Physics Letters B 660 (2008) 315–319

www.elsevier.com/locate/physletb

Determination of the ψ(3770), ψ(4040), ψ(4160)

and ψ(4415) resonance parameters

BES Collaboration

M. Ablikim a, J.Z. Bai a, Y. Ban l, 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, Y.P. Chu 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 a,1, C.D. Fu o, C.S. Gao a, Y.N. Gao o, S.D. Gu a, Y.T. Gu d, Y.N. Guo a,Z.J. Guo p,2, F.A. Harris p, K.L. He a, M. He m, Y.K. Heng a, J. Hou m, H.M. Hu a,∗, J.H. Hu c, T. Hu a,

G.S. Huang a,3, 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,Y.F. Lai a, G. Li a,4, H.B. 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.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,5, J.B. Liu a, J.P. Liu r, Jian Liu a, Q. Liu p, 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 b, L.L. Ma a,6, Q.M. Ma a, Z.P. Mao a, X.H. Mo a, J. Nie a,S.L. Olsen p, R.G. Ping a, N.D. Qi a, H. Qin a, J.F. Qiu a, Z.Y. Ren a, G. Rong a, X.D. Ruan d,

L.Y. Shan a, L. Shang a, C.P. Shen p, D.L. Shen a, X.Y. Shen a, H.Y. Sheng a, H.S. Sun a, S.S. Sun a,Y.Z. Sun a, Z.J. Sun a, X. Tang a, G.L. Tong a, G.S. Varner p, D.Y. Wang a,7, 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,8, Y.F. Wang a, Z. Wang a, Z.Y. Wang a,Zheng Wang a, C.L. Wei a, D.H. Wei a, Y. Weng 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, G.W. Yu a,C.Z. Yuan a, Y. Yuan a, S.L. Zang a, Y. Zeng g, 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.Y. Zhang m,Yiyun Zhang n, Z.X. Zhang l, 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,9, H.Q. Zheng l, J.P. Zheng a, Z.P. Zheng a, L. Zhou a, K.J. Zhu a,Q.M. Zhu a, Y.C. Zhu a, Y.S. Zhu a, 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 Chinas Zhejiang University, Hangzhou 310028, People’s Republic of China

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2007.11.100

316 BES Collaboration / Physics Letters B 660 (2008) 315–319

Received 31 May 2007; received in revised form 29 September 2007; accepted 20 November 2007

Available online 15 January 2008

Editor: W.-D. Schlatter

Abstract

R values measured with the BESII detector at center-of-mass energies between 3.7 and 5.0 GeV are fitted to determine resonance parameters(mass, total width, electron width) of the high mass charmonium states, ψ(3770), ψ(4040), ψ(4160) and ψ(4415). Various effects, including theinterferences and relative phases between the resonances, the energy-dependence of the full widths, and the initial state radiative correction, areexamined. The results are compared to previous studies.© 2008 Elsevier B.V. All rights reserved.

1. Introduction

The total cross section for hadron production in e+e− anni-hilation is usually parameterized in terms of the ratio R, whichis defined as R = σ(e+e− → hadrons)/σ (e+e− → μ+μ−),where the denominator is the lowest-order QED cross sectionof μ+μ−, σ 0

μμ = 4πα2/3s. The measured R values are con-sistent with the three-color quark model predictions. Above theopen flavor thresholds where resonance structures show up, themeasurements of R values are used to determine the parametersof resonances with JPC = 1−−. For the high mass charmo-nium resonances, the ψ(3770) was measured by MARK-I [1],DELCO [2], MARK-II [3] and BES [4,5]; the ψ(4040) andψ(4160) were measured by DASP [6]; and the ψ(4415) wasmeasured by DASP [6] and MARK-I [7]. There were also someother measurements of R values as reported in Refs. [8–10],but no attempt was made to determine resonance parameters.The resonance parameters in the Particle Data Group (PDG)’scompilation remained unchanged for more than 20 years untilthe 2004 edition [11]. The resonance parameters for the threehigh mass resonances were updated in PDG2006 [12], basedon Seth’s evaluation [13] using BESII [14,15] and Crystal Ball[10] data.

The R values between 2–5 GeV were determined by BESIIusing [15]:

(1)Rexp = Nobshad − Nbg

σ 0μμLεtrgεhad(1 + δobs)

,

* Corresponding author.E-mail address: huhm@ihep.ac.cn (H.M. Hu).

1 Current address: DESY, D-22607, Hamburg, Germany.2 Current address: Johns Hopkins University, Baltimore, MD 21218, USA.3 Current address: University of Oklahoma, Norman, OK 73019, USA.4 Current address: Universite Paris XI, LAL-Bat. 208–BP34, 91898 Orsay

Cedex, France.5 Current address: Max-Plank-Institut für Physik, Foehringer Ring 6, 80805

Munich, Germany.6 Current address: University of Toronto, Toronto M5S 1A7, Canada.7 Current address: CERN, CH-1211 Geneva 23, Switzerland.8 Current address: Laboratoire de l’Accélérateur Linéaire, Orsay F-91898,

France.9 Current address: University of Michigan, Ann Arbor, MI 48109, USA.

where Nobshad is the number of observed hadronic events, Nbg is

the number of residual background events, L is the integratedluminosity, (1 + δobs) is the effective initial state radiation(ISR) correction factor [10,16], εhad is the detection efficiencyfor hadronic events determined by the Monte Carlo simulationwithout bremsstrahlung, and εtrg is the trigger efficiency.

The measurement of R values and the determinations ofresonance parameters are intertwined: the factor (1 + δobs) con-tains contributions from the resonances and depends on theresonance parameters. Therefore, the calculation of (1 + δobs)

and R require a number of iterations before convergent resultsare obtained.

In this work, we perform a global fit over the center-of-mass energy region from 3.7 to 5.0 GeV covering the fourresonances, ψ(3770), ψ(4040), ψ(4160) and ψ(4415). We useenergy-dependent full widths, and introduce relative phases be-tween the resonances in the fit. Finally, the updated R valuesare compared with the previously published results.

2. Fitting of the resonant parameters

The phenomenological models and formulae used in our fit-ting are briefly described below.

2.1. Breit–Wigner amplitude

In quantum mechanics, the wave function for an unstablestate with central angular frequency ω ≡ M/h and lifetime τ ≡h/Γ is

(2)ψ(t) = ψ(0)e−iωt e−t/2τ = ∣∣ψ(0)∣∣eiδe−it (M−iΓ /2),

where eiδ is the initial phase factor at the moment of production.Based upon Eq. (2), the relativistic Breit–Wigner amplitude forthe process e+e− → resonance → hadronic final state f atcenter-of-mass energy W ≡ √

s can be derived

(3)T fr (W) = Mr

√Γ ee

r Γfr

W 2 − M2r + iMrΓr

eiδr ,

where the index r represents the resonance being considered,Mr is the nominal mass, Γr is the full width, Γ ee

r is the electron

width, Γfr is the hadronic width for the decaying channel f ,

and δr is the effective initial phase angle.

BES Collaboration / Physics Letters B 660 (2008) 315–319 317

High mass charmonium states can decay into several two-body final states f . According to the Eichten model [17] andexisting experimental data [18], the accessible decay channels(including their conjugate states) are:

ψ(3770) ⇒ DD;ψ(4040) ⇒ DD,D∗D∗,DD∗,DsDs;ψ(4140) ⇒ DD,D∗D∗,DD∗,DsDs,DsD

∗s ;

ψ(4415) ⇒ DD,D∗D∗,DD∗,DsDs,DsD∗s ,

D∗s D∗

s ,DD1,DD∗2 .

The total squared inclusive amplitude of the resonances is adouble summation: the inner one is the coherent sum for thesame state f decaying from different resonances r , and theouter one is the incoherent sum over all different decay chan-nels f ,

(4)|Tres|2 =∑f

∣∣∣∣∑

r

T fr (W)

∣∣∣∣2

.

The resonant cross section expressed as the R value is thengiven by

(5)Rres = σres

σ 0μμ

= 12π

s

[|Tψ ′ |2 + |Tres|2],

where the influence of the ψ(2S) tail on the fitting is included.

2.2. Energy-dependence of hadronic width

The full width of a broad resonance depends on the energy.A phenomenological model derived from quantum mechanicsis used to describe the behavior of Γ

fr (W). This depends on

the momentum and the orbital angular momentum L of the de-caying final state as [19]

(6)Γfr (W) = Γr

∑L

Z2L+1f

BL

,

where Γr is a parameter to be determined by fitting experimen-tal data, Zf ≡ ρPf , Pf is the decay momentum, ρ is the rangeof the interaction (on the order of a few fermis), the precisevalue of which is insensitive to the physical results. The energy-dependent partial wave functions BL are given in reference [19]or [20]:

B0 = 1, B1 = 1 + Z2, B2 = 9 + 3Z2 + Z4,

(7)B3 = 225 + 45Z2 + 6Z4 + Z6.

When the resonance decays to several different hadronic fi-nal states, the total hadronic width is the sum of all the partialwidths,

(8)Γ hadr (W) = 2Mr

Mr + W

∑f

Γfr (W),

where 2Mr/(Mr + W) is the relativistic correction factor [19].The total width of the resonance r is expressed as

(9)Γr(W) = Γ eer + Γ μμ

r + Γ ττr + Γ had

r (W),

where e–μ universality, Γ eer = Γ

μμr , is assumed, but kinematic

suppression factors for Γ ττr of 0.48, 0.66, 0.72, and 0.78 for

ψ(3773), ψ(4040), ψ(4160), and ψ(4415), respectively, areused. The value of the leptonic width includes the contributionfrom vacuum polarization.

2.3. Continuum background

The contribution from continuum production of hadronsoriginating from light quark pairs (uu, dd and ss) is welldescribed by pQCD above 2 GeV [12]. Since it is close tothe production threshold, continuum nonresonant cc produc-tion can only be described by phenomenological models orempirical expressions. Since there are many possible channelsabove the open-charm threshold, the continuum cross sectionsare expected to vary smoothly. For simplicity, we parameterizecontinuum production with a second-order polynomial,

(10)Rcon = C0 + C1(W − 2MD±) + C2(W − 2MD±)2,

where C0, C1 and C2 are free parameters, and MD± is the massof meson D±. C0 represents the contributions from light quarksR

(uds)con , C1 > 0 expresses the contribution of cc that increases

with energy, and C2 < 0 ensures the saturation of Rcon at ener-gies well above the charm threshold.

2.4. Fitting scheme

We fit the experimental data with the software packageMINUIT [21] using a least squares method that minimizes thefunction [22]

(11)χ2 =∑

i

[fcRexp(Wi) − Rthe(Wi)]2

[fc R(i)exp]2

+ (fc − 1)2

σ 2c

,

where Wi stands for the energy of the measured point. The ex-perimental and corresponding theoretical quantities are

(12)Rexp = Nobshad − Nbg

σ 0μμLεtrgεhad

,

and

(13)Rthe = (1 + δobs)Rthe,

where the calculation scheme used for the ISR factor (1 + δobs)

is the same as in Refs. [10] and [16]. If interference betweenthe continuum and resonant states is ignored, Rthe is given by

(14)Rthe = Rcon + Rres.

The term R(i)exp in Eq. (11) is the combined statistical and

non-common systematic errors (except for the ISR error), andRexp(Wi) is held constant during the fit. The error common toall the points σc(∼ 3.3%) is not included in Rexp(Wi). Thescale factor fc reflects the influence of the common error. Ineach iteration, the resonant parameters used in the recalculationof (1 + δobs) and Rthe are updated to the new values.

The free parameters in the fit are Mr , Γ eer , δr in Eq. (3),

Γr in Eq. (6), and C0, C1, C2 in Eq. (10). Since only rela-tive values of the phase-angles can be extracted, the phase of

318 BES Collaboration / Physics Letters B 660 (2008) 315–319

Table 1The resonance parameters of the high mass charmonia in this work together with the values in PDG2004 [11], PDG2006 [12] and Seth’s evaluations [13] based onCrystal Ball and BES data. The total width Γtot ≡ Γr (M) in Eq. (9)

ψ(3770) ψ(4040) ψ(4160) ψ(4415)

M (MeV/c2) PDG2004 3769.9±2.5 4040±10 4159±20 4415±6PDG2006 3771.1±2.4 4039±1 4153±3 4421±4CB (Seth) – 4037±2 4151±4 4425±6BES (Seth) – 4040±1 4155±5 4455±6BES (this work) 3772.0±1.9 4039.6±4.3 4191.7±6.5 4415.1±7.9

Γtot (MeV) PDG2004 23.6±2.7 52±10 78±20 43±15PDG2006 23.0±2.7 80±10 103±8 62±20CB (Seth) – 85±10 107±10 119±16BES (Seth) – 89±6 107±16 118±35BES (this work) 30.4±8.5 84.5±12.3 71.8±12.3 71.5±19.0

Γee (keV) PDG2004 0.26±0.04 0.75±0.15 0.77±0.23 0.47±0.10PDG2006 0.24±0.03 0.86±0.08 0.83±0.07 0.58±0.07CB (Seth) – 0.88±0.11 0.83±0.08 0.72±0.11BES (Seth) – 0.91±0.13 0.84±0.13 0.64±0.23BES (this work) 0.22±0.05 0.83±0.20 0.48±0.22 0.35±0.12

δ (degree) BES (this work) 0 130±46 293±57 234±88

ψ(3770) is set to zero. The parameters of the ψ(2S) in Eq. (5)are fixed to the values given in PDG2006.

3. Results and discussion

The values of the resonance parameters of the high masscharmonium states determined in this work, together with thosein PDG2004, PDG2006 and the results given in Ref. [13] arelisted in Table 1. The fitted parameters for the continuum com-ponent are C0 = 2.14 ± 0.10, C1 = (1.69 ± 0.23) × 10−3,and C2 = −(0.66 ± 0.25) × 10−6. And the scale factor isfc = 1.002 ± 0.033. The updated R values between 3.7 and5.0 GeV (the percentage errors are the same as in Refs. [14,15]) and the fitting curves are shown in Fig. 1. The quality ofthe global fitting is indicated by χ2/d.o.f. = 1.08 (the numberof energy-points is 78, the number of the free parameters is 19,and χ2 = 63.60) with a fit probability of 31.8%.

It should be noted that the ψ(4160) mass in this work isabout 30 MeV/c2 higher than the PDG2006 value, a differ-ence that is much larger than the quoted errors. If the interfer-ence terms in Eq. (4) all have their phase angles δr fixed to 0,then the obtained mass parameters of the resonances ψ(4040),ψ(4160), and ψ(4415) are 4048.4 ± 3.2, 4156.2 ± 4.4 and4405.2 ± 5.7 MeV, respectively, with a larger χ2/d.o.f. = 1.39corresponding to a probability of 2.3%. These comparisonsshow that the influence of the phase angles on the resonanceparameters is significant.

In order to understand the model-dependent uncertaintiesand to estimate the systematic errors, alternative choices andcombinations of Breit–Wigner forms, energy dependence of thefull width predicted by the quantum mechanics model [20] orthe effective interaction theory [23], and continuum charm pro-duction described by a second order polynomial or the phenom-enological form used by DASP [6] are used. We find the resultsare also somewhat sensitive to the form of the energy-dependenttotal width, but not sensitive to the continuum parameterization.

Fig. 1. The fit to the R values for the high mass charmonia structure. The dotswith error bars are the updated R values. The solid curve shows the best fit,and the other curves show the contributions from each resonance RBW, theinterference Rint, the summation of the four resonances Rres = RBW + Rint,and the continuum background Rcon respectively.

The DASP background function has six continuum productionchannels, while the effective interaction theory predicts a dif-ferent energy-dependent partial width for each one. However,in both cases the best fits give unreasonable values for some pa-rameters. This may be understood as being due to the fact thatthe inclusive data does not supply enough information to de-termine the relative width of different decay channels, nor thephase angles of the hadronic final states (if they exist). To un-derstand the detailed structure and components of the high masscharmonium states, it is necessary to collect data at each energypoint with sufficiently high statistics, and to develop more reli-able physical models. This is one of the physics tasks for a taucharm factory, and may be further studied with BESIII that isnow under construction.

BES Collaboration / Physics Letters B 660 (2008) 315–319 319

Fig. 2. (I) The comparison of R values between the values published in Ref. [15] (triangles: Rold) and the updated values in this work (points: Rnow). (II) Therelative differences between the two sets of R values.

It is worth noting that the change of the resonance para-meters affects the initial state radiative correction factors, and,thus, affects the R values. Fig. 2 shows a comparison betweenthe R values published in [15] and the values in this work; thedifferences vary with the resonant structure. In general, the rel-ative differences are within 3%, but for a few energy pointsthe maximum difference is about 5%. These values of R arein agreement with the previous determination within the givenerrors. The differences are mainly due to the retreatment ofthe radiative correction using the new resonance parametersobtained in this work and the inclusion of interferences (withnon-zero phase angle) between the resonances. A reanalysis ofthe R value measurements between 3.7 and 5.0 GeV based onthese new resonance parameter is in progress [24].

Acknowledgements

The BES Collaboration thanks the staff of BEPC and com-puting center for their hard efforts. This work is supported inpart by the National Natural Science Foundation of China un-der contracts Nos. 19991480, 19805009, 19825116, 10491300,10225524, 10225525, 10425523, the Chinese Academy of Sci-ences under contract No. KJ 95T-03, the 100 Talents Pro-gram of CAS under Contract Nos. U-11, U-24, U-25, andthe Knowledge Innovation Project of CAS under ContractNos. U-602, U-34 (IHEP), the National Natural Science Foun-dation of China under Contract No. 10225522 (Tsinghua Uni-versity), and the Department of Energy under Contract No.DE-FG02-04ER41291 (U. Hawaii).

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