Physics Letters B 642 (2006) 441–448

www.elsevier.com/locate/physletb

Partial wave analyses of J/ψ → γπ+π− and γπ0π0

BES Collaboration

M. Ablikim a, J.Z. Bai a, Y. Ban l, J.G. Bian a, X. Cai a, H.F. Chen q, H.S. Chen a, H.X. Chen a,J.C. Chen a, Jin Chen a, Y.B. Chen a, S.P. Chi b, Y.P. Chu a, X.Z. Cui a, Y.S. Dai s, L.Y. Diao i,Z.Y. Deng a, Q.F. Dong o, S.X. Du a, J. Fang a, S.S. Fang b, C.D. Fu a, C.S. Gao a, Y.N. Gao o,S.D. Gu a, Y.T. Gu d, Y.N. Guo a, Y.Q. Guo a, Z.J. Guo p, F.A. Harris p, K.L. He a, M. He m,

Y.K. Heng a, H.M. Hu a, T. Hu a, G.S. Huang a,1, X.T. Huang m, X.B. Ji a, X.S. Jiang a, X.Y. Jiang e,J.B. Jiao m, D.P. Jin a, S. Jin a, Yi Jin h, Y.F. Lai a, G. Li b, H.B. Li a, H.H. Li a, J. Li a, R.Y. Li a,

S.M. Li a, W.D. Li a, W.G. Li a, X.L. Li a, X.N. Li a, X.Q. Li k, Y.L. Li d, Y.F. Liang n, H.B. Liao a,B.J. Liu a, C.X. Liu a, F. Liu f, Fang Liu a, H.H. Liu a, H.M. Liu a, J. Liu l, J.B. Liu a, J.P. Liu r,Q. Liu a, R.G. Liu a, Z.A. Liu a, Y.C. Lou e, F. Lu a, G.R. Lu e, J.G. Lu a, C.L. Luo j, F.C. Ma i,

H.L. Ma a, L.L. Ma a, Q.M. Ma a, X.B. Ma e, Z.P. Mao a, X.H. Mo a, J. Nie a, S.L. Olsen p,H.P. Peng q,2, R.G. Ping a, N.D. Qi a, H. Qin a, J.F. Qiu a, Z.Y. Ren a, G. Rong a, L.Y. Shan a,

L. Shang a, C.P. Shen a, D.L. Shen a, X.Y. Shen a, H.Y. Sheng a, H.S. Sun a, J.F. Sun a, S.S. Sun a,Y.Z. Sun a, Z.J. Sun a, Z.Q. Tan d, X. Tang a, G.L. Tong a, G.S. Varner p, D.Y. Wang a, L. Wang a,

L.L. Wang a, L.S. Wang a, M. Wang a, P. Wang a, P.L. Wang a, W.F. Wang a,3, Y.F. Wang a, Z. Wang a,Z.Y. Wang a, Zhe Wang a, Zheng Wang b, C.L. Wei a, D.H. Wei a, N. Wu a, X.M. Xia a, X.X. Xie a,G.F. Xu a, X.P. Xu f, Y. Xu k, M.L. Yan q, H.X. Yang a, Y.X. Yang c, M.H. Ye b, Y.X. Ye q, Z.Y. Yi a,

G.W. Yu a, C.Z. Yuan a, J.M. Yuan a, Y. Yuan a, S.L. Zang a, Y. Zeng g, Yu Zeng a, B.X. Zhang a,B.Y. Zhang a, C.C. Zhang a, D.H. Zhang a, H.Q. Zhang a, H.Y. Zhang a, J.W. Zhang a, J.Y. Zhang a,S.H. Zhang a, X.M. Zhang a, X.Y. Zhang m, Yiyun Zhang n, Z.P. Zhang q, D.X. Zhao a, J.W. Zhao a,

M.G. Zhao a, P.P. Zhao a, W.R. Zhao a, Z.G. Zhao a,4, H.Q. Zheng l, J.P. Zheng a, Z.P. Zheng a,L. Zhou a, N.F. Zhou a,4, K.J. Zhu a, Q.M. Zhu a, Y.C. Zhu a, Y.S. Zhu a, Yingchun Zhu a,2,∗,

Z.A. Zhu a, B.A. Zhuang a, X.A. Zhuang a, B.S. Zou a

a Institute of High Energy Physics, Beijing 100049, People’s Republic of Chinab China Center for Advanced Science and Technology (CCAST), Beijing 100080, People’s Republic of China

c Guangxi Normal University, Guilin 541004, People’s Republic of Chinad Guangxi University, Nanning 530004, People’s Republic of China

e Henan Normal University, Xinxiang 453002, People’s Republic of Chinaf Huazhong Normal University, Wuhan 430079, People’s Republic of China

g Hunan University, Changsha 410082, People’s Republic of Chinah Jinan University, Jinan 250022, People’s Republic of China

i Liaoning University, Shenyang 110036, People’s Republic of Chinaj Nanjing Normal University, Nanjing 210097, People’s Republic of China

k Nankai University, Tianjin 300071, People’s Republic of Chinal Peking University, Beijing 100871, People’s Republic of China

m Shandong University, Jinan 250100, People’s Republic of Chinan Sichuan University, Chengdu 610064, People’s Republic of Chinao Tsinghua University, Beijing 100084, People’s Republic of China

p University of Hawaii, Honolulu, HI 96822, USAq University of Science and Technology of China, Hefei 230026, People’s Republic of China

r Wuhan University, Wuhan 430072, People’s Republic of China

0370-2693/$ – see front matter © 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2006.10.004

442 BES Collaboration / Physics Letters B 642 (2006) 441–448

s Zhejiang University, Hangzhou 310028, People’s Republic of China

Received 30 May 2006; received in revised form 2 October 2006; accepted 3 October 2006

Available online 17 October 2006

Editor: W.-D. Schlatter

Abstract

Results are presented on J/ψ radiative decays to π+π− and π0π0 based on a sample of 58M J/ψ events taken with the BES II detector.Partial wave analyses are carried out using the relativistic covariant tensor amplitude method in the 1.0 to 2.3 GeV/c2 ππ mass range. There areconspicuous peaks due to the f2(1270) and two 0++ states in the 1.45 and 1.75 GeV/c2 mass regions. The first 0++ state has a mass of 1466 ±6 ± 20 MeV/c2, a width of 108+14

−11 ± 25 MeV/c2, and a branching fraction B(J/ψ → γf0(1500) → γπ+π−) = (0.67 ± 0.02 ± 0.30) × 10−4.

Spin 0 is strongly preferred over spin 2. The second 0++ state peaks at 1765+4−3 ± 13 MeV/c2 with a width of 145 ± 8 ± 69 MeV/c2. If this 0++

is interpreted as coming from f0(1710), the ratio of its branching fractions to ππ and KK̄ is 0.41+0.11−0.17.

© 2006 Elsevier B.V. All rights reserved.

PACS: 12.39.Mk; 13.25.Gv; 14.40.Cs

1. Introduction

QCD predicts the existence of glueballs, the bound states ofgluons, and the observation of glueballs would provide a directtest of QCD. In the quenched approximation, lattice QCD cal-culations predict the lightest glueball to be a 0++ with the massbeing in the region from 1.4 to 1.8 GeV/c2 [1]. Although theidentification of a glueball is very complicated, there are sev-eral glueball candidates, including the f0(1500) and f0(1710).The properties of the f0(1500) and f0(1710) are reviewed indetail in the latest issue of the Particle Data Group (PDG) [2].

J/ψ radiative decays have been suggested as promisingmodes for glueball searches. The J/ψ → γπ+π− process wasanalyzed previously in the Mark III [3], DM2 [4] and BES I[5] experiments, in which there was evidence for f2(1270)

and an additional f2(1720). However, the high mass shoulderof the f2(1270), at about 1.45 GeV/c2, was unsettled. A re-vised amplitude analysis of Mark III data assigned the shoulderto be a scalar at ∼ 1.43 GeV/c2, and, in addition, found thepeak at ∼ 1.7 GeV/c2 to be scalar rather than tensor [6]. TheJ/ψ → γπ0π0 process was also studied by the Crystal Ball [7]and BES I experiments [8], but no partial wave analysis has yetbeen performed on this channel. In this Letter, the results ofpartial wave analyses on J/ψ → γπ+π− and γπ0π0 are pre-sented based on a sample of 58M J/ψ events collected by theupgraded Beijing Spectrometer (BES II) located at the BeijingElectron Positron Collider (BEPC).

2. BES detector

BES II is a large solid-angle magnetic spectrometer that isdescribed in detail in Ref. [9]. Charged particle momenta are

* Corresponding author.E-mail address: yczhu@mail.ihep.ac.cn (Y. Zhu).

1 Current address: Purdue University, West Lafayette, IN 47907, USA.2 Current address: DESY, D-22607, Hamburg, Germany.3 Current address: Laboratoire de l’Accélérateur Linéaire, Orsay, F-91898,

France.4 Current address: University of Michigan, Ann Arbor, MI 48109, USA.

determined with a resolution of σp/p = 1.78%√

1 + p2 (p inGeV/c) in a 40-layer cylindrical main drift chamber (MDC).Particle identification is accomplished by specific ionization(dE/dx) measurements in the drift chamber and time-of-flight(TOF) measurements in a barrel-like array of 48 scintillationcounters. The dE/dx resolution is σdE/dx = 8.0%; the TOFresolution is σTOF = 180 ps for Bhabha events. Outside of thetime-of-flight counters is a 12-radiation-length barrel showercounter (BSC) comprised of gas tubes operating in limitedstream mode. The BSC measures the energies of photons witha resolution of σE/E � 21%/

√E (E in GeV). Outside the

solenoidal coil, which provides a 0.4 T magnetic field over thetracking volume, is an iron flux return that is instrumented withthree double layers of counters that are used to identify muons.

In this analysis, a GEANT3 based Monte Carlo simulationprogram (SIMBES) [10] with detailed consideration of detectorperformance (such as dead electronic channels) is used. Theconsistency between data and Monte Carlo has been checked inmany high purity physics channels, and the agreement is quitereasonable [10].

3. Event selection

The first level of event selection of J/ψ → γπ+π− requirestwo charged tracks with total charge zero. Each charged track,reconstructed using MDC information, is required to be wellfitted to a three-dimensional helix, be in the polar angle region| cos θMDC| < 0.8, and have the point of closest approach of thetrack to the beam axis be within 2 cm of the beam axis andwithin 20 cm from the center of the interaction region along thebeam line.

More than one photon per event is allowed because of thepossibility of fake photons coming from the interactions ofcharged tracks with the shower counter or from electronic noisein the shower counter. A neutral cluster is considered to bea photon candidate when the energy deposited in the BSC isgreater than 50 MeV, the first hit is in the beginning six radi-ation lengths, the angle between the nearest charged track and

BES Collaboration / Physics Letters B 642 (2006) 441–448 443

the cluster is greater than 18◦, and the angle between the clus-ter development direction in the BSC and the photon emissiondirection is less than 30◦.

The total number of layers with hits associated with the twocharged particles in the muon counter is required to be less thanfour in order to remove γμ+μ− events. To remove the largebackgrounds from Bhabha events, we require that (i) the open-ing angle of the two tracks satisfies θop < 175◦ and (ii) the en-ergy deposit by each track in the BSC satisfies ESC < 1.0 GeV.We require θop > 10◦ to remove γ conversions that occur at lowπ+π− mass. In order to reduce the background from final stateswith kaons and electrons, both tracks are required to be identi-fied as pions by TOF or dE/dx when the momenta are lowerthan 0.7 GeV/c. In other cases, at least one track is required tobe identified as a pion by TOF.

Requirements on two variables, U and P 2tγ , are imposed

[11]. The variable U = (Emiss − | �Pmiss|) is required to sat-isfy |U | < 0.15 GeV. Here, Emiss and �Pmiss are, respec-tively, the missing energy and momentum of charged parti-cles. The variable P 2

tγ = 4| �Pmiss|2 sin2 θγ /2 is required to be

< 0.0045 (GeV/c)2, where θγ is the angle between the missingmomentum and the photon direction. The U cut removes mostbackground from events having multikaon or other neutral par-ticles, such as K∗(892)±K∓, γK+K− events. The cut on P 2

tγ

is used to reduce backgrounds with π0s.In order to reduce the dominant ρπ background, events

with more than one photon satisfying |Mγ1γ2 − Mπ0 | <

0.065 GeV/c2 are rejected. Here Mγ1γ2 is the invariant massof the two isolated photons with the smallest angle betweenthe plane determined by these two photons and the direction of�Pmiss in all possible photon combinations. Mγ1γ2 is calculated

using Pmiss and the angle between �Pmiss and the γ direction.The advantage of this method is that it uses the momenta ofthe charged tracks measured by the MDC, which has goodmomentum resolution, and is independent of photon energymeasurement.

Finally, the two charged tracks and photon in the event arekinematically fitted using four energy and momentum conser-vation constraints (4-C) under the J/ψ → γπ+π− hypothesisto obtain better mass resolution and to suppress backgroundsfurther by using the requirements χ2

γπ+π− < 15 and χ2γπ+π− <

χ2γK+K− . If there is more than one photon, the fit is repeated

using all permutations and the combination with the best fit toγπ+π− is retained.

For J/ψ → γπ0π0, the π0 mesons in the event are iden-tified through the decay π0 → γ γ . The isolated photon is re-quired to have the energy deposited in the BSC greater than 80MeV and come from the interaction point. The number of iso-lated photons is required to be greater than four and less thanseven. A 4-C kinematic fit to J/ψ → 5γ is performed, the com-bination of five photons with the smallest χ2 is selected, and akinematic fit chi-square χ2

5γ < 15 is required. For five photons,

there are 15 combinations from which to construct two π0s.To select π0s, we choose the combination with the smallest Δ,

where Δ =√

(Mγ1γ2 − Mπ0)2 + (Mγ3γ4 − Mπ0)2 and require

Fig. 1. Invariant mass spectrum of π+π− and the Dalitz plot forJ/ψ → γπ+π− , where the lightly and dark shaded histograms in the upperpanel correspond to J/ψ → π+π−π0 and other estimated backgrounds, re-spectively.

|Mπ01,2

− Mπ0 | < 40 MeV/c2. To reduce background with ωs,

events with the invariant mass of a π0 and one photon in theω mass interval |Mγπ0

1(2)− Mω| < 30 MeV/c2 are rejected. To

further suppress backgrounds with more than one neutral parti-cle recoling to the π0π0 system, the recoiling mass squared ofthe π0π0 system is required to be less than 4.8 (GeV/c2)2.

Fig. 1 shows the π+π− mass spectrum for the selectedevents, together with the corresponding background distribu-tions and the Dalitz plot. There is a strong ρ0(770) peak mainlydue to background from J/ψ → ρ0π0. A strong f2(1270)

signal, a shoulder on the high mass side of the f2(1270), anenhancement at ∼ 1.7 GeV/c2, and a peak at ∼ 2.1 GeV/c2

are clearly visible. The lightly shaded histogram in Fig. 1 cor-responds to the dominant background J/ψ → π+π−π0. Thedata taken at the e+e− center of mass energy of 3.07 GeV,with a luminosity of 2272.8 ± 36.4 nb−1, are used to determinethe continuum background. The sum of continuum backgroundand the other possible backgrounds, such as J/ψ → γ η′ (η′ →γρ0, ρ0 → π+π−), J/ψ → K∗(892)±K∓, . . . , is estimated tobe 3.8% of the data in the whole mass range and is shown asthe dark shaded histogram in Fig. 1.

Fig. 2 shows the π0π0 mass spectrum and the Dalitzplot. The shaded histogram corresponds to the sum of es-

444 BES Collaboration / Physics Letters B 642 (2006) 441–448

Fig. 2. Invariant mass spectrum of π0π0 and the Dalitz plot forJ/ψ → γπ0π0, where the shaded histogram in the upper panel correspondsto the estimated backgrounds.

Fig. 3. The π+π− invariant mass distribution from J/ψ → γπ+π− . Thecrosses are data, the full histogram shows the maximum likelihood fit, and theshaded histogram corresponds to the π+π−π0 background.

timated backgrounds determined using PDG branching ra-tios [2]. The backgrounds are mainly from J/ψ → ωπ0π0

(ω → γπ0,π0 → 2γ ), J/ψ → γ η, (η → 3π0,π0 → 2γ )

and J/ψ → γ ηπ0π0 (η → 2γ,π0 → 2γ ). The peak below0.5 GeV/c2 is mainly from J/ψ → γ η, (η → 3π0,π0 → 2γ ).The continuum background is also studied using the data taken

at the e+e− center of mass energy of 3.07 GeV. No events sur-vive selection requirements. In general, the π+π− and π0π0

mass spectra exhibit similar structures above 1.0 GeV/c2.

4. Partial wave analysis

We have carried out partial wave analyses for the ππ massrange from 1.0 to 2.3 GeV/c2 using relativistic covariant tensoramplitudes constructed from Lorentz-invariant combinations ofthe polarization and momentum 4-vectors of the initial and fi-nal state particles, with helicity ±1 for J/ψ initial states [12].Cross sections are summed over photon polarizations. The rel-ative magnitudes and phases of the amplitudes are determinedby a maximum likelihood fit.

For J/ψ → γπ+π−, the following channels are fitted to thedata:

J/ψ → γf2(1270)

→ γf0(1500)

→ γf0(1710)

→ γf2(1810)

→ γf0(2020)

→ γf2(2150)

→ γf4(2050).

Constant width Breit–Wigner functions are used for eachresonance. The form is described as follows:

BWX = mΓ

s − m2 + imΓ,

where s is the square of π+π− invariant mass, m and Γ are themass and width of intermediate resonance X, respectively.

The dominant background in J/ψ → γπ+π− comes fromJ/ψ → π+π−π0. From Mark III’s analysis on J/ψ →π+π−π0, described in terms of the amplitudes representing thesequential two-body decay process J/ψ → ρπ, ρ → ππ [13],the ρ(770) is dominant, but there are also contributions fromthe excited states of ρ(770). A preliminary PWA on BESIIJ/ψ → π+π−π0 also shows that a complete description of thedata requires not only the dominant ρ(770), but also the contri-butions from 1− states ρ(1450), ρ(1700) and ρ(2150), as wellas 3− state ρ(1690). Using the branching fraction measurementof BES II [14], which agrees with the result from the BaBarCollaboration [15], and the generator based on the publishedresults of Ref. [13], J/ψ → π+π−π0 events are generated andgiven the opposite log likelihood in the fit to cancel the back-ground events in the data. The other background contributionsare much smaller than J/ψ → π+π−π0 background in the 1.0to 2.3 GeV/c2 π+π− mass range and not considered in thepartial wave analysis.

Due to the limitation of the present statistics and the com-plexity of the large ρπ background, it is difficult to cleanlydistinguish components in the high mass region. The maingoal of this analysis will be to understand the structures be-low 2.0 GeV/c2. We use the four states f2(1810), f0(2020),f2(2150), and f4(2050), which are listed in PDG [2] in the fit

BES Collaboration / Physics Letters B 642 (2006) 441–448 445

Fig. 4. The mass projections of the individual components for J/ψ → π+π− . The crosses are data. The complete 0++ and 2++ contributions are also shown,including all interferences.

with the masses and widths fixed to those in the PDG, to de-scribe the contribution of the high mass states in the mass rangebelow 2.0 GeV/c2.

For the 2++ states, relative phases between different helicityamplitudes for a single resonance are theoretically expected tobe very small [16]. Therefore, these relative phases are set tozero in the final fit so as to constrain the intensities further.

After the mass and width optimization, the resulting fittedintensities are illustrated in Figs. 3 and 4. Angular distributionsin the whole mass range are shown in Fig. 5. Here, θγ is thepolar angle of the photon in the J/ψ rest frame, and θπ is thepolar angle of the pion in the π+π− rest frame.

From Figs. 3 and 5, we see that the fit agrees well with data.Fig. 4 shows the distributions of the individual componentsand full 0++ and 2++ contributions including interferences.A free fit to f2(1270) gives a fitted mass of 1262+1

−2 MeV/c2

and a width of 175+6−4 MeV/c2. The fitted masses and widths of

the f0(1500) and f0(1710) are Mf0(1500) = 1466 ± 6 MeV/c2,Γf0(1500) = 108+14

−11 MeV/c2 and Mf0(1710) = 1765+4−3 MeV/c2,

Γf0(1710) = 145 ± 8 MeV/c2, respectively. The branchingfractions of f2(1270), f0(1500), and f0(1710) determinedby the partial wave analysis fit are B(J/ψ → γf2(1270) →γπ+π−) = (9.14 ± 0.07) × 10−4, B(J/ψ → γf0(1500) →γπ+π−) = (6.65±0.21)×10−5, and B(J/ψ → γf0(1710) →γπ+π−) = (2.64 ± 0.04) × 10−4, respectively. For thef2(1270), we find the ratios of helicity amplitudes x = 0.89 ±0.02 and y = 0.46 ± 0.02 with correlation factor ρ = 0.26,where x = A1/A0, y = A2/A0, A0,1,2 correspond to the threeindependent production amplitudes with helicity 0, 1 and 2.The errors here are statistical errors. An alternative fit is triedby replacing f0(1500) with a 2++ resonance. There are threehelicity amplitudes fitted for spin 2, while only one amplitude

Fig. 5. Projections in cos θγ and cos θπ for the whole mass range. The crossesare data (J/ψ → γπ+π− sample), and the histograms are the fit results.

for spin 0, which means the number of degrees of freedom is in-creased by 2 in the JP = 2+ case. However, the log likelihoodis worse by 108. This indicates that f0(1500) with JP = 0+ isstrongly favored. If the f0(1710) is removed from the fit, thelog likelihood is worse by 379, which corresponds to a signalsignificance much larger than 5σ .

446 BES Collaboration / Physics Letters B 642 (2006) 441–448

The partial wave analysis of J/ψ → γπ0π0 is performedindependently. The components used are the same as those ofJ/ψ → γπ+π−. Due to the limited statistics, we take the par-tial wave analysis fit results of J/ψ → γπ0π0 as a cross-checkof the ones obtained in the charged channel. The backgrounddistribution is close to flat in the mass interval 1.0–2.3 GeV/c2,so we use 1/p2(m) multiplied by phase space to approximatelydescribe such a flat contribution, where the 2nd order polyno-mial function p2(m) is obtained by fitting the π0π0 mass phasespace distribution of J/ψ → γπ0π0 Monte Carlo simulation.

The fitted intensities as a function of π0π0 mass are il-lustrated in Fig. 6. A free fit to f2(1270) gives a fitted massof 1261 ± 6 MeV/c2 and a width of 188+18

−16 MeV/c2. Thefitted masses and widths of the f0(1500) andf0(1710) areMf0(1500) = 1485 ± 21 MeV/c2, Γf0(1500) = 178+60

−40 MeV/c2

and Mf0(1710) = 1755 ± 14 MeV/c2, Γf0(1710) = 155+30−26

MeV/c2, respectively. The errors shown here are statistical.

Fig. 6. The π0π0 invariant mass distribution from J/ψ → γπ0π0. The crossesare data, the full histogram shows the maximum likelihood fit, and the shadedhistogram corresponds to the background.

Besides the above global fit, a bin-by-bin fit is applied toJ/ψ → γπ+π− data using the method described in Ref. [17].A strong f2(1270) is observed, and the S-wave π+π− massdistribution shows a large signal at ∼ 1.45 GeV/c2, a signif-icant signal at ∼ 1.75 GeV/c2, and a peak at ∼ 2.1 GeV/c2.In general, the bin-by-bin fit gives similar features as those ofthe global fit, and the results of these two fits are approximatelyconsistent with each other.

5. Systematic errors

The systematic errors for the partial wave analysis fit toJ/ψ → γπ+π− data are estimated by varying the masses andwidths of the f2(1270), f0(1500), and f0(1710) within thefitted errors; varying the masses and widths of the f2(1810),f0(2020), f2(2150), and f4(2050) within the PDG errors [2];adding a small component f2(1565); varying the backgroundfraction within reasonable limit and replacing the f2(1810)

with the f2(1950). They also include the uncertainties in thenumber of J/ψ events analyzed, the efficiency of photon de-tection, the efficiency of MDC tracking, and the kinematicfit. Different generators, one is based on the published resultsof Ref. [13] and another is from BESII preliminary PWA re-sults which include a dominant ρ(770) and its excited statesρ(1450), ρ(1700), ρ(2150) and ρ3(1690), are used for theestimation of the background J/ψ → π+π−π0. The PWA re-sults with different backgrounds agree with each other withinthe error. Their difference is taken as the systematic error too.The systematic errors in this analysis do not include all model-dependent effects, such as the difference of using simple Breit–Wigner formalism and K-matrix formalism. Table 1 shows thesummary of the systematic errors for the global fit.

For the f2(1270), the total systematic errors are 0.10 and0.19 for x and y, respectively. The correlation factor betweenthe x and y systematic errors is 0.29 which is calculated withρ = ∑ ρiσxiσyi , where i runs over all systematic errors.

i σxσy

Table 1Estimation of systematic errors for the J/ψ → γπ+π− in the global fit. For the mass and width, what shown are the absolute errors in MeV/c2. For the branchingfraction B and the ratios of helicity amplitudes, x and y, the listed are the relative errors. ρ is the correlation factor between x and y

f2(1270) f0(1500) f0(1710)

M Γ B (%) x (%) y (%) ρ M Γ B (%) M Γ B (%)

M and Γ of f2(1270) 1.4 0.2 2.0 0.26 0 0 2.3 0 0 0.1M and Γ of f0(1500) 1 0 0.1 0.1 2.2 0.26 10.1 0 1 1.9M and Γ of f0(1710) 0 0 0.1 0.5 0.6 0.26 0 1 2.6 8.9M and Γ of f2(1810) 0 1 1.4 2.9 0.5 0.26 1 7 4.2 3 15 6.9M and Γ of f0(2020) 1 3 1.0 1.6 1.2 0.27 6 2 3.0 5 10 5.8M and Γ of f2(2150) 1 2 1.2 1.2 0.4 0.26 3 0 3.2 1 3 3.3M and Γ of f4(2050) 0 1 0.1 0.4 0.7 0.26 1 2 0.4 0 0 0.7Add f2(1565) 2 5 0.2 6.9 34.6 0.35 3 7 18.6 3 10 2.9MDC tracking and kinematic fit 4 4 3.5 6.2 12.8 0.38 14 18 31.0 8 1 13.4Background fraction 3 2 1.8 3.4 4.3 0.33 4 2 13.9 3 10 7.8Background with different generator 3 3 11.8 1.7 19.5 0.41 2 13 14.6 5 7 9.0Replace f2(1810) with f2(1950) 4 5 8.7 2.1 4.4 0.27 12 4 11.0 5 65 16.4δNJ/ψ

4.7 4.7 4.7

Detection efficiency of photon 2.0 2.0 2.0Total systematic error 8 10 16.2 10.9 42.3 0.29 20 25 44.8 13 69 28.3

BES Collaboration / Physics Letters B 642 (2006) 441–448 447

Table 2Fit results for J/ψ → γπ+π− . The first error is statistical, and the second issystematic

J/ψ → γX,X → π+π−

Mass (MeV/c2) Γ (MeV/c2) B (×10−4)

f2(1270) 1262+1−2 ± 8 175+6

−4 ± 10 9.14 ± 0.07 ± 1.48

f0(1500) 1466 ± 6 ± 20 108+14−11 ± 25 0.67 ± 0.02 ± 0.30

f0(1710) 1765+4−3 ± 13 145 ± 8 ± 69 2.64 ± 0.04 ± 0.75

Table 3The branching fraction measurements of J/ψ → γπ0π0, where the massesand widths of the resonances are fixed to the values determined from J/ψ →γπ+π−. The first error is statistical, and the second is systematic

J/ψ → γX,X → π0π0

Mass (MeV/c2) Γ (MeV/c2) B (×10−4)

f2(1270) same as charged channel 4.00 ± 0.09 ± 0.58f0(1500) same as charged channel 0.34 ± 0.03 ± 0.15f0(1710) same as charged channel 1.33 ± 0.05 ± 0.88

Table 2 shows the mass, width, and branching fraction mea-surements for f2(1270), f0(1500), and f0(1710), where thefirst error is statistical and the second is systematic, determinedby adding all sources in quadrature. In order to compare thebranching fractions of f2(1270), f0(1500), and f0(1710) inγπ+π− and γπ0π0, we fix the mass and width of each compo-nent in γπ0π0 to those of the charged channel and re-calculatethe branching fractions and estimate the systematic errors. Theresults are shown in Table 3. The branching fractions deter-mined from the two channels agree with each other within er-rors after considering isospin corrections.

6. Discussion

The measured mass of the f2(1270), 1262+1−2 ± 8 MeV/c2,

is lower than the PDG value, and the branching fraction ofJ/ψ → γf2(1270), f2(1270) → π+π− is a bit higher than thePDG value [2]. A fit with the PDG mass and width is visiblypoorer, and the log likelihood is worse than the optimum fitby 44. In this analysis, the S-wave contribution on the highmass shoulder of the large f2(1270) peak is well separated,which may explain this mass difference. The ratios of the he-licity amplitudes of the f2(1270) from the present analysis arex = 0.89 ± 0.02 ± 0.10 and y = 0.46 ± 0.02 ± 0.19 with cor-relation ρstat = 0.26 and ρsys = 0.29. The values of x and y arein agreement with predictions [16,18] within the errors. Com-paring to the results determined by DM2 [4], Mark III [3], andCrystal Ball [19], there is a difference in the value of y. Themain reason for this difference may be that we consider influ-ences from other states in the 1.0 to 2.3 GeV/c2 ππ mass range,while previous analyses by DM2 and Mark III ignored theseand only considered the f2(1270) in the 1.15–1.4 GeV/c2 ππ

mass range.The most remarkable feature of the above results is that three

0++ states at the ππ mass ∼ 1.45, 1.75, and 2.1 GeV/c2 are ob-served from the partial wave analysis. For the high mass state at∼ 2.1 GeV/c2, we use the f0(2020), which is listed in the PDG,

to describe it. No further efforts are made on the measurementsof its resonant parameters due to the difficulties described inSection 4.

The lower 0++ state peaks at a mass of 1466 ± 6 ±20 MeV/c2 with a width of 108+14

−11 ± 25 MeV/c2, which isconsistent with the scalar glueball candidate, f0(1500). Spin0 is strongly preferred over spin 2. Therefore we interpretthe small but definite shoulder on the high mass side of thef2(1270) in the π+π− mass distribution as originating from thef0(1500), which interferes with nearby resonances in the par-tial wave analysis. However, due to the uncertainties of the massand width determinations and large interferences between theS-wave states, the existence of the f0(1370) in J/ψ → γππ

cannot be excluded by present data.Strong production of the f0(1710) was observed in the par-

tial wave analysis of J/ψ → γKK̄ , with a mass of 1740 ±4+10−25 MeV/c2, a width of 166+5+15

−8−10 MeV/c2, and a branch-ing fraction of J/ψ → γf0(1710), f0(1710) → KK̄ of (9.62±0.29+2.11+2.81

−1.86−0.00) × 10−4 [17]. Interpreting the 0++ state in themass region ∼ 1.75 GeV/c2 as coming from the f0(1710)

and using the branching fraction of f0(1710) → ππ deter-mined from f0(1710) → π+π− after isospin correction and thebranching fraction of f0(1710) → KK̄ in Ref. [17], we obtainthe ratio of ππ to KK̄ branching fractions for the f0(1710) as

Γ (f0(1710) → ππ)

Γ (f0(1710) → KK̄)= 0.41+0.11

−0.17.

The ratio is consistent with the PDG value (0.200 ± 0.024 ±0.036) [2] within the errors. An alternative interpretation forthis 0++ state is the f0(1790). Data on J/ψ → φπ+π− andφK+K− show a definite peak in π+π− at 1790 MeV/c2 butno significant signal in K+K− [20]. If the mass and width ofthe 0++ state are fixed to 1790 MeV/c2 and 270 MeV/c2 foundin [20], the log likelihood is worse by 47. This 0++ state mayalso be a superposition of f0(1710) and f0(1790).

Due to the uncertainty of the high mass region, we do analternative fit removing the f4(2050) and re-optimizing themasses and widths of f2(1270), f0(1500), and f0(1710) for theJ/ψ → γπ+π− sample. The log likelihood is worse by 160.The fit gives a f2(1270) mass of 1259 ± 2 ± 6 MeV/c2 and awidth of 175+4

−5 ± 8 MeV/c2. The measured masses and widthsof the f0(1500) and f0(1710) are Mf0(1500) = 1466 ± 6 ±17 MeV/c2, Γf0(1500) = 118+14

−15 ± 28 MeV/c2 and Mf0(1710) =1768+5

−4 ± 15 MeV/c2, Γf0(1710) = 112+10−8 ± 52 MeV/c2, re-

spectively. The branching fractions of the f2(1270), f0(1500),and f0(1710) are B(J/ψ → f2(1270) → γπ+π−) = (9.04 ±0.07 ± 0.89) × 10−4, B(J/ψ → γf0(1500) → γπ+π−) =(0.68 ± 0.02 ± 0.28) × 10−4, and B(J/ψ → γf0(1710) →γπ+π−) = (1.40 ± 0.03 ± 0.55) × 10−4, respectively.

The light-meson spectroscopy of scalar states in the massrange 1–2 GeV/c2, which has long been a source of contro-versy, is still very complicated. Overlapping states interferewith each other differently in different production and decaychannels. More experimental data are needed to clarify theproperties of these scalar states.

448 BES Collaboration / Physics Letters B 642 (2006) 441–448

7. Summary

In summary, the partial wave analyses of J/ψ → γπ+π−and J/ψ → γπ0π0 using 58M J/ψ events of BES II showstrong production of f2(1270) and evidence for two 0++ statesin the 1.45 and 1.75 GeV/c2 mass regions. For the f2(1270),the branching ratio determined by the partial wave analysis fitis B(J/ψ → γf2(1270) → γπ+π−) = (9.14±0.07±1.48)×10−4. The ratios of the helicity amplitudes of the f2(1270)

are determined to be x = 0.89 ± 0.02 ± 0.10 and y = 0.46 ±0.02 ± 0.19 with correlations ρstat = 0.26 and ρsys = 0.29. Thef0(1500) has a mass of 1466 ± 6 ± 20 MeV/c2, a width of108+14

−11 ± 25 MeV/c2, and a branching fraction B(J/ψ →γf0(1500) → γπ+π−) = (0.67 ± 0.02 ± 0.30) × 10−4. The0++ state in the ∼ 1.75 GeV/c2 mass region has a mass of1765+4

−3 ± 13 MeV/c2 and a width of 145 ± 8 ± 69 MeV/c2. Ifthis 0++ state is interpreted as coming from f0(1710), the ra-tio of the ππ to KK̄ branching fractions is 0.41+0.11

−0.17. This mayhelp in understanding the properties of f0(1500) and f0(1710).

Acknowledgements

The BES Collaboration thanks the staff of BEPC and com-puting center for their hard efforts. We wish to thank Prof.David Bugg for contributions to the early stage of this analy-sis. This work is supported in part by the National NaturalScience Foundation of China under contracts Nos. 10491300,10225524, 10225525, 10425523, the Chinese Academy of Sci-ences under contract No. KJ 95T-03, the 100 Talents Programof CAS under Contract Nos. U-11, U-24, U-25, and the Knowl-edge Innovation Project of CAS under Contract Nos. KJCX2-

SW-N10, U-602, U-34 (IHEP), the National Natural ScienceFoundation of China under Contract No. 10225522 (TsinghuaUniversity), and the Department of Energy under ContractNo. DE-FG02-04ER41291 (U Hawaii).

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