This is the accepted manuscript made available via CHORUS. The article has been
published as:
Amplitude analysis of the χ_{c1}→ηπ^{+}π^{-} decays
M. Ablikim et al. (BESIII Collaboration)
Phys. Rev. D 95, 032002 — Published 13 February 2017
DOI: 10.1103/PhysRevD.95.032002
Amplitude analysis of the + −χc1 → ηπ π decays
M. Ablikim1, M. N. Achasov9,e, S. Ahmed14, X. C. Ai1, O. Albayrak5, M. Albrecht4, D. J. Ambrose44,
A. Amoroso49A,49C, F. F. An1, Q. An46,a, J. Z. Bai1, O. Bakina23, R. Baldini Ferroli20A, Y. Ban31, D. W. Bennett19,
J. V. Bennett5, N. Berger22, M. Bertani20A, D. Bettoni21A, J. M. Bian43, F. Bianchi49A,49C , E. Boger23,c,
I. Boyko23, R. A. Briere5, H. Cai51, X. Cai1,a, O. Cakir40A, A. Calcaterra20A, G. F. Cao1, S. A. Cetin40B ,
J. Chai49C , J. F. Chang1,a, G. Chelkov23,c,d, G. Chen1, H. S. Chen1, J. C. Chen1, M. L. Chen1,a, S. Chen41,
S. J. Chen29, X. Chen1,a, X. R. Chen26, Y. B. Chen1,a, H. P. Cheng17, X. K. Chu31, G. Cibinetto21A, H. L. Dai1,a,
J. P. Dai34, A. Dbeyssi14, D. Dedovich23, Z. Y. Deng1, A. Denig22, I. Denysenko23, M. Destefanis49A,49C ,
F. De Mori49A,49C , Y. Ding27, C. Dong30, J. Dong1,a, L. Y. Dong1, M. Y. Dong1,a, Z. L. Dou29, S. X. Du53,
P. F. Duan1, J. Z. Fan39, J. Fang1,a, S. S. Fang1, X. Fang46,a, Y. Fang1, R. Farinelli21A,21B, L. Fava49B,49C ,
F. Feldbauer22, G. Felici20A, C. Q. Feng46,a, E. Fioravanti21A, M. Fritsch14,22, C. D. Fu1, Q. Gao1, X. L. Gao46,a,
Y. Gao39, Z. Gao46,a, I. Garzia21A, K. Goetzen10, L. Gong30, W. X. Gong1,a, W. Gradl22, M. Greco49A,49C ,
M. H. Gu1,a, Y. T. Gu12, Y. H. Guan1, A. Q. Guo1, L. B. Guo28, R. P. Guo1, Y. Guo1, Y. P. Guo22, Z. Haddadi25,
A. Hafner22, S. Han51, X. Q. Hao15, F. A. Harris42, K. L. He1, F. H. Heinsius4, T. Held4, Y. K. Heng1,a,
T. Holtmann4, Z. L. Hou1, C. Hu28, H. M. Hu1, J. F. Hu49A,49C , T. Hu1,a, Y. Hu1, G. S. Huang46,a, J. S. Huang15,
X. T. Huang33, X. Z. Huang29, Y. Huang29, Z. L. Huang27, T. Hussain48, W. Ikegami Andersson50, Q. Ji1,
Q. P. Ji15, X. B. Ji1, X. L. Ji1,a, L. W. Jiang51, X. S. Jiang1,a, X. Y. Jiang30, J. B. Jiao33, Z. Jiao17, D. P. Jin1,a,
S. Jin1, T. Johansson50, A. Julin43, N. Kalantar-Nayestanaki25, X. L. Kang1, X. S. Kang30, M. Kavatsyuk25,
B. C. Ke5, P. Kiese22, R. Kliemt10, B. Kloss22, O. B. Kolcu40B,h, B. Kopf4, M. Kornicer42, A. Kupsc50, W. Kühn24,
J. S. Lange24, M. Lara19, P. Larin14, H. Leithoff22, C. Leng49C , C. Li50, Cheng Li46,a, D. M. Li53, F. Li1,a,
F. Y. Li31, G. Li1, H. B. Li1, H. J. Li1, J. C. Li1, Jin Li32, K. Li33, K. Li13, Lei Li3, P. R. Li41, Q. Y. Li33, T. Li33,
W. D. Li1, W. G. Li1, X. L. Li33, X. N. Li1,a, X. Q. Li30, Y. B. Li2, Z. B. Li38, H. Liang46,a, Y. F. Liang36,
Y. T. Liang24, G. R. Liao11, D. X. Lin14, B. Liu34, B. J. Liu1, C. X. Liu1, D. Liu46,a, F. H. Liu35, Fang Liu1,
Feng Liu6, H. B. Liu12, H. H. Liu1, H. H. Liu16, H. M. Liu1, J. Liu1, J. B. Liu46,a, J. P. Liu51, J. Y. Liu1, K. Liu39,
K. Y. Liu27, L. D. Liu31, P. L. Liu1,a, Q. Liu41, S. B. Liu46,a, X. Liu26, Y. B. Liu30, Y. Y. Liu30, Z. A. Liu1,a,
Zhiqing Liu22, H. Loehner25, Y. F. Long31, X. C. Lou1,a,g, H. J. Lu17, J. G. Lu1,a, Y. Lu1, Y. P. Lu1,a, C. L. Luo28,
M. X. Luo52, T. Luo42, X. L. Luo1,a, X. R. Lyu41, F. C. Ma27, H. L. Ma1, L. L. Ma33, M. M. Ma1, Q. M. Ma1,
T. Ma1, X. N. Ma30, X. Y. Ma1,a, Y. M. Ma33, F. E. Maas14, M. Maggiora49A,49C, Q. A. Malik48, Y. J. Mao31,
Z. P. Mao1, S. Marcello49A,49C , J. G. Messchendorp25, G. Mezzadri21B, J. Min1,a, T. J. Min1, R. E. Mitchell19,
X. H. Mo1,a, Y. J. Mo6, C. Morales Morales14, N. Yu. Muchnoi9,e, H. Muramatsu43, P. Musiol4, Y. Nefedov23,
F. Nerling10, I. B. Nikolaev9,e, Z. Ning1,a, S. Nisar8, S. L. Niu1,a, X. Y. Niu1, S. L. Olsen32, Q. Ouyang1,a,
S. Pacetti20B, Y. Pan46,a, P. Patteri20A, M. Pelizaeus4, H. P. Peng46,a, K. Peters10,i, J. Pettersson50, J. L. Ping28,
R. G. Ping1, R. Poling43, V. Prasad1, H. R. Qi2, M. Qi29, S. Qian1,a, C. F. Qiao41, L. Q. Qin33, N. Qin51,
X. S. Qin1, Z. H. Qin1,a, J. F. Qiu1, K. H. Rashid48, C. F. Redmer22, M. Ripka22, G. Rong1, Ch. Rosner14,
X. D. Ruan12, A. Sarantsev23,f , M. Savrié21B, C. Schnier4, K. Schoenning50, S. Schumann22, W. Shan31,
M. Shao46,a, C. P. Shen2, P. X. Shen30, X. Y. Shen1, H. Y. Sheng1, M. Shi1, W. M. Song1, X. Y. Song1,
S. Sosio49A,49C , S. Spataro49A,49C , G. X. Sun1, J. F. Sun15, S. S. Sun1, X. H. Sun1, Y. J. Sun46,a, Y. Z. Sun1,
Z. J. Sun1,a, Z. T. Sun19, C. J. Tang36, X. Tang1, I. Tapan40C , E. H. Thorndike44, M. Tiemens25, I. Uman40D,
G. S. Varner42, B. Wang30, B. L. Wang41, D. Wang31, D. Y. Wang31, K. Wang1,a, L. L. Wang1, L. S. Wang1,
M. Wang33, P. Wang1, P. L. Wang1, W. Wang1,a, W. P. Wang46,a, X. F. Wang39, Y. Wang37, Y. D. Wang14,
Y. F. Wang1,a, Y. Q. Wang22, Z. Wang1,a, Z. G. Wang1,a, Z. H. Wang46,a, Z. Y. Wang1, Z. Y. Wang1, T. Weber22,
D. H. Wei11, P. Weidenkaff22, S. P. Wen1, U. Wiedner4, M. Wolke50, L. H. Wu1, L. J. Wu1, Z. Wu1,a, L. Xia46,a,
L. G. Xia39, Y. Xia18, D. Xiao1, H. Xiao47, Z. J. Xiao28, Y. G. Xie1,a, Q. L. Xiu1,a, G. F. Xu1, J. J. Xu1, L. Xu1,
Q. J. Xu13, Q. N. Xu41, X. P. Xu37, L. Yan49A,49C , W. B. Yan46,a, W. C. Yan46,a, Y. H. Yan18, H. J. Yang34,j,
H. X. Yang1, L. Yang51, Y. X. Yang11, M. Ye1,a, M. H. Ye7, J. H. Yin1, Z. Y. You38, B. X. Yu1,a, C. X. Yu30,
J. S. Yu26, C. Z. Yuan1, W. L. Yuan29, Y. Yuan1, A. Yuncu40B,b, A. A. Zafar48, A. Zallo20A, Y. Zeng18, Z. Zeng46,a,
B. X. Zhang1, B. Y. Zhang1,a, C. Zhang29, C. C. Zhang1, D. H. Zhang1, H. H. Zhang38, H. Y. Zhang1,a, J. Zhang1,
J. J. Zhang1, J. L. Zhang1, J. Q. Zhang1, J. W. Zhang1,a, J. Y. Zhang1, J. Z. Zhang1, K. Zhang1, L. Zhang1,
S. Q. Zhang30, X. Y. Zhang33, Y. Zhang1, Y. Zhang1, Y. H. Zhang1,a, Y. N. Zhang41, Y. T. Zhang46,a,
Yu Zhang41, Z. H. Zhang6, Z. P. Zhang46, Z. Y. Zhang51, G. Zhao1, J. W. Zhao1,a, J. Y. Zhao1, J. Z. Zhao1,a,
Lei Zhao46,a, Ling Zhao1, M. G. Zhao30, Q. Zhao1, Q. W. Zhao1, S. J. Zhao53, T. C. Zhao1, Y. B. Zhao1,a,
Z. G. Zhao46,a, A. Zhemchugov23,c, B. Zheng47, J. P. Zheng1,a, W. J. Zheng33, Y. H. Zheng41, B. Zhong28,
L. Zhou1,a, X. Zhou51, X. K. Zhou46,a, X. R. Zhou46,a, X. Y. Zhou1, K. Zhu1, K. J. Zhu1,a, S. Zhu1, S. H. Zhu45,
2
X. L. Zhu39, Y. C. Zhu46,a, Y. S. Zhu1, Z. A. Zhu1, J. Zhuang1,a, L. Zotti49A,49C , B. S. Zou1, J. H. Zou1
(BESIII Collaboration)
1 Institute of High Energy Physics, Beijing 100049, People’s Republic of China
2 Beihang University, Beijing 100191, People’s Republic of China
3 Beijing Institute of Petrochemical Technology, Beijing 102617, People’s Republic of China
4 Bochum Ruhr-University, D-44780 Bochum, Germany
5 Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA
6 Central China Normal University, Wuhan 430079, People’s Republic of China
7 China Center of Advanced Science and Technology, Beijing 100190, People’s Republic of China
8 COMSATS Institute of Information Technology, Lahore, Defence Road, Off Raiwind Road, 54000 Lahore, Pakistan
9 G.I. Budker Institute of Nuclear Physics SB RAS (BINP), Novosibirsk 630090, Russia
10 GSI Helmholtzcentre for Heavy Ion Research GmbH, D-64291 Darmstadt, Germany
11 Guangxi Normal University, Guilin 541004, People’s Republic of China
12 Guangxi University, Nanning 530004, People’s Republic of China
13 Hangzhou Normal University, Hangzhou 310036, People’s Republic of China
14 Helmholtz Institute Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany
15 Henan Normal University, Xinxiang 453007, People’s Republic of China
16 Henan University of Science and Technology, Luoyang 471003, People’s Republic of China
17 Huangshan College, Huangshan 245000, People’s Republic of China
18 Hunan University, Changsha 410082, People’s Republic of China
19 Indiana University, Bloomington, Indiana 47405, USA
20 (A)INFN Laboratori Nazionali di Frascati, I-00044, Frascati,
Italy; (B)INFN and University of Perugia, I-06100, Perugia, Italy
21 (A)INFN Sezione di Ferrara, I-44122, Ferrara, Italy; (B)University of Ferrara, I-44122, Ferrara, Italy
22 Johannes Gutenberg University of Mainz, Johann-Joachim-Becher-Weg 45, D-55099 Mainz, Germany
23 Joint Institute for Nuclear Research, 141980 Dubna, Moscow region, Russia
24 Justus-Liebig-Universitaet Giessen, II. Physikalisches Institut, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany
25 KVI-CART, University of Groningen, NL-9747 AA Groningen, The Netherlands
26 Lanzhou University, Lanzhou 730000, People’s Republic of China
27 Liaoning University, Shenyang 110036, People’s Republic of China
28 Nanjing Normal University, Nanjing 210023, People’s Republic of China
29 Nanjing University, Nanjing 210093, People’s Republic of China
30 Nankai University, Tianjin 300071, People’s Republic of China
31 Peking University, Beijing 100871, People’s Republic of China
32 Seoul National University, Seoul, 151-747 Korea
33 Shandong University, Jinan 250100, People’s Republic of China
34 Shanghai Jiao Tong University, Shanghai 200240, People’s Republic of China
35 Shanxi University, Taiyuan 030006, People’s Republic of China
36 Sichuan University, Chengdu 610064, People’s Republic of China
37 Soochow University, Suzhou 215006, People’s Republic of China
38 Sun Yat-Sen University, Guangzhou 510275, People’s Republic of China
39 Tsinghua University, Beijing 100084, People’s Republic of China
40 (A)Ankara University, 06100 Tandogan, Ankara, Turkey; (B)Istanbul Bilgi
University, 34060 Eyup, Istanbul, Turkey; (C)Uludag University, 16059 Bursa,
Turkey; (D)Near East University, Nicosia, North Cyprus, Mersin 10, Turkey
41 University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China
42 University of Hawaii, Honolulu, Hawaii 96822, USA
43 University of Minnesota, Minneapolis, Minnesota 55455, USA
44 University of Rochester, Rochester, New York 14627, USA
45 University of Science and Technology Liaoning, Anshan 114051, People’s Republic of China
46 University of Science and Technology of China, Hefei 230026, People’s Republic of China
47 University of South China, Hengyang 421001, People’s Republic of China
48 University of the Punjab, Lahore-54590, Pakistan
3
49 (A)University of Turin, I-10125, Turin, Italy; (B)University of Eastern
Piedmont, I-15121, Alessandria, Italy; (C)INFN, I-10125, Turin, Italy
50 Uppsala University, Box 516, SE-75120 Uppsala, Sweden
51 Wuhan University, Wuhan 430072, People’s Republic of China
52 Zhejiang University, Hangzhou 310027, People’s Republic of China
53 Zhengzhou University, Zhengzhou 450001, People’s Republic of China
a Also at State Key Laboratory of Particle Detection and
Electronics, Beijing 100049, Hefei 230026, People’s Republic of China
b Also at Bogazici University, 34342 Istanbul, Turkey
c Also at the Moscow Institute of Physics and Technology, Moscow 141700, Russia
d Also at the Functional Electronics Laboratory, Tomsk State University, Tomsk, 634050, Russia
e Also at the Novosibirsk State University, Novosibirsk, 630090, Russia
f Also at the NRC ”Kurchatov Institute”, PNPI, 188300, Gatchina, Russia
g Also at University of Texas at Dallas, Richardson, Texas 75083, USA
h Also at Istanbul Arel University, 34295 Istanbul, Turkey
i Also at Goethe University Frankfurt, 60323 Frankfurt am Main, Germany
j Also at Institute of Nuclear and Particle Physics, Shanghai Key Laboratory for
Particle Physics and Cosmology, Shanghai 200240, People’s Republic of China
Using 448.0 × 106 ψ(3686) events collected with the BESIII detector, an amplitude analysis is
performed for ψ(3686) → γχ + −c1, χc1 → ηπ π decays. The most dominant two-body structure
observed is a0(980)
±π∓; a0(980)
±
→ ηπ±. The a0(980) line shape is modeled using a dispersion
relation, and a significant non-zero a0(980) coupling to the η
′π channel is measured. We observe
χc1 → a2(1700)π production for the first time, with a significance larger than 17σ. The production
of mesons with exotic quantum numbers, JPC = 1−+, is investigated, and upper limits for the
branching fractions χ π (1400)±π∓, χ π (1600)±π∓, and χ π (2015)±c1 → 1 c1 → 1 c1 → 1 π
∓, with
subsequent π (X)± ηπ±1 → decay, are determined.
PACS numbers: 13.25.Gv, 14.40.Be, 14.40.Pq, 14.40.Rt
I. INTRODUCTION exotic signal in χ ′ + −c1 → η π π decays, consistent with
π1(1600) → η′π production [10]. However, other possi-
ble exotic signals that could be expected have not been
Charmonium decays provide a rich laboratory for light observed in either χ + − ′ + −c1 → ηπ π or χc1 → η π π de-
meson spectroscopy. Large samples of charmonium states cays. With a more than 15 times larger data sample at
with JPC = 1−−, like the J/ψ and ψ(3686), are easily BESIII, there is an opportunity to search for the pro-
produced at e+e− colliders, and their transitions pro- duction of π1 exotic mesons. In this work we investi-
vide sizable charmonium samples with other JPC quan- gate possible production of exotic mesons in the mass
tum numbers, like the χc1 (1
++). The χc1 → ηππ region (1.3-2.0) GeV/c2, decaying into the ηπ+ + c.c. fi-
decay is suitable for studying the production of exotic nal state, namely the π (1400), π (1600), and π (2015),
mesons with JPC = 1−+
1 1 1
, which could be observed de- using χ + −c1 → ηπ π decays. Charge conjugation and
caying into the ηπ final state. The lowest orbital ex- isospin symmetry are assumed in this analysis.
citation of a two-body combination in χc1 decays to
three pseudoscalars, for instance χc1 → ηππ, is the S- Additional motivation for studying these decays is that
wave transition, in which if a resonance is produced,
PC a very prominent a0(980) → ηπ signal of high purityit has to have J = 1−+. Several candidates with
PC + was observed in χc1 → ηπ+π−, by CLEO-c [10]. TheJ = 1− , decaying into different final states, such as a0(980) was discovered several decades ago, but its na-
ηπ, η′π, f1(1270)π, b1(1235)π and ρπ, have been reported ture was puzzling from the beginning, leading to the
by various experiments, and these have been thoroughly hypothesis that it is a four-quark rather than an ordi-
reviewed in Ref. [1]. The lightest exotic meson candidate nary qq̄ state [11–13]. The first coupled meson-meson
is the π1(1400) [2], reported only in the ηπ final state (ηπ,KK̄, η′π) scattering amplitudes based on lattice
by GAMS [3], KEK [4], Crystal Barrel [5], and E852 [6], QCD calculations [14] indicate that the a0(980) might
but its resonance nature is controversial [7]. The most
PC + be a resonance strongly coupled to ηπ and KK̄ channels,promising J = 1− candidate, the π1(1600) [2], could which does not manifest itself as a symmetric bump in
also couple to the ηπ, since it has been observed in the the spectra. Recent theoretical work based on the chiral
η′π channel by VES [8] and E852 [9]. unitarity approach also points that the a0(980), as well
The CLEO-c Collaboration reported evidence of an as the σ and f0(980) states, could be dynamically gener-
4
ated through meson-meson interactions, for example in events. The background is studied using an inclusive MC
heavy-meson decays: χc1 → ηππ [15] and ηc → ηππ [16]. sample of 106×106 generic ψ(3686) events.
However, there is still no consensus on the exact role that
meson-meson loops play in forming of the a BESIII is a conventional solenoidal magnet detector0(980), which
is now generally accepted as a four-quark object, see [17] that has almost full geometrical acceptance, and four
and reference therein. main components: the main drift chamber (MDC), elec-
tromagnetic calorimeter (EMC), time-of-flight detector
The a0(980) indeed decays dominantly into ηπ and (TOF), all enclosed in 1 T magnetic field, and the muon
KK̄ final states; the latter has a profound influence on chamber. The momentum resolution for majority of
the a0(980) line shape in the ηπ channel, due to the prox- charged particles is better than 0.5%. The energy reso-
imity of the KK̄ threshold to the a0(980) mass. Differ- lution for 1.0 GeV photons in the barrel (end-cap) region
ent experiments, E852 [18], Crystal Barrel [19, 20] and of the EMC is 2.5% (5%). For majority of photons in
CLEO-c [10] analyzed data to determine the couplings of the barrel region, with the energy between 100 and 200
the a0(980) to the ηπ (gηπ) and KK̄ final states (gKK̄), MeV, the energy resolution is better than 4%. Details of
in order to help resolve the true nature of the a0(980). the BESIII detector and its performance can be found in
This is not an exhaustive list of analyses: it points that Ref [25].
the values obtained for the a0(980) parameters vary con-
siderably among various analyses. Good photon candidates are selected from isolated
EMC showers with energy larger than 25 (50) MeV in
Another channel of interest is a0(980) → η′π, with the barrel (end-cap) region, corresponding to the polar
the threshold more than 100 MeV/c2 above the a0(980) angle, θ, satisfying | cos θ| < 0.80 (0.86 < | cos θ| < 0.92).
mass. The first direct observation of the decay a0(980) → The timing of good EMC showers is required to be within
η′π was reported by CLEO-c [10], using a sample of
× 700 ns of the trigger time. Charged tracks must satisfy26 106 ψ(3686) decays. The a0(980) coupling to the | cos θ| < 0.93, and the point of closest approach of a
η′π channel, gη′π, was determined from χ
+
c1 → ηπ π− track from the interaction point along the beam direc-
decays, although the analysis was not very sensitive to tion is required to be within 20 cm and within 2 cm
the a0(980) → η′π component in the a0(980) → ηπ in- perpendicular to the beam direction. All charged tracks
variant mass distribution, and gη′π was found to be con- are assumed to be pions, and the inclusive MC sample is
sistent with zero. In many analyses of a0(980) couplings, used to verify that the kaon contamination in the final
gη′π has not been measured. For example, its value was sample is negligible in each of the η channels. We require
fixed in Ref. [20] based on SU(3) flavor-mixing predic- two charged tracks for the η → γγ and η → 3π0 channels,
tions. Using a clean sample of χc1 produced in the radia- and four tracks for the η → π+π−π0 channel, with zero
tive transition ψ(3686) → γχc1 at BESIII, we investigate
→ net charge. For η → γγ and η → π
+π−π0, at least three
the χ +c1 ηπ π
− decays to test if the a0(980) → ηπ in- photon candidates are required, and for η → 3π0 at least
variant mass distribution is sensitive to η′π production. 7 photon candidates. The invariant mass of two-photon
Dispersion integrals in the description of the a0(980) line combinations is kinematically constrained to the π0 or η
shape are used to determine the a0(980) parameters, its mass.
invariant mass, ma (980), and three coupling constants:0
gηπ, gKK̄ and gη′π. This information might help in de- The sum of momenta of all final-state particles, for
termining the quark structure of the a0(980). a given final state topology, is constrained to the ini-
tial ψ(3686) momentum. If multiple combinations for an
In this χc1 decay mode, it is also possible to study
→ → event are found, the one with the smallest χ
2 is re-
χc1 a2(1700)π; a2(1700) ηπ production. The NCtained. Here NC refers to the number of constraints,
a2(1700) has been reported in this decay mode by Crys- which is four plus the number of two-photon π0 and η
tal Barrel [21] and Belle [22], but still is not accepted
candidates in the final state (see Table I).
as an established resonance by the Particle Data Group
(PDG) [2].
TABLE I. Characteristics of the η decay channels used to re-
construct the ψ(3686) γηπ+π−→ decays: branching fraction
II. EVENT SELECTION B, final state topology, number of constraints (NC) in the
kinematic fit, and reconstruction efficiency, ε, according to
exclusive phase-space MC.
For our studies we use (448.0 ± 3.1) × 106 ψ(3686)
events, collected in 2009 [23] and 2012 [24] with the BE- Decay B [%] [2] Final state NC ε [%]
SIII detector [25]. We select 95% of possible η decays, in η → γγ 39.41±0.20 3γ 1(π+π−) 5 26.58
the η → γγ, η → π+π−π0 and η → π0π0π0 decay modes. η → π+π−π0 22.92 0.28 3γ 2(π+ −± π ) 5 16.46
For each ψ(3686) → γηπ+π− 0 0 0 + −final state topology, exclu- η → π π π 32.68±0.23 7γ 1(π π ) 7 5.64
sive Monte Carlo (MC) samples are generated according Total 95.01±0.71 16.91
to the relative branching fractions given in Table I, equiv-
alent to a total of 2×107 ψ(3686) → γχ + −c1; χc1 → ηπ π
5
A. χc1 → ηπ
+π− event selection case of η three-pion decays, the η signal region, defined
by Eq. (1), is indicated by dash-dotted bars in Fig. 1.
→ Although the mass distribution of three neutral pions,The χ ηπ+π−c1 candidates in η three-pion decays Fig. 1(c), is wider than the corresponding distribution
are selected by requiring that the invariant mass of three from the charged channel, Fig. 1(b), we use the same
pions satisfy: selection criteria for both η decays, which keeps the ma-
0
0.535 < m(3π) < 0.560 GeV/c2. (1) jority of good η → 3π candidates and results in similar
background levels in the two channels. The effects of in-
For the η → γγ candidates, we require that the mass cluding more data from the tails of these distributions
constraint fit for η → γγ satisfies χ2γγ < 15. The are taken into account in the systematic uncertainties.
χ2 obtained from four-momenta kinematic constraint The invariant mass plot representing η → γγ candidates,NC
fits are required to satisfy χ2 25C < 40, χ5C < 40 and Fig. 1(a), is used only to select η sidebands for back-
χ2 < 56 for η → γγ, η → π+π−π07 and η → 3π0, ground subtraction. Table I lists channel efficiencies andC
respectively. These selection criteria effectively remove the effective efficiency for all channels.
kaon and other charged track contamination, justifying The ηπ+π− invariant mass distribution, when events
the assumption that all charged tracks are pions. To
→ from all η channels are combined, is shown in Fig. 2. Inselect the χc1 candidates from the ψ(3686) γχc1 tran- the signal region, indicated by vertical bars, there are
sition, we require the energy of the radiative photon to 33919 events, with the background of 497 events esti-
satisfy 0.155 < Eγ < 0.185 GeV. mated from the η sidebands. The sideband background
does not account for all the background, and after the η-
sideband background is subtracted, the remaining back-
1. Background suppression ground is estimated by fitting the invariant mass distri-
bution. The fit is shown by the solid distribution, Fig. 2.
For the χc1 signal, a double-sided Crystal-Ball distribu-The major background for all final states comes from
→ → tion (dotted) is used, and for the background, a linearψ(3686) ηJ/ψ, while in the η γγ case the back-
→ function along with a Gaussian corresponding to the χground from ψ(3686) γγJ/ψ decays is also significant. c2
→ contribution (dashed) are used. The signal purity esti-The background from ψ(3686) ππJ/ψ is negligible,
mated from the fit is P = (98.5±0.3)%, where the error is
once a good η candidate is found.
obtained from fluctuations in the background when using
To suppress ψ(3686) → ηJ/ψ background for all three different fitting ranges and shapes of the background.
η decays, the system recoiling against the η, with respect
to the ψ(3686), must have its invariant mass separated
at least 20 MeV/c2 from the J/ψ mass. B. Two-body structures in the χ +c1 → ηπ π−
Additional selection criteria are used in the η → γγ decays
channel to suppress π0 contamination and ψ(3686) →
γγJ/ψ production. The former background is suppressed
The Dalitz plot for selected signal events is shown
by rejecting events in which any two-photon combination
2 in Fig. 3(a). Two-body structures reported in previoussatisfies 0.110 < m(γγ) < 0.155 GeV/c . The latter
analyses of the χ → ηπ+c1 π− decays, by BESII [26] andbackground is suppressed by vetoing events for which a
CLEO [10, 27], the a0(980)π, a2(1320)π and f2(1270)η,two-photon combination not forming an η has a total
are indicated by the long-dash-dotted, dashed and dash-
energy between 0.52 GeV < Eγγ < 0.60 GeV. This range dotted arrows pointing into the Dalitz space, respectively.
of energies is associated with the doubly radiative decay
→ → One feature of this distribution is the excess of events inψ(3686) γχcJ ;χcJ γJ/ψ, for which the energy sum the upper left corner of the Dalitz plot (a), pointed to
of two transitional photons is Eγγ ≈ 0.560 GeV. by the dotted arrows, which cannot be associated with
known structures observed in previous analyses of this
χc1 decay. We hypothesize this is due to a2(1700) pro-
2. Background subtraction duction. The expected Dalitz plot of a a2(1700)π signal is
shown in Fig. 3(b), obtained assuming that the a2(1700)
+
The background estimated from the inclusive MC after is the only structure produced. The a2(1700) → ηπ
−
all selection criteria are applied is below 3% in each chan- and a2(1700) → ηπ components cannot be easily iden-
nel. The background from η sidebands is subtracted, and tified along the dotted arrows in the Dalitz plot, Fig 3(a),
Fig. 1 shows the invariant mass distributions of η candi- but their crossing in the plot shown in Fig. 3(b) visually
dates with vertical dotted bars showing the η sideband matches the excess of events in the upper left corner of
regions. The sideband regions for the two-photon and the Dalitz plot of Fig. 3(a).
three-pion modes are defined as 68 < |m(γγ) − mη| < The distributions of the square of the invariant mass
113 MeV/c2 and 37 < |m(3π) − mη| < 62 MeV/c2, re- are shown in Fig. 3(c) for ηπ and (d) for π+π−. Struc-
spectively, where mη is the nominal η mass [2]. In the tures that correspond to a0(980), a2(1320) and f2(1270)
6
2500 800
a) η→ γ γ b)  η→π+π-π0 c)  η→π0π0π0
2000 2000
600
1500 1500
400
1000 1000
500 200500
0 0 0
0.4 0.45 0.5 0.55 0.6 0.65 0.7 0.45 0.5 0.55 0.6 0.65 0.45 0.5 0.55 0.6 0.65
 M(γγ) [GeV/c2] M(π+π-π0) [GeV/c2] M(π0π0π0) [GeV/c2]
FIG. 1. The invariant mass distribution of the η candidates, where dotted (red) lines indicate regions used for background
subtraction, while dash-dotted bars (blue) show η-signal boundaries for the three-pion η decay cases. There are no blue bars
on plot a) since the η → γγ signal is selected using the γγ kinematic constraint (color online).
production are evident, as well as a low-mass ππ peak,
sometimes referred to as the σ state. In each of these 5000
two distributions there is a visible threshold effect. In data
the ππ distribution, there is a structure above the KK̄ fit4000
threshold, which is too broad to result from the f (980) signal0
alone. In the ηπ distribution, the broadening of the background
a (980) peak around 1.2 GeV2/c40 could be associated 3000
with the η′π threshold. By examining various regions in
the Dalitz space, we conclude that the cross-channel con- 2000
tamination, or reflections, are not associated with these
threshold effects in the data. In order to eliminate back-
ground as the source of these peculiar line-shapes, back- 1000
ground studies are performed. Namely, we increased the
background level by relaxing the kinematic constraint to 0
χ2 3.45 3.5 3.55NC/NC < 10 and also suppressed more background by
requiring χ2NC/NC < 5. In addition we varied the limits M(ηπ+π-) [GeV/c2]
on tagging η and χc1 candidates, as explained in Sec. V.
It is possible that the ππ line shape results from a FIG. 2. Invariant mass of the χc1 candidates, after the η
destructive interference between the f0(980) and other sideband background is subtracted. Vertical bars indicate the
components of the ππ S-wave. It has been known for region used to select the χc1 candidates. See text for the fit
some time that the a0(980) → ηπ line shape is affected discussion (color online).
by the proximity of the KK̄ threshold to the a0(980)
mass [28]. If the a (980) → η′0 π coupling appears to be
important for describing the a0(980) → ηπ distribution, orbital angular momentum L with respect to the bache-
this would be an example when a virtual channel is influ- lor meson, hb. For resonances with J > 0, there are two
encing the distribution of another decay channel, despite possible values of L that satisfy the quantum number
its threshold being far away from the resonance peak. We conservation for the 1++ → (JPC)0− transition.
use an Amplitude Analysis (AA), described in the next L
section, to help in answering the above questions, and We use the extended maximum likelihood technique to
to determine the nature and significance of the “crossing find a set of amplitudes and their production coefficients
structure” discussed. that best describe the data. The method and complete
description of amplitudes constructed using the helicity
formalism are given in Ref. [10], with two exceptions.
III. AMPLITUDE ANALYSIS
The first difference is that the events from the
η−sidebands are subtracted in the likelihood function L,
To study the substructures observed in the χc1 → with equal weight given to the left-hand and right-hand
ηπ+π− decays, we use the isobar model, in which it is sides, using a weighting factor ω = −0.5. The second
assumed that the decay proceeds through a sequence of difference with respect to Ref. [10] is that we deviate
two-body decays, χc1 → Rhb; R → h1h2, where either from the strict isobar model by allowing production am-
an isospin-zero (R → ππ) or isospin-one (R → ηπ) res- plitudes to be complex. Isospin symmetry for ηπ± reso-
onance is produced, with the total spin J , and relative nances is imposed.
eevveennttss  //  22  MMeeVV//cc22
eevveennttss  //  22  MMeeVV//cc22
events / 2 MeV/c2
eevveennttss  //  22  MMeeVV//cc22
7
10 20 10 1.2
a) b)
18
1
8 a (980) 16 80
a2(1320)
14 a2(1700) 0.8
6 f2(1270) 12 6
a2(1700)
10 0.6
4 8 4
0.4
6
2 4 2 0.2
2
0 0 0 0
0 2 4 6 8 10 0 2 4 6 8 10
 M2(ηπ+) [GeV/c2]2  M2(ηπ+) [GeV/c2]2
1400
a0(980) c)
700 σππ d)
1200 600
1000 M2(η’π) 500 M2(K K) f2(1270)
800 400
600 300
400 200
a2(1320)
200 100
0 0
0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 2.5 3
2 + - 2 2
 M2(ηπ) [GeV/c2]2  M (π π ) [GeV/c ]
FIG. 3. Dalitz plots obtained from selected χc1 candidates from (a) data and (b) exclusive MC, assuming the a2(1700) is the
only structure produced. The (c) ηπ and (d) π+π− projections show various structures, which can also be identified by arrows
in the Dalitz plot (a). Vertical dotted lines in plots (c) and (d) indicate the thresholds for producing the η′π or KK̄ in the ηπ
or ππ space, respectively (color online).
In the minimization process of the expression −2 lnL, corrected number of χc1 events, adjusted by subtracted
the total amplitude intensity, I(x), constructed from the background contributions. In this case, ξ(x) ≡ 1 for all
coherent sum of relevant amplitudes, is bound to the MC events. Fractional contributions, Fα, from specific
number of observed χc1 candidates by using the integral amplitudes, Aα, are obtained by restricting the coherent
∫ sum in I(x) to Iα(x), so t∫hatNχ = ξ(x)I(x)dx, (2)c1 F Iα(x)dxα = ∫ . (3)
I(x)dx
where x represents the kinematic phase space, while ξ(x)
is the acceptance function, with the value of one (zero) The numerator represents acceptance-corrected yield of a
for accepted (rejected) exclusive MC events. The proper given substructure, used to calculate relevant branching
normalization of different η channels is ensured by using fractions, Bα. Errors are obtained from the covariance
exclusive MC samples, generated with sample sizes pro- matrix using proper error propagation, so for a given
portional to the η branching fractions, listed in Table I. substructure, the errors on Bα and Fα are not necessarily
If the complete generated exclusive MC set is used in the the same.
MC integration, then Eq. (2) provides the acceptance The decay chain ψ(3686) → γχ + −c1; χc1 → ηπ π is de-
2 entries / 50 (MeV/c2)2  M2(π+π-) [GeV/c2]2
2 + - 2 2
events / 50 (MeV/c2)2  M (π π ) [GeV/c ]
8
scribed by amplitudes constructed to take into account In the above expressions ρch(s) is the available phase
the spin alignment of the initial state and the helicity of space for a given channel, obtained from the cor√respond-
the radiated photon. Linear combinations of helicity am- ing decay momentum qch(s): ρch(s) = 2qch(s)/ s. The
plitudes can be used to construct amplitudes in the mul- integral in Eq. (6) is divergent when s → ∞, so the phase
tipole basis, matching the electric dipole (E1) and mag- 2space is modified by a form factor F −βq (s)ch(s) = e ch ,
netic quadrupole (M2) transitions. The ψ(3686) → γχc1 where the parameter β is related to the root-mean-square
decay is dominated by the E1 transition (CLEO) [29], (RMS) size of an emitting source [20]. We use β =
and a small M2 contribution (≈ 3%) can be treated as a 2.0 [GeV/c2]−2 corresponding to RMS = 0.68 fm, and
systematic uncertainity. we verify that our results are not sensitive to the value of
β. The integration in Eq. (6) starts from the threshold for
a particular channel, sch, which conveniently solves the
A. Mass dependent terms, T (s) problem of the analytical continuation in special cases ofα
final state configurations like the a ′0(980) → η π, when
the decay momentum below the threshold (s < mη′+mπ)
The dependence of amplitude Aα on the energy can becomes real again for s < mη′ −mπ. Figure 4 shows the
be separated from its angular dependence, employing a shapes of (a) ImΠch(s) and (b) ReΠch(s) (b), for the
general form pLqJTα(s), if the width of the χc1 is ne- KK̄ and η′π channels, for arbitrary values of the cou-
glected. Here, p and q are decay momenta for decays pling constants. In the final form, the real parts in the
χc1 → RJhb and RJ → h1h2 in the rest frame of the χc1 denominator of Eq. (4) are adjusted by ReΠch(m0) terms:
and a resonance RJ , respectively, while s = m
2
12 is the ReΠch(s) → ReΠch(s)− ReΠch(m0).
squared invariant mass of the corresponding isobar prod-
ucts (ππ or ηπ). For most resonances, we use relativistic
Breit-Wigner (BW) distributions, with spin-dependent
Blatt-Weisskopf factors [30]. For the a (980) and ππ S- 2. ππ S-wave model0
wave line shapes, we use different prescriptions explained
below. The ππ S-wave parametrization follows the prescrip-
To account for the non-resonant process χ → ηπ+π−c1 , tion given in Ref. [10], in which two independent pro-
we use an amplitude constructed as the sum of all pos- cesses for producing a ππ pair are considered: direct
sible final state combinations of helicity amplitudes con- (ππ)S → (ππ)S , and production through kaon loops,
strained to have the same production strength, with no (KK̄)S → (ππ)S . Amplitudes corresponding to these
dependence on the invariant mass of the respective two- scattering processes, labeled Sππ(s) and SKK̄(s), are
body combinations. based on di-pion phases and intensities obtained from
scattering data [32], which cover the ππ invariant mass
region up to 2 GeV/c2. The Sππ(s) component is adapted
in Ref. [10] to account for differences in the ππ production
1. Parametrization of a0(980)
through scattering and decay processes, using the denom-
inator, D(s), extracted from scattering experiments. The
Instead of using the usual Flatté formula [28] to de- Sππ(s) amplitude in this analysis takes the form:
scribe the a0(980) line-shape, we use dispersion integrals, ∑ ∑
following the prescription given in Ref. [20]. We consider Sππ(s) = c S
0
0 (s)+ c z
i
i s (s)S
0(s)+ c′iz
i
s′(s)S
0(s).
KK̄
three a0(980) decay channels, the ηπ, KK̄, and η
′π, with i=1 i=1
corresponding coupling constants, gch, and use an appro- (7)
priate dispersion relation to avoid the problem of a false The common term in the above expression, S
0(s) =
singularity [31] present in the η′π mode (see discussion 1/D(s), is expanded using conformal transformations of
at the end of this section). The a0(980) amplitude is the type
constructed using the following denominator: √ √
∑ s+ s0 − sth − sz (s) = √ √ , (8)
2 − − sthDα(s) = m s Πch(s), (4) s+ s0 + sth − s0
ch
which is a complex function for s > sth. Equation (7) fea-
where m0 is the a0(980) mass and Πch(s) in the sum over tures two threshold-functions, zs (s), one corresponds toth
channels is a complex function, with imaginary part KK̄ production with sKK̄ = 4m
2
K , while another with
sth = s
′ could be used to examine other possible thresh-
ImΠ 2ch(s) = gchρch(s)Fch(s), (5) old effects in di-pion production. The ci, i = 1, 2 are
production coefficients to be determined.
while real parts are given by principal value integrals
∫ Figure 5 shows the (a) phase and (b) intensity of vari-
1 ∞ ImΠ (s′ch )ds
′
ous components used in constructing the ππ S-wave am-
ReΠch(s) = P − . (6)π (s′s s) plitude based on two functions given by Eq. (8), withch
9
0.4 a) 
b) 
ImΠ (s) 0.5KK ReΠKK(s)
ImΠη 0.4’π(s) ReΠη’π(s)
0.3 0.3
0.2
0.2
0.1
0.1 0
-0.1
0
0.5 1 1.5 2 2.5 3 3.5 4 0.5 1 1.5 2 2.5 3 3.5 4
s [GeV2/c4] s [GeV2/c4]
FIG. 4. Line shapes of (a) ImΠ(s) and (b) ReΠ(s) for the KK̄ and η′π production with arbitrary normalization.
different thresholds: zKK̄(s) and zs′(s). The follow- As indicated earlier, the threshold used to construct the
ing convention is used: Siππ(s) = z
i S0(s), S′i (s) = S1(s) term is s = 4m2 . The threshold for the S′i(s)
KK̄ ππ KK̄ K
i 0 components (i = 1, 2) is s′ 2 2√zs′S (s). Components are arbitrarily scaled, and we set = 2.23 [GeV/c ] , which is
s′ ∼ 1500 MeV/c2, similarly to the value used later in close to the mass of the f0(1500), and it is responsible
analysis. The parameter s = 1.5 (GeV/c2)2 can be used for the peaking of the Sππη amplitude in this region,0
′i
to adjust the left-hand cut in the complex plane, and the Fig. 6(b). In fact, the S (s) components are used instead
same value is used in all components. of the f0(1500)η amplitude, which would be needed in the
optimal solution if only threshold functions zi (s) were
KK̄
used in the expansion of the Sππ(s)η amplitude. With
these additional terms, the contribution and significance
IV. RESULTS
of ππ scalars, the f0(1370), f0(1500) and f0(1710), is
negligible, for each. Although this particular set of am-
We present results from the amplitude analysis of the plitudes respects the unitarity of the ππ S-wave, we use
full decay ψ(3686) → γχc1; χ + −c1 → ηπ π , reconstructed the sum of BW to model other spins and final states,
in three major η decay modes. The optimal solution namely the f2(1270), f4(2050), a2(1320) and a2(1700).
to describe the data is found by using amplitudes with Our approach provides reasonable modeling of the ππ
fractional contributions larger than 0.5% and significance line shape, and the sum of all ππ S-wave components,
larger than 5σ. The significance for each amplitude α is SKK̄ and Sππ, is reported in Table II.
determined from the change in likelihood with respect
− L L Besides the f0(1370), f0(1500), and f0(1710), otherto the null-hypothesis, ∆Λ = 2 ln 0/ α. The null- conventional resonances are probed, including the
hypothesis for a given amplitude is found by excluding it f0(1950), f2(1525), f2(2010), and a0(1450), with param-
from the base-line fit, and the corresponding amplitude eters fixed to PDG values [2]. They do not pass the
significance is calculated taking into account the change tests for significance and fractional contribution. The
in the number of degrees of freedom, which is two (four) non-resonant χ → ηπ+π−c1 production is found to be
for J = 0 (J > 0) amplitudes. negligible. The search for possible 1−+ resonances in the
The most dominant amplitude in this reaction is ηπ final state will be presented below.
a0(980)π, as evident from the ηπ projection of the Dalitz
plot, Fig. 3(c). Other amplitudes used in our base-
line fit include the SKK̄η, Sππη, f2(1270)η, f4(2050)η, A. The a2(1700) signature
a2(1320)π and a2(1700)π, where masses and widths of
resonances described by BW functions are taken from
the PDG [2], while the a2(1700) and a0(980) parameters All structures listed in Table II have been already re-
are free parameters to be determined by the fit in this ported in the decay χ → ηπ+ −c1 π , except the a2(1700)π.
work. The mass projections are shown in Fig. 6, and the Its fractional contribution is around 1%, and the signifi-
corresponding fractional contributions and significances cance of each orbital momentum component is more than
are listed in Table II. For amplitudes with spin J > 0 10σ. Detailed background studies are performed to en-
both orbital momentum components are included. sure that the background, remaining after η-sideband
The following components form the S (s) amplitude: subtraction, is not affecting the significance and frac-ππ
tional contribution of the a2(1700). Results of fitting
Sππ(s) = c S
0(s)+ c S1 ′ ′10 1 ππ(s)+ c1Sππ(s)+ c
′ ′2
2Sππ(s). (9) the mass and width of the a2(1700), shown in Table III,
10
3
400 a) b) S0ππ
300 2.5 S1ππ
200 2 S
2
ππ
1
100 1.5
S’ππ
S’2ππ
0 1 SKK
-100 0.5
-200 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5
s  [GeV2/c4] s 2 4ππ ππ [GeV /c ]
FIG. 5. The (a) phase and (b) intensity of the ππ S-wave components. Red (dash) histograms represent the SKK̄ amplitude,
blue histograms (dot and dash-dot) are obtained using Si = zi S0ππ KK̄ ππ terms, while purple (long-dash-dot and dash-three-dot)
represent S′i = zi S0ππ s′ ππ terms (color online).
TABLE II. Fractional intensities F , and significances of amplitudes in the base-line fit, with the first and second errors being
statistical and systematic, respectively. The third error for the branching fractions for the χ + −c1 → ηπ π decay and decays into
significant conventional isobars is external (see text). For exotic mesons only statistical errors on their fractional contributions
are provided. The upper limits for exotic meson candidates, which include both statistical and systematic uncertainties, are
at the 90% confidence-level. The coherent sum of all ππ S-wave components, (π+π−)Sη, is included in this report. Note, the
branching fractions for amplitudes of the type Aαη, involving isobars decaying into π
+π−, are the products of χc1 → Aαη and
A π+π−α → rates. Branching fractions for isobars decaying into ηπ include charge conjugates.
Decay F [%] Significance [σ] B(χ + −c1 → ηπ π ) [10
−3]
ηπ+π− - - 4.67 ± 0.03 ± 0.23 ± 0.16
a0(980)
+π− 72.8 ± 0.6 ± 2.3 > 100 3.40 ± 0.03 ± 0.19 ± 0.11
a (1320)+π−2 3.8 ± 0.2 ± 0.3 32 0.18 ± 0.01 ± 0.02 ± 0.01
a (1700)+π−2 1.0 ± 0.1 ± 0.1 20 0.047 ± 0.004 ± 0.006 ± 0.002
SKK̄η 2.5 ± 0.2 ± 0.3 22 0.119 ± 0.007 ± 0.015 ± 0.004
Sππη 16.4 ± 0.5 ± 0.7 > 100 0.76 ± 0.02 ± 0.05 ± 0.03
(π+π−)Sη 17.8 ± 0.5 ± 0.6 - 0.83 ± 0.02 ± 0.05 ± 0.03
f2(1270)η 7.8 ± 0.3 ± 1.1 > 100 0.36 ± 0.01 ± 0.06 ± 0.01
f4(2050)η 0.6 ± 0.1 ± 0.2 9.8 0.026 ± 0.004 ± 0.008 ± 0.001
Exotic candidates U.L. [90% C.L.]
π1(1400)
+π− 0.58±0.20 3.5 < 0.046
π1(1600)
+π− 0.11±0.10 1.3 < 0.015
π1(2015)
+π− 0.06±0.03 2.6 < 0.008
are consistent with the values listed by the PDG [2]. To The a0(980) errors are shown in Table IV. Variations
check how the a2(1700) parameters and fractional con- in the shape of the ππ-S wave amplitude are taken into
tributions are affected by the f2(1270) and a2(1320), we account by changing terms in the expansion, Eq. (9).
also fitted their masses and widths, which are provided We also test the significance of the a2(1700) includ-
in Table III with statistical uncertainties only. The mass ing alternative states with the same mass and width, but
(width) of the f2(1270) is lower (higher) than its nominal different spins: J = 0, 1, 4. In all cases, the significance
value [2], maybe because of interference with underlying of the a2(1700) in the presence of an alternative state
ππ S-wave components or threshold effects, other than exceeds 17σ. The statistical significance of the a2(1700)
those for the KK̄ or f0(1500) production. signal alone is 20σ. This result confirms our hypothesis
The systematic uncertainties for the a2(1700) mass and based on a visual inspection of the Dalitz plot, Fig. 3(a),
width are obtained by varying parameters of other am- that the excess of events in the upper left corner of the
plitudes within respective uncertainties listed in Ref. [2], Dalitz space results from the a2(1700) production, and
and taking into account variations listed in Table III. it is associated with the crossing of the a2(1700)
+π−
 δππ [deg]
 intensty [arb. unit]
11
600 a0(980)π a (980)πa) a (1320)π 600 b) 02 a2(1320)π
500 a2(1700)π a2(1700)πSKK→ππη 500 SKK→ππη
Sππ→ππη Sππ→ππη
400 f2(1270)η f2(1270)η
f4(2050)η 400 f4(2050)η
300 300
200 200
100 100
0 0
0.5 1 1.5 2 2.5 3 3.5 0.5 1 1.5 2 2.5 3
 M(ηπ) [GeV/c2]  M(π+π-) [GeV/c2]
FIG. 6. Projections in the (a) ηπ and (b) π+π− invariant mass from data, compared with our base-line fit (solid curve) and
corresponding amplitudes (various dashed and dotted lines). All features of the data, including structures discussed in Sec. II B
are reproduced rather well.
and a2(1700)
−π+ components. Further, Fig. 7 shows the we repeat the analysis with gη′π = 0, and let the values
ηπ mass distribution in the region around the expected of the other parameters free. The results of this fit are
a2(1700) peak, where data points are compared with a also given in Table IV. The likelihood change when the
fit when the a2(1700)π amplitude is excluded. η
′π channel is ignored shows that the significance of a
non-zero gη′π measurement is 8.9σ. The same result is
obtained when the analysis is performed in the presence
of the a0(1450). The values of the two ratios based on
B. a0(980) parameters the SU(3) expectation are
g2 /g2 = 1/(2 cos2 φ) = 0.886± 0.034, (10)
When determining the a0(980) parameters we use the
KK̄ ηπ
ratios R 221 = g /g
2 2 2
KK̄ ηπ
, and R31 = gη′π/gηπ. The result-
ing values are listed in Table IV, where systematic uncer- g2 2 2η′π/gηπ = tan φ = 0.772± 0.068, (11)
tainties are obtained by fitting the a0(980) parameters
under different conditions. The level of background is which depend on the choice of the η−η′ mixing angle; φ =
varied by changing selection criteria described in Sec. II, (41.3±1.2)◦ in this case [20]. Our result is consistent with
and by changing the amount of background subtracted Eq. (11) within 1.5σ, based on the quoted uncertainties.
from the η sidebands. Effects of the line shapes of the
a2(1320), a2(1700), f2(1270) and f4(2050) resonances are
taken into account by varying their masses and widths C. Search for ηπ P -wave states
within the respective uncertainties [2], and using values
from Table III. The effect of the ππ S-wave shape is
examined in similar way as for the a2(1700). The pres- We examine possible exotic meson production in the
ence of alternative conventional and exotic resonances is ηπ invariant mass region from 1.4 to 2.0 GeV/c2. Ta-
also taken into account. Our result is not sensitive to the ble II lists fractional contributions and significances of
value of the parameter β in Eqs. (5) and (6), within the three JPC = 1−+ candidates, added one at the time to
range of values: β = (2.0± 1.0) [GeV/c2]2. our nominal fit. Two possible orbital-momentum config-
urations for an exotic amplitude are the S-wave and D-
For comparison we list two previous results, one from
wave, and the significance of each is tested individually.
a similar experiment, CLEO-c, and the other obtained
We find that the significance of the S-wave is marginal,
using Crystal Barrel data. There is a general agreement
less than 2σ for every π1, and the reported significancesbetween different analyses for the a0(980) mass and R21. in Table II result from using the S and D waves together
The ratio R31 was fixed in Ref. [20] to the theoretical in the fit. The most significant of the three possible ex-
value provided by Eq.(11), while it was consistent with
otic states is the π (1400), with a significance of 3.5σ and
zero in the CLEO-c analysis, possibly because of smaller 1
fractional contribution less than 0.6%. This represents a
statistics. It is not easy to comment on the difference in
weak evidence for the existence of the π1(1400) becausevalues for the ηπ coupling, which could be affected by
in alternative amplitude configurations, when parame-
different normalizations used by different analyses.
ters of other amplitudes are varied, the significance of
This analysis provides the first non-zero measurement this state becomes < 3σ. In the nominal amplitude con-
of the coupling constant gη′π. To test the sensitivity of figuration, the significance of each π1(1400) component is
the a0(980) → ηπ line shape to the decay a0(980) → η′π, less then 3σ, and when taken together, the contribution
 events / 10 MeV/c2
 events / 20 MeV/c2
12
350 a0(980)π
a2(1320)π
300 SKK→ππη
Sππ→ππη
f
250 2
(1270)η
f4(2050)η
200
150
100
50
0
1.4 1.6 1.8 2 2.2 2.4
 M(ηπ) [GeV/c2]
FIG. 7. The ηπ invariant mass projection from data in the region (1.3, 2.4) GeV/c2, compared with the fit without the
a2(1700)η amplitude (solid curve). Other amplitudes are plotted (various dashed and dotted lines) for comparison, while the
peak that is associated with the a2(1700) is evident.
TABLE III. The mass and width of the a2(1700), with statistical and systematic uncertainties. Only statistical uncertainties
from the f2(1270) and a2(1320) fits are listed. Comparison with the PDG [2] values is provided, with all units in GeV/c
2.
BESIII PDG [2]
Resonance M Γ M Γ
a2(1700) 1.726±0.012±0.025 0.190±0.018±0.030 1.732±0.016 0.194±0.040
f2(1270) 1.258±0.003 0.206±0.008 1.275±0.001 0.185±0.003
a2(1320) 1.317±0.002 0.090±0.005 1.318±0.001 0.107±0.005
of the S-wave is much smaller than the D-wave contri- Nχ = 192658 ± 1075, where the error is from the co-c1
bution, pointing that the evidence for the π1(1400) is variance matrix. The efficiency in Eq. (12) is ǫ ≡ 1, by
circumstantial. construction.
Masses and widths of the three exotic candidates are Table II lists the branching fraction for the χc1 →
not very well constrained by previous analyses, and we ηπ+π−, and branching fractions for subsequent reso-
±
vary the respective parameters within listed limits [2]. nance production in respective isospin states, ηπ or
Our conclusion is that there is no significant evidence for π+π−, where the first and second errors are statistical
an exotic ηπ structure in the χ + −c1 → ηπ π decays, and and systematic, respectively. The branching fraction for
we determine upper limits at the 90% confidence level for a given substructure is effectively a product:
the production of each π1 candidate. B + −α = Fα × B(χc1 → ηπ π ), (13)
obtained using generated exclusive MC in accordance
with Eq. (3). The third error is external, associated with
D. Branching fractions
uncertainties in the branching fractions for the radiative
transition ψ(3686) → γχc1 and η decays. We also show
+ −
The branching fraction for the χ → ηπ+π− decay is the total π π S-wave contribution, obtained from thec1
given by coherent sum of the SKK̄ and Sππ components. Statis-
tical errors, as well as systematic ones, for a given frac-
P ∗ N + tional contribution and branching fraction differ, becauseB → χ →ηπ π−(χc1 ηπ+π−) = c1 , (12)
N B B ǫ common systematic uncertainties for all amplitudes can-ψ(3686) ψ(3686)→γχc1 η cel when fractions are calculated, which will be discussed
where the branching fractions Bψ(3686)→γχ and B are
below.
c1 η
from Ref. [2]; the latter is listed in Table I. The num- The upper limits for the production of the π1(1400)π,
ber of ψ(3686), Nψ(3686), [23, 24] is provided in Sec II. π1(1600)π, and π1(2015)π are shown in Table II. The
The signal purity, P , given in Sec. II A 1, takes into ac- limits are determined by including the corresponding am-
count that the number of χc1 obtained from the ampli- plitude in the nominal fit, one at a time. The analysis is
tude analysis includes the background not accounted for repeated by changing other amplitude line shapes, and
by the sideband subtraction. Using Eq. (2) we obtain the background level, in a similar fashion used for deter-
 events / 20 MeV/c2
13
TABLE IV. Parameters of the a0(980) determined from the fit using the dispersion relation of Eqs. (4-6), compared to results
from previous analyses. Bold values indicate quantities that are fixed in the fit.
Data m [GeV/c2] g2 [GeV/c2 20 ηπ ] g
2 2 2 2
KK̄/gηπ gη′π/gηπ
CLEO-c [10] 0.998 ± 0.016 0.36 ± 0.04 0.872 ± 0.148 0.00± 0.17
C.Barrel [20] 0.987 ± 0.004 0.164 ± 0.011 1.05 ± 0.09 0.772
BESIII 0.996±0.002±0.007 0.368±0.003±0.013 0.931±0.028±0.090 0.489±0.046±0.103
BESIII (R231 ≡ 0) 0.990±0.001 0.341±0.004 0.892±0.022 0.0
mining systematic uncertainties of nominal amplitudes χc1 → ηπ+π− branching fraction stem from uncertain-
(see Sec. V). Masses and widths of exotic candidates are ties in charged track and shower reconstruction efficien-
also varied within limits provided by the PDG [2]. The cies, the contribution of the M2 multipole transition,
largest positive deviation of the exotic candidate yield amplitude modeling, the background contribution, and
with respect to the corresponding yield from the mod- the uncertainty in the number of ψ(3686) produced at
ified nominal fit is effectively treated as the systematic BESIII [23, 24]. External sources of uncertainty include
error, summed in quadrature with the statistical error the branching fraction B(ψ(3686) → γχc1) and the frac-
on a given exotic state yield. The resulting uncertainty tion of η decays, B(η) in Eq. (12). The external error
is used to determine the 90% confidence level deviation, affects only branching fractions, not fractional contribu-
and added to the ’nominal’ yield of an exotic candidate tions, and it is reported as a separate uncertainty.
to obtain the corresponding upper limit for the branching
fraction B(χ → π+π−). Systematic uncertainties associated with the trackingc1 1 efficiency and shower reconstruction are 1% per track and
The branching fractions for the substructures in χc1 → 1% per photon. Because of different final states used
ηπ+π− decays reported by the PDG [2] are compared in in this analysis, tracking and photon uncertainties are
Table V with the values measured in this work, and with weighted according to the product of branching fractions
the previous most precise measurement (CLEO-c) [10]. and efficiencies of the different η channels, as listed in Ta-
The measurement for the f2(1270) production is adjusted ble I. The resulting systematic uncertainties for charged
to account for the measured relative f2(1270) → π+π− tracks and photons are 2.47% and 3.92%, respectively.
width. There is a rather large discrepancy between the
values for the two most dominant substructures listed The electromagnetic transition ψ(3686) → γχc1 is
by the PDG and the two most recent measurements. dominated by the E1 multipole amplitude with a small
There is very good agreement between the last two mea- fraction of the M2 transition [29]. The nominal fit takes
surements, suggesting that the PDG values on two-body only the E1 multipole amplitude. Adding a small con-
structures observed in χ → ηπ+ −c1 π need to be updated. tribution of the M2 helicity amplitude, of 2.9%, we find
a difference in the branching fraction of 0.62%. This is
taken as a systematic uncertainty.
TABLE V. Comparison between recent measurements of the When considering the effects of modeling line-shapes
branching fractions B(χ ηπ+π−c1 → ), and with the PDG of different amplitudes, we repeat the analysis changing
values.
the mass and width of resonances, a2(1320), f2(1270),
(χ ηπ+π−) [10−3] and f4(2050), within respective uncertainties, and changeB c1 → ×
Decay BESIII CLEO-c [10] PDG [2] the a0(980) and a2(1700) parameters within the lim-
ηπ+π− 4.67 ± 0.28 4.97 ± 0.31 4.9 ± 0.5 its of their statistical uncertainties, given in Tables IV
a (980)+π− 3.40 ± 0.23 3.29 ± 0.22 1.8 ± 0.6 and III. We also change BW line shapes by replacing0
f2(1270)η 0.64 ± 0.11 0.66 ± 0.11 2.7 ± 0.8 spin-dependent widths with fixed widths, and take into
account the χc1 width and centrifugal barrier as another
systematic error. The largest effect from all these sources
is taken as a systematic uncertainty for the branching
fractions and fractional contributions.
V. SYSTEMATIC UNCERTAINTIES The effect of background is estimated by varying
the kinematic-constraint requirement, changing limits on
tagging η and χc1 candidates, changing the level of sup-
Tables VI summarizes various contributions to the sys- pression of the J/ψ and π0 productions, and the level
tematic uncertainties in determining the χc1 → ηπ+π− of background subtraction. As a general rule, selection
branching fraction, and Table VII shows the systemat- criteria were changed to allow for ≈ 1σ additional back-
ics on the fractional contributions of amplitudes in the ground events, based on the numbers from the inclusive
nominal fit. Systematic uncertainties in determining the MC. We use χ2NC/NC < 9 in all three modes when
14
varying the kinematic constraint. Based on these vari- An amplitude analysis of the ψ(3686) → γχc1; χc1 →
ations, we conclude that the systematic uncertainty as- ηπ+π− decay is performed, and the parameters of the
sociated with the assumption that all charged tracks are a0(980) are determined using a dispersion relation. The
pions is negligible. To select χc1 candidates, we use pho- a0(980) line shape in its ηπ final state appears to be sen-
ton energy ranges of (0.152-0.187) GeV, in the η → γγ sitive to the details of the a0(980) → η′π production, and
channel, and (0.150-0.190) GeV, in two η → 3π chan- for the first time, a significant non-zero coupling of the
nels. The mass window for the η selection is changed to a (980) to the η′0 π mode is measured with a statistical
(0.530-0.565) GeV/c2. The π0 suppression window is re- significance of 8.9σ.
duced to (0.120-0.150) GeV/c2 and the J/ψ suppression We also report a2(1700)π production in the χc1 →
is reduced by vetoing two-photon energy within (0.525- ηπ+π− decays for the first time, with the mass and width
0.595) GeV. We also determine the branching fractions in agreement with world average values, and this analy-
without background subtraction from η-sidebands, and sis provides both qualitative and quantitative evidence
the largest effect is listed in Tables VI and VII. for the existence of the a2(1700). First, the signature of
Some uncertainties that are common for all ampli- the a2(1700) in the Dalitz space is consistent with the ob-
tudes, like tracking, shower reconstruction, and Nψ(3686) served Dalitz plot distribution. Second, the a2(1700) sig-
errors, cancel out in the fractional contributions. How- nificance from the amplitude analysis is larger than 17σ,
ever, they are taken into account when branching frac- compared to alternative spin assignments, even though
tions are determined. the fractional yield of the a2(1700)π is only 1%. This
may help in listing the a2(1700) as an established reso-
nance by the the PDG [2].
TABLE VI. Systematic uncertainties in determining the
branching fraction B(χc1 → ηπ
+π−). The systematic un- We examine the production of exotic mesons that
certainty per track is 1.0%, and for photons it is 1.0% per might be expected in the χc1 → ηππ decays: the
shower. π1(1400), π1(1600) and π1(2015). There is only weak
evidence for the π1(1400) while other exotic candidates
Contribution Relative uncertainty (%) are not significant, and we determine the upper limits on
MDC tracking 2.5 the respective branching fractions.
photon detection 3.9
M2/E1 0.6
Background 1.6
Amplitude modeling 0.1 ACKNOWLEDGMENTS
Nψ(3686) 0.7
Total 5.0
External 3.4 The BESIII collaboration thanks the staff of BEPCII
and the IHEP computing center for their strong sup-
port. This work is supported in part by National
Key Basic Research Program of China under Contract
No. 2015CB856700; National Natural Science Founda-
TABLE VII. Systematic uncertainties in fractional contribu- tion of China (NSFC) under Contracts Nos. 11235011,
tions, in percent, for the base-line amplitudes used to model 11322544, 11335008, 11425524; the Chinese Academy
the χ ηπ+c1 → π
− decays. of Sciences (CAS) Large-Scale Scientific Facility Pro-
gram; the CAS Center for Excellence in Particle Physics
Source M2/E1 Bckg. Tα(s) Total (CCEPP); the Collaborative Innovation Center for Par-
a0(980)π 0.2 0.5 3.1 3.2 ticles and Interactions (CICPI); Joint Large-Scale Scien-
a2(1320)π 0.5 5.6 5.6 7.9
a (1700)π 1.4 3.8 12 13 tific Facility Funds of the NSFC and CAS under Con-2
Skkη 3.7 2.2 11 11.5 tracts Nos. U1232201, U1332201; CAS under Con-
S η 1.1 1.1 4.3 4.6 tracts Nos. KJCX2-YW-N29, KJCX2-YW-N45; 100pp
ππSη 1.5 1.1 3.0 3.6 Talents Program of CAS; National 1000 Talents Pro-
f2(1270)η 0.5 2.3 14 15 gram of China; INPAC and Shanghai Key Laboratory
f4(2050)η 5.6 25 18 32 for Particle Physics and Cosmology; Istituto Nazionale
di Sic Nucleare, Italy; Joint Large-Scale Scientific Fa-
cility Funds of the NSFC and CAS under Contract No.
U1532257; Joint Large-Scale Scientific Facility Funds of
the NSFC and CAS under Contract No. U1532258;
VI. SUMMARY Koninklijke Nederlandse Akademie van Wetenschap-
pen (KNAW) under Contract No. 530-4CDP03; Min-
istry of Development of Turkey under Contract No.
We analyze the world’s largest χ → ηπ+ −c1 π sample, DPT2006K-120470; The Swedish Resarch Council; U.
selected with very high purity, and find a very promi- S. Department of Energy under Contracts Nos. DE-
nent a0(980) peak in the ηπ
± invariant mass distribution. FG02-05ER41374, DE-SC-0010504, DE-SC0012069; U.S.
15
National Science Foundation; University of Groningen forschung GmbH (GSI), Darmstadt; WCU Program of
(RuG) and the Helmholtzzentrum fuer Schwerionen- National Research Foundation of Korea under Contract
No. R32-2008-000-10155-0
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