PHYS. REV. D
104,
072002 (2021)
072002-2
60
University of Texas at Austin, Austin, Texas 78712, USA
61
University of Texas at Dallas, Richardson, Texas 75083, USA
62a
INFN Sezione di Torino, I-10125 Torino, Italy
62b
Dipartimento di Fisica, Universit`
a di Torino, I-10125 Torino, Italy
63
INFN Sezione di Trieste and Dipartimento di Fisica, Universit`
a di Trieste, I-34127 Trieste, Italy
64
IFIC, Universitat de Valencia-CSIC, E-46071 Valencia, Spain
65a
Institute of Particle Physics, Victoria, British Columbia V8W 3P6, Canada
65b
University of Victoria, Victoria, British Columbia V8W 3P6, Canada
66
Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom
67
University of Wisconsin, Madison, Wisconsin 53706, USA
(Received 10 June 2021; revised 9 August 2021; accepted 9 September 2021; published 7 October 2021)
We study the processes
γγ
→
η
c
→
η
0
K
þ
K
−
,
η
0
π
þ
π
−
, and
ηπ
þ
π
−
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. This is the first observation of
the decay
η
c
→
η
0
K
þ
K
−
and we measure the branching fraction
Γ
ð
η
c
→
η
0
K
þ
K
−
Þ
=
ð
Γ
ð
η
c
→
η
0
π
þ
π
−
Þ¼
0
.
644
0
.
039
stat
0
.
032
sys
. Significant interference is observed between
γγ
→
η
c
→
ηπ
þ
π
−
and the
nonresonant two-photon process
γγ
→
ηπ
þ
π
−
. A Dalitz plot analysis is performed of
η
c
decays to
η
0
K
þ
K
−
,
η
0
π
þ
π
−
, and
ηπ
þ
π
−
. Combined with our previous analysis of
η
c
→
K
̄
K
π
, we measure the
K
0
ð
1430
Þ
parameters and the ratio between its
η
0
K
and
π
K
couplings. The decay
η
c
→
η
0
π
þ
π
−
is dominated by the
f
0
ð
2100
Þ
resonance, also observed in
J=
ψ
radiative decays. A new
a
0
ð
1700
Þ
→
ηπ
resonance is observed
in the
η
c
→
ηπ
þ
π
−
channel. We also compare
η
c
decays to
η
and
η
0
final states in association with scalar
mesons as they relate to the identification of the scalar glueball.
DOI:
10.1103/PhysRevD.104.072002
I. INTRODUCTION
Scalar mesons remain a puzzle in light meson spectros-
copy: they have complex structure, and there are too many
states to be accommodated within the quark model without
difficulty
[1]
. In particular, the structure of the isospin I
¼
1
2
K
π
S
wave is still poorly understood, which limits the
precision of measurements involving a
K
π
system in the
final state, including recent searches for
CP
violation in
B
meson decay
[2]
, and studies of new exotic resonances
[3]
and charmed mesons
[4]
.
Decays of the
η
c
, the lightest pseudoscalar
c
̄
c
state,
provide a window on light meson states. The
BABAR
experiment first performed a Dalitz plot analysis of
η
c
→
K
þ
K
−
π
0
and
η
c
→
K
þ
K
−
η
using an isobar model
[5]
. The analysis reported the first observation of
K
0
ð
1430
Þ
→
K
η
, and observed that
η
c
decays into three
pseudoscalars are dominated by intermediate scalar mes-
ons. This newly observed
K
0
ð
1430
Þ
decay mode was
expected to be small and in fact was not observed in the
study of
K
−
p
→
K
−
η
p
interactions
[6]
. More recently, the
BABAR
experiment performed a measurement of the I
¼
1
2
K
π
S
-wave amplitude from a Dalitz plot analyses of
η
c
→
K
̄
K
π
[7]
. Further information on the properties of
the
K
0
ð
1430
Þ
resonance has been obtained by the CLEO
experiment in an analysis of the
D
þ
→
K
−
π
þ
π
þ
decay
[8]
,
and by the BESIII experiment, which observed its decay to
K
η
0
using
χ
c
1
decays to
η
0
K
þ
K
−
[9]
.
The existence of gluonium states is still an open issue for
quantum chromodynamics (QCD). Lattice QCD calcula-
tions predict the lightest gluonium states to have quantum
numbers
J
PC
¼
0
þþ
and
2
þþ
and to be in the mass region
below
2
.
5
GeV
=c
2
[10]
. In particular, the
J
PC
¼
0
þþ
glueball is predicted to have a mass around
1
.
7
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=
ψ
[11,12]
and
Υ
ð
1
S
Þ
[13]
.
However, despite intense experimental searches, there has
been no conclusive experimental observation
[14,15]
. The
identification of the scalar glueball is further complicated
by possible mixing with standard
q
̄
q
states. The broad
*
Deceased.
†
Also at Wuhan University, Wuhan 430072, China.
‡
Also at Universit`
a di Bologna and INFN Sezione di Bologna,
I-47921 Rimini, Italy.
§
Also at King
’
s College, London, WC2R 2LS, United Kingdom.
∥
Also at University of Huddersfield, Huddersfield HD1 3DH,
United Kingdom.
¶
Also at University of South Alabama, Mobile, Alabama
36688, USA.
**
Also at Universit`
a di Sassari, I-07100 Sassari, Italy.
††
Also at Gran Sasso Science Institute, I-67100 L
’
Aquila, Italy.
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI. Funded by SCOAP
3
.
LIGHT MESON SPECTROSCOPY FROM DALITZ PLOT
...
PHYS. REV. D
104,
072002 (2021)
072002-3
f
0
ð
500
Þ
,
f
0
ð
1370
Þ
[16]
,
f
0
ð
1500
Þ
[17,18]
,
f
0
ð
1710
Þ
[19,20]
, and possibly
f
0
ð
2100
Þ
[21]
have been suggested
as scalar glueball candidates. In the BESIII partial wave
analysis of the radiative
J=
ψ
decay to
ηη
[21]
, the authors
conclude that the production rates of
f
0
ð
1710
Þ
and
f
0
ð
2100
Þ
are both about one order of magnitude larger
than that of the
f
0
ð
1500
Þ
and no clear evidence is found for
f
0
ð
1370
Þ
. A feature of the scalar glueball is that its
s
̄
s
decay
mode should be favored with respect to
u
̄
u
or
d
̄
d
[22,23]
.
In the present analysis, we consider the three-body
η
c
decays to
η
0
K
þ
K
−
,
η
0
π
þ
π
−
, and
ηπ
þ
π
−
, using two-photon
interactions,
e
þ
e
−
→
e
þ
e
−
γ
γ
→
e
þ
e
−
η
c
. If both of the
virtual photons are quasireal, then the allowed
J
PC
values
of any produced resonances are
0
þ
;
2
þ
;
4
þ
...
[24]
.
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
η
0
meson, due to the
so-called gluon anomaly, has been discussed in recent years
[25,26]
. A comparison of the
η
and
η
0
content of
η
c
decays
might yield information on the possible gluonic content of
resonances decaying to
π
þ
π
−
or
K
þ
K
−
. The
γγ
→
η
0
π
þ
π
−
process has been recently studied by the Belle experiment
[27]
, 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
η
c
branching fraction. In Sec.
VI
we describe the Dalitz plot
analysis methodology, and in Secs.
VII
,
VIII
, and
IX
we
analyze
η
c
decays to
η
0
K
þ
K
−
,
η
0
π
þ
π
−
, and
ηπ
þ
π
−
, respec-
tively. 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
þ
e
−
collider located at SLAC, and
correspond to an integrated luminosity of
519
fb
−
1
[28]
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 in Ref.
[29]
. 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
L
mesons are detected in the instrumented
flux return of the magnet. Monte Carlo (MC) simulated
events
[30]
, 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
[31]
. In this article,
the inclusion of charge-conjugate processes is implied,
unless stated otherwise.
III. EVENT RECONSTRUCTION AND SELECTION
A. Reconstruction of the
η
0
h
+
h
−
final state
We first study the reactions
γγ
→
η
0
h
þ
h
−
;
ð
1
Þ
where
h
þ
h
−
indicates a
π
þ
π
−
or
K
þ
K
−
system. The
selection criteria are optimized for the
η
c
signal, as
described below. The
η
0
is reconstructed in the two decay
modes
η
0
→
ρ
0
γ
,
ρ
0
→
π
þ
π
−
, and
η
0
→
ηπ
þ
π
−
,
η
→
γγ
.To
reconstruct these final states we select events in which the
e
þ
and
e
−
beam particles are scattered at small angles, and
hence are undetected, ensuring that both virtual photons
are quasireal. We consider photon candidates with recon-
structed energy in the electromagnetic calorimeter greater
than 100 MeV. All pairs of photon candidates are com-
bined, assuming they originate from the
e
þ
e
−
interaction
region, and pairs with invariant-mass within
20
MeV
=c
2
(
150
MeV
=c
2
) of the neutral pion (
η
meson) mass are
considered
π
0
(
η
) candidates. We consider events with
exactly four 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
þ
e
−
interaction region and have a
χ
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
η
0
→
ρ
0
γ
selection, we allow the presence of only
two
γ
candidates, where
π
0
candidates are excluded. For the
η
0
→
ηπ
þ
π
−
we require exactly one
η
candidate, no more
than three additional background photon candidates, and no
π
0
candidate in the event. These selections are optimized on
the data using as reference the
η
c
signal.
To reconstruct
η
0
→
ρ
0
γ
decays, we consider
π
þ
π
−
pairs
in the mass region
0
.
620
<m
ð
π
þ
π
−
Þ
<
0
.
875
GeV
=c
2
.
Each of these
ρ
0
candidates is combined with all
γ
candidates, and any combination with invariant mass in
the range
0
.
935
<m
ð
ρ
0
γ
Þ
<
0
.
975
GeV
=c
2
is considered
an
η
0
candidate. We compute the angle
θ
γ
, defined as the
angle between the
π
þ
and the
γ
in the
π
þ
π
−
rest frame. The
distribution of
θ
γ
is expected to be proportional to sin
2
θ
γ
[32]
. We thus scan the
ρ
0
γ
mass spectrum with varying
selection on
j
cos
θ
γ
j
and obtain a small reduction of the
J. P. LEES
et al.
PHYS. REV. D
104,
072002 (2021)
072002-4
combinatorial background by requiring
j
cos
θ
γ
j
<
0
.
85
.
The above selection reduces the
η
0
signal and background
yields by 3% and 17%, respectively.
To improve the mass experimental resolution, the
η
0
four-
momentum is constructed by adding the momenta of the
π
þ
,
π
−
, and
γ
, and computing the
η
0
energy by assigning
the Particle Data Group (PDG)
[33]
nominal mass. This
method, tested on MC simulations, improves the resolution
by
≈
20%
.
To reconstruct
η
0
→
ηπ
þ
π
−
decays, we perform a kin-
ematic fit to the
η
candidate, and require the
ηπ
þ
π
−
mass to
be within
2
σ
of the fitted
η
0
mass
ð
956
.
8
0
.
5
Þ
MeV
=c
2
,
where
σ
¼
2
.
9
MeV
=c
2
is the width of the resolution
function describing the
η
0
signal. Similarly, to improve
the experimental resolution, the
η
0
four momentum is
constructed by adding the momenta of the
π
þ
,
π
−
, and
η
, and computing the
η
0
energy by assigning the PDG mass.
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 radi-
ation (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
¼
1
−−
resonance
[34]
.We
discriminate against ISR events by requiring the recoil
mass
M
2
rec
≡
ð
p
e
þ
e
−
−
p
rec
Þ
2
>
2
GeV
2
=c
4
, where
p
e
þ
e
−
is
the four-momentum of the initial state
e
þ
e
−
and
p
rec
is the reconstructed four-momentum of the candidate
η
0
ð
η
Þ
h
þ
h
−
system.
We define
p
T
as the magnitude of the transverse
momentum of the
η
0
h
þ
h
−
system, in the
e
þ
e
−
rest frame,
with respect to the beam axis. Well reconstructed two-
photon events with quasireal photons are expected to
have low values of
p
T
. Substantial background arises
from
γγ
→
2
h
þ
2
h
−
events, combined with a background
photon candidate. These are removed by requiring
p
T
ð
2
h
þ
2
h
−
Þ
>
0
.
1
GeV
=c
.
We retain events with
p
T
below a maximum value that
is optimized with respect to the
η
c
signal for each decay
mode. We produce
η
0
h
þ
h
−
invariant-mass spectra with
different maximum
p
T
values, and fit them to extract
the number of
η
c
signal events (
N
s
) (defined as the
2
.
93
–
3
.
03
GeV
=c
2
interval) and the number of background
events underneath the
η
c
signal (
N
b
). We then compute the
purity, defined as
P
¼
N
s
=
ð
N
s
þ
N
b
Þ
, the figure of merit
S
¼
N
s
=
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
N
s
þ
N
b
p
, and their product,
PS
.
1. Reconstruction of the
η
0
π
+
π
−
final state
For the final selection of the
η
0
π
þ
π
−
final state, we
require all four charged tracks to be positively identified
as pions, using an algorithm based on multivariate
analysis
[35]
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
ð
η
0
ð
η
Þ
h
þ
h
−
Þ
>
2
.
7
GeV
=c
2
. In the case of
η
0
→
ρ
0
γ
an upper mass
requirement
m
ð
η
0
π
þ
π
−
Þ
<
3
.
5
GeV
=c
2
is applied because
of the large number of combinations produced by
thepresenceofthe
γ
. The data are compared with
expectations from
η
c
signal MC simulations; a signal
from two-photon production is observed in the data in
both cases, and is particularly clean for
η
0
→
ηπ
þ
π
−
.Ina
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
GeV
=c
for the
η
0
→
ρ
0
γ
decay
candidates, and a maximum of
PS
at
0
.
15
GeV
=c
for
the
η
0
→
ηπ
þ
π
−
candidates. We require
p
T
<
0
.
05
GeV
=c
and
p
T
<
0
.
15
GeV
=c
, respectively, as indicated by the
dashed lines in the figures.
Figures
2(a)
and
2(b)
show the
ρ
0
γ
and
ηπ
þ
π
−
invariant-
mass distributions, respectively, for events satisfying all
selection criteria except that on these masses. Clear
η
0
signals are visible, and the shaded regions indicate the
selection windows,
ð
0
.
935
–
0
.
975
Þ
GeV
=c
2
for
η
0
→
ρ
0
γ
and
ð
0
.
948
–
0
.
966
Þ
GeV
=c
2
for
η
0
→
ηπ
þ
π
−
. Figures
3(a)
and
3(b)
show the
η
0
π
þ
π
−
invariant-mass spectra for the
selected events in the data. Prominent
η
c
signals are
observed, and there is some activity in the
η
c
ð
2
S
Þ
mass
region.
If there are multiple candidates in the same event, then
we retain them all. The fraction of events having two
combinations in the
η
c
mass region is 3% (and 3.4% in
η
c
signal MC simulations) for
η
c
→
η
0
π
þ
π
−
with
η
0
→
ρ
0
γ
.
No multiple candidates are found for
η
0
→
π
þ
π
−
π
0
or any
of the other final states discussed below.
2. Reconstruction of the
η
0
K
+
K
−
final state
For the
η
0
K
þ
K
−
final state, we require the two charged
tracks assigned to the
η
0
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 charmo-
nium region, compared with MC
η
c
signal simulations, are
shown in Figs.
1(c)
–
1(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
η
0
π
þ
π
−
final state are used,
indicated by the dashed lines in the figures.
The corresponding
η
0
signals for this final state are
shown in Figs.
2(c)
–
2(d)
, and the
η
0
K
þ
K
−
invariant-mass
spectra are shown in Figs.
3(c)
–
3(d)
. Prominent
η
c
signals
with low background are present in both invariant-mass
spectra with possible weak activity in the
η
c
ð
2
S
Þ
mass
region. The decay
η
c
→
η
0
K
þ
K
−
is observed here for the
first time.
LIGHT MESON SPECTROSCOPY FROM DALITZ PLOT
...
PHYS. REV. D
104,
072002 (2021)
072002-5
B. Reconstruction of the
ηπ
+
π
−
final state
We study the reaction
γγ
→
ηπ
þ
π
−
;
ð
2
Þ
where
η
→
γγ
and
η
→
π
þ
π
−
π
0
.
1.
η
→
γγ
For reaction
(2)
, where
η
→
γγ
, we again consider well-
measured charged-particle tracks with transverse momenta
greater than
0
.
1
GeV
=c
and photons with energy greater
than
0
.
1
GeV, and each pair of
γ
s is kinematically fitted to
the
π
0
and
η
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
χ
2
probability of the fit to be greater than 0.1%. We retain
events having exactly one
η
candidate, no
π
0
candidates,
and no more than three background
γ
s.
The two charged tracks are required to be loosely
identified as pions. Most ISR events are removed by
requiring
M
2
rec
≡
ð
p
e
þ
e
−
−
p
rec
Þ
2
>
2
GeV
2
=c
4
. Further
background is due to the presence of ISR events from
ψ
ð
2
S
Þ
→
η
J=
ψ
→
ημ
þ
μ
−
, where the two muons are mis-
identified as pions. This background is efficiently removed
by vetoing events having two loosely identified muons.
Background from the process
γγ
→
π
þ
π
−
is removed by
requiring
p
T
ð
π
þ
π
−
Þ
>
0
.
05
GeV
=c
.
The
p
T
distribution for such events in the charmonium
mass region is compared with
η
c
signal MC simulation in
Fig.
4(a)
, where a clear signal of the two-photon reaction
is observed. Optimizing the
η
c
figure of merit (
S
) and
purity (
P
), we require
p
T
<
0
.
1
GeV
=c
. The resulting
ηπ
þ
π
−
invariant-mass spectrum is shown in Fig.
6(a)
,
where the
η
c
signal can be observed together with some
weak activity in the
η
c
ð
2
S
Þ
mass region.
2.
η
→
π
+
π
−
π
0
For reaction
(2)
, where
η
→
π
þ
π
−
π
0
, we require exactly
four well-measured charged-particle tracks with the
vertex
χ
2
fit probability greater than 0.1%. In order to
have sensitivity to low momentum
π
0
mesons, we consider
photons with energy greater than
30
MeV
=c
2
. We allow no
) (GeV/c)
-
π
+
π
'
η
(
T
p
0
0.1
0.2
0.3
0.4
0.5
events/5 (MeV/c)
0
500
1000
1500
γ
0
ρ
→
'
η
(a)
) (GeV/c)
-
π
+
π
'
η
(
T
p
0
0.1
0.2
0.3
0.4
0.5
events/5 (MeV/c)
0
100
200
−
π
+
π
η
→
'
η
(b)
) (GeV/c)
-
K
+
'K
η
(
T
p
0
0.1
0.2
0.3
0.4
0.5
events/5 (MeV/c)
0
100
200
300
γ
0
ρ
→
'
η
(c)
) (GeV/c)
-
K
+
'K
η
(
T
p
0
0.1
0.2
0.3
0.4
0.5
events/5 (MeV/c)
0
20
40
60
80
−
π
+
π
η
→
'
η
(d)
FIG. 1. Distributions of the transverse momenta of the (a),(b)
η
0
π
þ
π
−
and (c),(d)
η
0
K
þ
K
−
systems for events satisfying all other
selection criteria, in which the
η
0
is reconstructed in the (a),(c)
ρ
0
γ
and (b),(d)
ηπ
þ
π
−
decay modes. The data are represented by points
with error bars, and the
η
c
MC simulation by solid (red) histograms with arbitrary normalization. The (blue) dashed lines indicate the
selection used to isolate two-photon event candidates.
J. P. LEES
et al.
PHYS. REV. D
104,
072002 (2021)
072002-6
more than two kinematically fitted
π
0
candidates and no
more than five background
γ
s. Candidate
γγ
→
2
π
þ
2
π
−
events are removed by requiring
p
T
ð
2
π
þ
2
π
−
Þ
>
0
.
05
GeV
=c
. Background ISR events are removed by
requiring
M
2
rec
≡
ð
p
e
þ
e
−
−
p
rec
Þ
2
>
2
GeV
2
=c
4
. All four
charged tracks are required to be loosely identified as pions.
The
η
candidates are reconstructed by combining every
pair of oppositely charged tracks with each of the
π
0
candidates in the event. The resulting
π
þ
π
−
π
0
invariant-
mass spectrum is shown in Fig.
5
. A clean
η
signal can be
seen; we select candidates in the mass region
538
<
m
ð
π
þ
π
−
π
0
Þ
<
557
MeV
=c
2
. The
η
is then reconstructed
by adding the momentum three-vectors of the three
pions and computing the
η
energy using its nominal
PDG mass.
The
p
T
distribution for such events in the charmonium
mass region is compared with
η
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
p
T
<
0
.
1
GeV
=c
. The resulting
ηπ
þ
π
−
invariant-mass
spectrum is shown in Fig.
6(b)
,wherethe
η
c
signal
can be observed together with some weak activity in the
η
c
ð
2
S
Þ
mass region.
IV. EFFICIENCY AND
η
c
INVARIANT-MASS
RESOLUTION
To compute the reconstruction and selection efficiency,
MC signal events are generate
d using a detailed detector
simulation
[30,31]
in which the
η
c
mesons decay uni-
formly in phase space. These simulated events are
reconstructed and analyzed in the same manner as data.
We define the helicity angle
θ
H
as the angle formed
by the
h
þ
(where
h
¼
π
,
K
), in the
h
þ
h
−
rest frame, and
the
η
0
(
η
) direction in the
h
þ
h
−
η
0
(
h
þ
h
−
η
)restframe.
For each final state, we compute the raw efficiency in
50
×
50
intervals of the invariant-mass,
m
ð
h
þ
h
−
Þ
,and
cos
θ
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
θ
H
in each of the
100
MeV
=c
2
wide
intervals of
m
ð
h
þ
h
−
Þ
, using Legendre polynomials up to
L
¼
12
:
ε
ð
cos
θ
H
Þ¼
X
12
L
¼
0
a
L
ð
m
Þ
Y
0
L
ð
cos
θ
H
Þ
;
ð
3
Þ
)
2
) (GeV/c
γ
0
ρ
m(
0.85
0.9
0.95
1
1.05
)
2
events/2 (MeV/c
0
200
400
600
(a)
)
2
) (GeV/c
-
π
+
π
η
m(
0.9
0.95
1
)
2
events/1 (MeV/c
0
100
200
300
400
(b)
)
2
) (GeV/c
γ
0
ρ
m(
0.85
0.9
0.95
1
1.05
)
2
events/4 (MeV/c
0
100
200
300
(c)
)
2
) (GeV/c
-
π
+
π
η
m(
0.9
0.95
1
)
2
events/1 (MeV/c
0
50
100
(d)
FIG. 2. Invariant-mass distributions of (a)
ρ
0
γ
and (b)
ηπ
þ
π
−
for
γγ
→
η
0
π
þ
π
−
candidates satisfying all other selection criteria.
Corresponding (c)
ρ
0
γ
and (d)
ηπ
þ
π
−
invariant-mass distributions for
γγ
→
η
0
K
þ
K
−
candidates. The shaded areas indicate the
η
0
selections.
LIGHT MESON SPECTROSCOPY FROM DALITZ PLOT
...
PHYS. REV. D
104,
072002 (2021)
072002-7
where
m
denotes the
h
þ
h
−
invariant mass. For a given value
of
m
ð
h
þ
h
−
Þ
, the efficiency is interpolated linearly between
adjacent mass intervals.
Figure
7
shows the resulting efficiency maps
ε
ð
m;
cos
θ
H
Þ
for the four
η
0
h
þ
h
−
final states, and Fig.
8
shows the maps for the two
ηπ
þ
π
−
final states. The small
regions of very low efficiency near
j
cos
θ
H
j
∼
1
are the
result of the difficulty of reconstructing
K
mesons
with laboratory momentum less than
≈
200
MeV
=c
,
and
π
mesons with laboratory momentum less than
≈
100
MeV
=c
, due to energy loss in the beam pipe and
inner-detector material.
)
2
) (GeV/c
-
π
+
π
'
η
m(
2.5
3
3.5
4
)
2
events/10 (MeV/c
0
100
200
300
400
γ
0
ρ
→
'
η
(a)
)
2
) (GeV/c
-
π
+
π
'
η
m(
2.5
3
3.5
4
)
2
events/10 (MeV/c
0
50
100
150
-
π
+
π
η
→
'
η
(b)
)
2
) (GeV/c
-
K
+
'K
η
m(
2.5
3
3.5
4
)
2
events/10 (MeV/c
0
50
100
γ
0
ρ
→
'
η
(c)
)
2
) (GeV/c
-
K
+
'K
η
m(
2.5
3
3.5
4
)
2
events/10 (MeV/c
0
20
40
-
π
+
π
η
→
'
η
(d)
FIG. 3. The
η
0
π
þ
π
−
invariant-mass spectra for selected events with (a)
η
0
→
ρ
0
γ
and (b)
η
0
→
ηπ
þ
π
−
. The
η
0
K
þ
K
−
invariant-mass
spectra for selected events with (c)
η
0
→
ρ
0
γ
and (d)
η
0
→
ηπ
þ
π
−
.
) (GeV/c)
-
π
+
π
η
(
T
p
0
0.1
0.2
0.3
0.4
0.5
events/5 (MeV/c)
0
1000
2000
3000
γ
γ
→
η
(a)
) (GeV/c)
-
π
+
π
η
(
T
p
0
0.1
0.2
0.3
0.4
0.5
events/5 (MeV/c)
0
500
1000
0
π
-
π
+
π
→
η
(b)
FIG. 4. Distributions of the transverse momentum
p
T
ð
ηπ
þ
π
−
Þ
for selected
γγ
→
ηπ
þ
π
−
candidates with (a)
η
→
γγ
and
(b)
η
→
π
þ
π
−
π
0
, in the charmonium mass region. The data are represented by the points with error bars, and the
η
c
MC simulation
as solid (red) histograms with arbitrary normalizations. The dashed (blue) lines indicate the selection used to isolate two-photon event
candidates.
J. P. LEES
et al.
PHYS. REV. D
104,
072002 (2021)
072002-8
The mass resolution is determined from the distribution
of the difference (
Δ
m
) between the generated and recon-
structed
η
0
h
þ
h
−
or
ηπ
þ
π
−
invariant-mass values. The
Δ
m
distributions are parametrized by the sum of a Crystal
Ball
[36]
and a Gaussian function, which describe well
the distributions, and have root-mean-squared values
of the following:
11
.
5
MeV
=c
2
for
η
0
π
þ
π
−
,
η
0
→
ρ
0
γ
;
13
.
9
MeV
=c
2
for
η
0
π
þ
π
−
,
η
0
→
ηπ
þ
π
−
;
8
.
2
MeV
=c
2
for
η
0
K
þ
K
−
,
η
0
→
ρ
0
γ
;
12
.
2
MeV
=c
2
for
η
0
K
þ
K
−
,
η
0
→
ηπ
þ
π
−
;
15
.
9
MeV
=c
2
for
ηπ
þ
π
−
,
η
→
γγ
; and
13
.
8
MeV
=c
2
for
ηπ
þ
π
−
,
η
→
π
þ
π
−
π
0
.
V. YIELDS AND BRANCHING FRACTIONS
In this section, we fit the invariant-mass distributions to
obtain the numbers of selected
η
c
events,
N
η
0
K
þ
K
−
,
N
η
0
π
þ
π
−
,
and
N
ηπ
þ
π
−
, for each
η
0
or
η
decay mode. We then use
the
η
0
K
þ
K
−
and
η
0
π
þ
π
−
yields to compute the ratio of
branching fractions for
η
c
to the
η
0
K
þ
K
−
and
η
0
π
þ
π
−
final
states. This ratio is computed as
R
¼
B
ð
η
c
→
η
0
K
þ
K
−
Þ
B
ð
η
c
→
η
0
π
þ
π
−
Þ
;
¼
N
η
0
K
þ
K
−
N
η
0
π
þ
π
−
ε
η
0
π
þ
π
−
ε
η
0
K
þ
K
−
ð
4
Þ
for each
η
0
decay mode, where
ε
η
0
K
þ
K
−
and
ε
η
0
π
þ
π
−
are
the corresponding weighted efficiencies described in the
following Sec.
VB
.
A. Fits to the invariant-mass spectra
We determine
N
K
þ
K
−
η
0
and
N
π
þ
π
−
η
0
from
η
c
decays by
performing binned
χ
2
fits to the
η
0
K
þ
K
−
and
η
0
π
þ
π
−
invariant-mass spectra, in the
2
.
7
–
3
.
3
GeV
=c
2
mass region,
separately for the two
η
0
decay modes. In these fits, the
η
c
signal contribution is described by a simple Breit-Wigner
(BW) function convolved with a fixed resolution function
described above, with
η
c
parameters fixed to PDG values
[33]
. An additional BW function is used to describe the
residual background from ISR
J=
ψ
events, and the remain-
ing background is parametrized by a second order poly-
nomial. The fitted
η
0
h
þ
h
−
invariant-mass spectra are
shown in Fig.
9
. The fits generally describe the data well,
although the fit to the
η
0
K
þ
K
−
invariant-mass spectrum for
η
0
→
ηπ
þ
π
−
[Fig.
9(d)
], which has low statistics, appears
to the eye to have a somewhat distorted line shape. 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
η
c
signal yields are obtained by integrating the
data over the
η
c
signal region after subtracting the fitted
backgrounds.
Statistical errors on the
η
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
η
c
subtracted yields are fitted using a
)
2
) (GeV/c
0
π
-
π
+
π
m(
0.5
0.55
0.6
)
2
events/1 (MeV/c
0
500
1000
1500
FIG. 5. Distribution of the reconstructed
π
þ
π
−
π
0
mass for
selected
γγ
→
ηπ
þ
π
−
candidate events. The shaded area indicates
the
η
selection region.
)
2
) (GeV/c
-
π
+
π
η
m(
2.5
3
3.5
4
)
2
events/10 (MeV/c
0
500
1000
γ
γ
→
η
(a)
)
2
) (GeV/c
-
π
+
π
η
m(
2.5
3
3.5
4
)
2
events/10 (MeV/c
0
100
200
300
400
0
π
-
π
+
π
→
η
(b)
FIG. 6. The
ηπ
þ
π
−
invariant-mass spectra for selected events with (a)
η
→
γγ
and (b)
η
→
π
þ
π
−
π
0
.
LIGHT MESON SPECTROSCOPY FROM DALITZ PLOT
...
PHYS. REV. D
104,
072002 (2021)
072002-9