Measurements of the cross sections for $e^+e^- \to {\rm hadrons}$ at 3.650, 3.6648, 3.773 GeV and the branching fraction for $\psi(3770)\to {\rm non-}D\bar D$

Using the BES-II detector at the BEPC Collider, we measured the lowest order cross sections and the $R$ values ($R=\sigma^0_{e^+e^- \to {\rm hadrons}}/\sigma^0_{e^+e^- \to \mu^+\mu^-}$) for inclusive hadronic event production at the center-of-mass energies of 3.650 GeV, 3.6648 GeV and 3.773 GeV. The results lead to $\bar R_{uds}=2.224\pm 0.019\pm 0.089$ which is the average of these measured at 3.650 GeV and 3.6648 GeV, and $R=3.793\pm 0.037 \pm 0.190$ at $\sqrt{s}=3.773$ GeV. We determined the lowest order cross section for $\psi(3770)$ production to be $\sigma^{\rm B}_{\psi(3770)} = (9.575\pm 0.256 \pm 0.813)~{\rm nb}$ at 3.773 GeV, the branching fractions for $\psi(3770)$ decays to be $BF(\psi(3770) \to D^0\bar D^0)=(48.9 \pm 1.2 \pm 3.8)%$, $BF(\psi(3770) \to D^+ D^-)=(35.0 \pm 1.1 \pm 3.3)%$ and $BF(\psi(3770) \to D\bar{D})=(83.9 \pm 1.6 \pm 5.7)%$, which result in the total non-$D\bar D$ branching fraction of $\psi(3770)$ decay to be $BF(\psi(3770) \to {\rm non}-D\bar D)=(16.1 \pm 1.6 \pm 5.7)%$.


I. INTRODUCTION
proceeds via quark-antiquark pair production where the photon couples directly to the charge of the pointlike quarks. A consequence of this picture is that the total lowest order cross section, σ B had , for inclusive hadron production in e + e − annihilation must be proportional to the lowest order cross section, σ B µ + µ − , for muon pair production, which results in the relation where Q i is the charge of the ith quark; the factor of 3 accounts for three different colors of quarks; the sum runs over all quark flavors, N f , involved, for which the quark pair production thresholds are below the e + e − annihilation energy. The Eq. (1) indicates the ratio to be constant as long as the c.m. (center-of-mass) energy E cm does not overlap with resonances or the threshold of the production of new quark flavors. It also indicates that the R uds value for continuum light hadron (containing u, d and s quarks) production should tend to be constant in the energy region above 2 GeV. This naive theoretical prediction for the R value has to be modified to take into account the finite mass of the quarks and the emission of the gluons by the quarks. In principle, the R values can be computed in the pQCD (perturbative QCD) with these corrections. So precise measurements of R values at low energy region are important for the test of the prediction by the pQCD in this energy region. Moreover the R values at all energies are needed to calculate the effects of vacuum polarization on the parameters of the Standard Model. For example, the dominant uncertainties in the quantities α(M 2 Z ), the QED running coupling constant evaluated at the mass of Z 0 , and a µ = (g −2)/2, the anomalous magnetic moment of the muon, are due to the calculation of hadronic vacuum polarization [1]. A large part of uncertainty in the calculation arises from the uncertainties in the measured inclusive hadronic cross sections in open charm threshold region, in which many resonances overlap. To get credible measurements of R and various lowest order cross sections in this region, the overlapping effects have to be clarified clearly.
On the other hand, the measurements of the R values below and above the threshold of DD production can be used to determine the branching fractions for ψ(3770) → D 0D0 , D + D − , DD, and for ψ(3770) → non−DD with the measured cross sections for the D 0D0 and D + D − together. The ψ(3770) resonance is believed to decay predominantly into DD [2]. However, there are discrepancies between the measurements of the DD cross section and the measurements of ψ(3770) cross section which can be obtained from ψ(3770) resonance parameters. In recent days, there are some publications to report the observation of non−DD decays of ψ(3770) res-measurements of the R values at the c.m. energies of 3.650, 3.6648 and 3.773 GeV. With the measured R values and the previously measured cross sections for DD production at 3.773 GeV [6], we determine the branching fractions for ψ(3770) → D 0D0 , D + D − , DD, and for ψ(3770) → non−DD

II. BES-II DETECTOR
The BES-II is a conventional cylindrical magnetic detector that is described in detail in Ref. [7]. A 12-layer Vertex Chamber (VC) surrounding the beryllium beam pipe provides input to the event trigger, as well as coordinate information. A forty-layer main drift chamber (MDC) located just outside the VC yields precise measurements of charged particle trajectories with a solid angle coverage of 85% of 4π; it also provides ionization energy loss (dE/dx) measurements which are used for particle identification. Momentum resolution of 1.7% 1 + p 2 (p in GeV/c) and dE/dx resolution of 8.5% for Bhabha scattering electrons are obtained for the data taken at √ s = 3.773 GeV. An array of 48 scintillation counters surrounding the MDC measures the time of flight (TOF) of charged particles with a resolution of about 180 ps for electrons. Outside the TOF, a 12 radiation length, leadgas barrel shower counter (BSC), operating in limited streamer mode, measures the energies of electrons and photons over 80% of the total solid angle with an energy resolution of σ E /E = 0.22/ √ E (E in GeV) and spatial resolutions of σ φ = 7.9 mrad and σ Z = 2.3 cm for electrons. A solenoidal magnet outside the BSC provides a 0.4 T magnetic field in the central tracking region of the detector. Three double-layer muon counters instrument the magnet flux return and serve to identify muons with momentum greater than 500 MeV/c. They cover 68% of the total solid angle.

III. MEASUREMENT OF THE OBSERVED HADRONIC CROSS SECTIONS
For a sample of data taken at c.m. energy E cm,i , the observed cross section for the inclusive hadronic event production is obtained by where i is the ith energy point at which the data were collected, N had (E cm,i ) is the number of the inclusive hadronic events observed at this energy; L(E cm,i ) is the integrated luminosity of the data collected; ǫ had (E cm,i ) is the efficiency for detection of the inclusive hadronic events, and ǫ trig had is the trigger efficiency for collecting the

A. Measurement of luminosity
The integrated luminosities of the data sets are determined by where N e + e − (E cm,i ) and n b are the number of the selected Bhabha events and the number of background events respectively, ǫ e + e − (E cm,i ) is the efficiency for detection of the Bhabha events, ǫ trig e + e − is the trigger efficiency for collecting the Bhabha events in on-line data acquisition. For the data used in the analysis, the trigger efficiency is ǫ trig e + e − = (100.0 +0.0 −0.5 )% (see subsection D). To select the candidates for Bhabha scattering e + e − → (γ)e + e − , it is first required that exactly two charged tracks with total charge zero be well reconstructed. For each track, the point of the closest approach to the beam line must have the radius ≤ 1.5 cm and |z| ≤ 15 cm where |z| is measured along the beam line from the nominal beam crossing point. Furthermore, each track is required to satisfy | cos θ| ≤ 0.7, where θ is the polar angle of the charged track, to ensure that it is contained within the barrel region of the detector. Next, it is required that the energy deposited for each charged track in BSC be greater than 1.1 GeV (i.e. E track BSC > 1.1 GeV) and at least the magnitude of one charged track momentum be greater than 0.9E b , where E b is the beam energy. Figure 1 shows the distribution of the energies deposited for muons (hatched histogram) and electrons or positrons (points with error bars) in the BSC, where the data sample of the muons and the electrons or positrons are selected from the decays of ψ(2S) → J/ψπ + π − , and J/ψ → µ + µ − or e + e − . From Fig. 1 one can see that the criterion E track BSC > 1.1 GeV separates the e + e − → (γ)µ + µ − from the Bhabha scattering effectively. In addition, because the Monte Carlo simulation does not model the energy deposited well in the rib regions of the BSC, any charged track from the selected Bhabha events is required to hit one of the four regions of the BSC (selected z regions in BSC): (1) z BSC ≤ −1.04 m, The two oppositely charged tracks go in opposite directions in the R − φ plane. Because the tracks are bent in the magnetic field, the positions of the two shower clusters in the R − φ plane of the BSC are deviated from the back-to-back directions. We define the angle difference of the two clusters by δφ = |φ 1 −φ 2 |−180 o in degrees, where the φ 1 and φ 2 are the azimuthal angles of the two clusters in the BSC. Figure 2 shows the δφ distribution for the events which satisfy the selection criteria for the Bhabha events. These events are from a portion of the data taken at 3.773 GeV. Using a double Gaussian function plus a second order polynomial to fit the distribution, we obtain the number of the candidates for e + e − → (γ)e + e − . The contaminations by visual scan. The detailed scans for the accepted e + e − → (γ)e + e − events show that about 0.5% of the accepted events are due to background contamination. After subtracting the background, the pure number of e + e − → (γ)e + e − events is retained.
The detection efficiency ǫ e + e − for the Bhabha scattering e + e − → (γ)e + e − is determined by analyzing the Monte Carlo events of e + e − → (γ)e + e − . These events are generated with the radiative Bhabha generator [8] written by R. Kleiss et al., which includes hard photon emission and α 3 radiative correction.
Using the pure number selected, the visible cross section σ e + e − read from the generator, the detection efficiency for e + e − → (γ)e + e − obtained by Monte Carlo simulation, and the trigger efficiency, we can determine the integrated luminosity of the data from Eq. (4). Applying the procedure to the data sets taken at the three energy points, we get the measured integrated luminosities of the data sets. The second column of table I lists the integrated luminosities of the data sets, where the errors are combined from statistical and systematic errors.
The systematic uncertainty in the measured values of the luminosities arises mainly from the difference between the data and Monte Carlo simulation. Table II  TABLE I: Summary of the luminosities of the data sets, the numbers of the selected candidates for e + e − → hadrons and the estimated numbers of the events of the processes e + e − → l + l − (l = τ, e, µ), e + e − → e + e − l + l − and e + e − → e + e − hadrons which were misidentified as the events of e + e − → hadrons. In order to effectively remove the e + e − → (γ)e + e − and e + e − → (γ)µ + µ − events from the selected hadronic event sample, the hadronic events are required to have more than 2 good charged tracks, each of which is required to satisfy the following selection criteria: • the charged track must be with a good helix fit and the number of dE/dx hits per charged track is required to be greater than 14; • the point of the closest approach to the beam line must have radius ≤ 2.0 cm; • | cos θ| ≤ 0.84, where θ is the polar angle of the charged track; where p is the charged track momentum and E b is the beam en- the time-of-flight of the charged particle, and T p is the expected time-of-flight of proton with the given momentum; • the charged track must not be identified as a muon; • for the charged track, the energy deposited in BSC should be less than 1.0 GeV.
In addition, the total energy deposited in BSC should be greater than 28% of the beam energy. Furthermore, the selected tracks must not all point into the same hemisphere in the z direction. No criterion for the number of the observed photons is applied to the selected hadronic events. Some beam-gas associated background events can also satisfy above selection criteria. However, the beam-gas associated background events are produced at random z positions, while the hadronic events are produced around z = 0. This characteristic can be used to distinguish the hadronic events from the beam-gas associated background events. To this end, the averaged z of the charged tracks in each event is calculated. Figure 3 shows the distribution of the averaged z. These events are from a portion of the data taken at 3.773 GeV. In Fig. 3, the points with error bars show the events from the Monte Carlo sample which is generated with the generator [9] described in Section III.C and simulated with the GEANT3-based Monte Carlo package [10], the histogram shows the events from the data, and the shadowed histogram shows the events from the separated beam data. Using a Gaussian function plus a second order polynomial to fit the averaged z distribution of the events, we obtain the number, N zfit had , of the candidates for hadronic events. The third column of table I lists N zfit had obtained from the data sets taken at each of the energy points, where the errors are combined from statistical and systematic errors. This number of candidates contains some contaminations from some background events such as e + e − → τ + τ − , e + e − → (γ)e + e − , e + e − → (γ)µ + µ − and two-photon exchange processes. The number of the background events can be estimated by means of Monte Carlo simulation   Carlo sample, the histogram shows the events from the data, and the shadowed histogram shows the events from the separated beam data; the curves give the best fit to the z distribution.
The systematic uncertainty in measuring the produced hadronic events due to the hadronic event selection criteria is estimated to be about 2.5%. Table III summarizes the relative systematic uncertainties in selecting the pro-C. Monte Carlo method and the efficiency ǫ had Due to ISR (Initial State Radiation), the effective c.m. energy for the e + e − annihilation is is the total energy of the emitted photons and √ s is the nominal c.m. energy. For a certain energy point in experiment, the experimentally observed hadronic events are not only produced at the √ s, but produced in the full energy range from the √ s to ∼ 0.28 GeV (for production of two pions). To determine the efficiency for detection of hadronic events produced in the full energy range, we developed a special Monte Carlo generator [9] in which the initial state radiative correction to α 2 order is taken into account. Figure 4 shows the differential cross section dσ/dE eff for the inclusive hadronic event production when setting the nominal c.m. energy to be at 3.80 GeV. At an effective c.m. energy, the final hadronic states are produced by calling the sub-generators such as LUND model [11] [12], and the resonance generators including ψ(3770), ψ(2S), J/ψ [13], φ(1020), ρ(770), and ω(782) etc. according to the corresponding lowest order cross sections of these processes, respectively. The CMD-2 ππ production data [14] with Gounaris-Sakurai parameterization [15] are used to simulate the spectrum of ρ(770) and the ρ-ω mixing in the energy range below 1.2 GeV. The resonances are set to decay into all possible final states according to the known decay modes and branching fractions. These generated events are simulated with the GEANT3-based Monte Carlo simulation package. The reconstructed Monte Carlo events are then fed into the analysis program to determine the efficiencies, ǫ had , for measurements of the observed cross sections for inclusive hadronic event production at each of the three energy points.
For simulations of the inclusive hadron production, parameters in the LUND generator are tuned using an inclusive hadronic event sample of 5.5 × 10 5 events from the data taken at 3.65 GeV with the BES-II detector. The parameters are adjusted to reproduce good agreeable distributions of some main kinematic variables between data and Monte Carlo sample. The uncertainty in ǫ had due to the adjusted parameters is estimated to be ∼ 0.6%, while the uncertainties due to the errors of the ψ(3770) and ψ(2S) resonance parameters are estimated to be ∼ 1.5% and ∼ 1.2%, respectively. Combining these uncertainties in quadrature yields the systematic uncertainty in the efficiency ǫ had to be about 2%.
The second column of table IV lists the efficiencies for detection of the inclusive hadronic events at three energy points in the case of setting the continuum R value to be at 2.26 (see section IV).

D. Trigger efficiency
The requirements of the trigger for recording the data  data for the work [16] and the work [17]. However, for the ψ(3770) data acquisition, we slightly modified the trigger requirements for the charged tracks, which results in a little bit improvement in recording the two charged trak events. The trigger efficiencies are obtained by comparing the responses to different trigger requirements in the data taken at 3.097 GeV during the time period taking the data at √ s = 3.773 GeV. The trigger efficiencies are measured to be 100.0% for both the e + e − → (γ)e + e − and e + e − → hadrons events, with an uncertainty of +0.0 −0.5 %.

E. Observed hadronic event cross sections
The observed cross section for inclusive hadronic event where N b is the number of the background events, such as e + e − → τ + τ − , e + e − → (γ)e + e − , e + e − → (γ)µ + µ − and two-photon exchange processes. The number of the background events can be estimated by using the theoretical cross sections of these processes, the rates of misidentifying these processes as hadronic events and the total integrated luminosities of the data sets, which is given by where σ l + l − , σ e + e − l + l − , σ e + e − π + π − and σ e + e − hadrons are the cross sections for e + e − → l + l − , e + e − → e + e − l + l − , e + e − → e + e − π + π − , and e + e − → e + e − hadrons processes, respectively; while η l + l − , η e + e − l + l − , η e + e − π + π − , and η e + e − hadrons are the corresponding misidentification rates.
In the calculation of the τ + τ − cross section, we consider the contributions from ψ(2S) decay, the QED production and their interference; we also consider the effects of the initial and final state radiative corrections and Coulomb interaction on the cross section [18]. For e + e − → (γ)e + e − [8], e + e − → (γ)µ + µ − [19] and twophoton processes e + e − → e + e − l + l − [20], the cross sections are read from respective generator outputs. As for the estimate of the total cross section for e + e − → e + e − hadrons, we employ the equivalent photon approximation formalisms to deal with the γ-γ collision subprocess [21] [22]. In the sub-process, the energy dependence of the total hadronic cross section can be described well by the formula of Donnachie-Landshoff parameterization [23] above the three pion threshold. For the contribution in the low energy region below the three pion threshold, it is good enough to use the simple point-like π + π − production cross section in the calculation [21].
The rates of misidentifying the above processes as the hadronic events are obtained from Monte Carlo simulation with the generators mentioned above. For the two-photon process e + e − → e + e − hadrons, we use the Monte Calro generator described in [20] to simulate the process and determine the rates of misidentifying the processes as the hadronic events. The fourth and fifth columns of Table I give the estimated numbers (n l + l − , n e + e − l + l − and n e + e − h ) of the background events from the e + e − → l + l − and the two-photon exchange processes, which are misidentified as the inclusive hadronic events.
Inserting N zfit had , N b , L, ǫ had and ǫ trig had in Eq. (3), we obtain the observed cross sections for the inclusive hadronic event production at each of the three energy points, which are summarized in table IV, where the first error is statistical and second point-to-point systematic error arising from the uncertainty in ǫ e + e − (0.6%), uncertainty in N e + e − (∼ (0.2 ∼ 0.5%)) and uncertainty in ǫ had (0.5%). The common systematic uncertainty is not

A. Radiative corrections
To get the lowest order cross section for the inclusive hadronic event production in e + e − annihilation, the observed cross section has to be corrected for the radiative effects including the initial state radiative corrections and the vacuum polarization corrections. The correction factor, (1 + δ(s)), is given by where σ exp (s) is the expected cross section and σ B (s) lowest order cross section for the inclusive hadronic event production.
The expected cross section for hadronic event production can be written as where σ B (s(1 − x)) is the total lowest order cross section in the energy range from 0.28 GeV to √ s (or from 3.729 GeV to √ s in the case of considering the DD production), F (x, s) is a sampling function and 1 |1−Π(s(1−x))| 2 is the correction factor for the effects of vacuum polarization including both the leptonic and hadronic terms in QED [24], with the effects of hadronic vacuum polarization can be calculated via the dispersion integral [25] Π had (s where m l is the lepton mass. In the structure function approach by Kuraev and Fadin [24], where β is the electron equivalent radiator thickness, In above expressions, m e is the mass of electron and α is the fine structure constant. For the resonances, such as ψ(2S), J/ψ, φ and ω, we use the Breit-Wigner formula to calculate the lowest order cross section, where M and Γ are the mass and the total width of the resonance, and Γ ee = Γ 0 ee |1 − Π(s ′ )| −2 and Γ h are the partial widths to the e + e − channel and to the inclusive hadronic final state, respectively. For the ψ(3770) resonance, we use to calculate the lowest order cross section, where Γ tot (s ′ ) is chosen to be energy dependent and normalized to the total width Γ tot at the peak of the resonance. The Γ tot (s ′ ) is defined as where Γ D 0D0 (s ′ ), Γ D + D − (s ′ ) and Γ non−DD (s ′ ) are the partial widths for ψ(3770) → D 0D0 , ψ(3770) → D + D − and ψ(3770) → non − DD, respectively, which are taken in the form and where p 0 D and p D are the momenta of the D mesons produced at the peak of ψ(3770) and at the c.m. energy √ s ′ , respectively; Γ 0 is the total width of the ψ(3770) at the peak, and r is the interaction radius of the cc, which is set to be 1.0 fm; B 00 and B +− are the branching fractions for ψ(3770) → D 0D0 and ψ(3770) → D + D − , respectively; θ(E cm − 2M D 0 ) and θ(E cm − 2M D + ) are the step functions to account for the thresholds of the DD production.
In the calculation of the lowest order cross section, the ψ(3770) resonance parameters M = 3772.3 ± 1.0 MeV, Γ 0 = 25.5 ± 3.1 MeV and Γ ee = 0.224 ± 0.031 keV measured by BES Collaboration [30] are used. Inserting the resonance parameters of J/ψ, ψ(2S) quoted from PDG [26] and the R = 2.26 ± 0.14 [6] [16] for the light hadron production in the energy range from 2.0 to 3.0 GeV measured by BES Collaboration in Eqs. (6)-(25), we obtain the radiative correction factors at the three energy points, which are summarized in table V. In determination of the value of (1 + δ(s)), the input of R value in calculating the cross section for continuum hadronic event production affects the value of (1 + δ(s)). Varying the input R by ±10% causes a variation of ±1.3% in (1 + δ(s)), which results in a variation of the product ǫ had (1 + δ(s)) by only ±0.4%. The most uncontrolled cross sections in the calculation of (1 + δ(s)) come from the hadronic cross sections in the energy range from 1.2 to 2.0 GeV. However, the whole contribution of the cross section from this energy range is less than 5% of the total observed cross section in our case. Since the efficiency is quite low for detection of the hadronic events from this energy range (ǫ had < 10%), the amount of the product ǫ had (1 + δ(s)) would also be rather stable with error less than 0.4%. Taking into account the uncertainty in the measured hadronic event production and the errors of the resonance parameters together, the total uncertainty in R measurement due to the calculation of (1 + δ(s)) is then estimated to be less than 1.5% in this work.

B. Lowest order cross sections and R values
The lowest order cross section for inclusive hadronic event production is obtained by where in which σ B e + e − →hadrons (s) is the cross section for inclusive hadronic event production through one photon annihilation, σ B Res,i (s) is the cross section for the ith resonance, such as J/ψ, ψ(2S), ψ(3770) etc. which decays into hadronic final states.
To obtain the lowest order cross section σ 0 h (s) = R uds · σ B µ + µ − (s) = σ B e + e − →hadrons (s) for the hadronic event production through one photon annihilation at the energies of 3.650 and 3.6648 GeV, and the lowest order cross section σ 0 h (s) = (R uds + R ψ(3770) ) · σ B µ + µ − = σ B e + e − →hadrons (s) + σ B ψ(3770) (s) for both one photon annihilation and ψ(3770) production at 3.773 GeV, the amount of the cross section due to the resonance production at the energies of 3.650 and 3.6648 GeV, and the amount of the cross section due to the resonance production but ψ(3770) at 3.773 GeV have to be subtracted out. The third column of table V summarizes the lowest order cross sections σ 0 h (s). Dividing the σ 0 h (s) by the lowest order cross section for µ + µ − production at the same c.m. energy, we obtain the R values, which are summarized in the fourth column of the table. The first error in the measured lowest order cross section and the R value listed in table V is statistical, the second is the pointto-point systematic and the third is common systematic error.
The common systematic error arises from the uncertainty in luminosity (∼ 1.8%), in selection of hadronic event (∼ 2.5%), in Monte Carlo Modeling (∼ 2.0%), in radiative correction (∼ 1.5%) for the measured cross sections and R values at the three energy points, and the uncertainty in ψ(3770) resonance parameters (∼ 2.7%) for those at 3.773 GeV only. Adding these uncertainties in quadrature yields the total systematic uncertainties to be ∼ 4.0% and ∼ 4.9% for the measured hadronic cross sections and R values for the data taken below the DD threshold and at 3.773 GeV, respectively.
Averaging the R values measured at the first two energy points (3.650 and 3.6648 GeV) by weighting the combined statistical and point-to-point systematic errors, we obtainR uds = 2.224 ± 0.019 ± 0.089, where the first error is combined from statistical and point-to-point systematic errors, and the second is common systematic. ThisR uds excludes the contribution from resonances and reflects the lowest order cross section for the inclusive light hadronic event production through one photon annihilation of e + e − . So it can be directly compared with those calculated based on the pQCD. The value is consistent withR uds = 2.26 ± 0.14 obtained by fitting those [12] measured in the energy region between 2.0 and 3.0 GeV [6] and withR uds = 2.21 ± 0.13 obtained from fitting to the inclusive hadronic cross sections for both the ψ(2S) and ψ(3770) resonances in the energy region from 3.666 to 3.897 GeV [30]. Figure 5 displays the values of R from this measure-  tion [12] [16], MARK-I Collaboration [27], γγ2 Collaboration [28] and PLUTO Collaboration [29] in the energy region between 2.85 and 3.90 GeV. The error bars shown in the figure are obtained by combining statistical and systematic errors in quadrature. Using the measured R value at 3.773 GeV listed in table V and theR uds value for light hadron production measured below the DD threshold we obtain the R ψ(3770) due to ψ(3770) decays to be R ψ(3770) = 1.569 ± 0.042 ± 0.133, (28) where the first error is combined from statistical and point-to-point systematic error and the second common systematic. In estimation of the systematic uncertainty, we assumed that the same amount of the systematic uncertainties in the measured values of the R and theR uds is canceled in subtracting theR uds from the R. The corresponding lowest order cross section for ψ(3770) production is Assuming that there are no other new structure and effects except the ψ(3770) resonance and the continuum hadron production in the energy region from 3.70 GeV to 3.86 GeV, the branching fraction for ψ(3770) → DD can be determined by where σ obs DD and σ B ψ(3770) are the observed and lowest order production cross sections for DD and inclusive hadronic events, respectively; (1 + δ) DD is the radiative correction factor for DD production. Inserting the ψ(3770) resonance parameters (M = 3772.3 ± 1.0 MeV; Γ tot = 25.5 ± 3.1 MeV and Γ ee = 0.224 ± 0.031 keV) measured by BES Collaboration [30] in Eqs. (6) and (7) with combining the Eqs. (7)-(25) together, we obtain the radiative correction factor (1 + δ) DD = 0.764 ± 0.014, (31) where the error is the uncertainty arising from the errors of the ψ(3770) resonance parameters, the uncertainty in vacuum polarization correction and the uncertainty arising from varying the branching fraction for ψ(3770) → DD from 84% to 100%. BES Collaboration measured the observed cross sections for D 0D0 and D + D − production at c.m. energy √ s = 3.773 GeV to be σ D 0D0 = (3.58 ± 0.09 ± 0.31) nb and σ D + D − = (2.56 ± 0.08 ± 0.26) nb [6]. These observed cross sections were obtained by analyzing the same data set from which the R value at √ s = 3.773 GeV is measured.
which results in the non-DD branching fraction of ψ(3770) to be ¯ where the first error is statistical and the second systematic arising from uncanceled systematic uncertainties. The uncanceled relative systematic uncertainties are ∼ 6.2%, ∼ 8.4% and ∼ 3.2% for the σ obs D 0D0 , σ obs D + D − and the σ B ψ(3770) , respectively. The systematic error also includes the common uncertainty of ∼ 2.7% arising from the statistical uncertainty in the measured lowest order cross section for ψ(3770) production and the uncertainty (∼ 1.8%) in radiative correction factor (1 + δ) DD . Table VI summarizes sources of the uncanceled systematic uncertainties for the measured σ obs DD and σ ψ(3770) . The uncertainties in luminosity (∼ 1.8%), in ψ(3770) resonance parameters (∼ 2.7%) and in radiative correction (∼ 1.5%) are canceled out in the estimation of the systematic uncertainty in the measured branching fractions.

VI. DISCUSSION ABOUT INTERFERENCE EFFECTS
The measured R values discussed in above sections are obtained based on the same treatment on the measurements of inclusive hadronic cross sections in which no interference between the inclusive hadronic final states of the resonance decays and the inclusive hadronic final states from non-resonance annihilation of e + e − is taken into account [12][16] [31]. However, since the c.m. energies of 3.650 GeV, 3.6648 GeV and 3.773 GeV are close to the ψ(2S) resonance, there may be interference effects between the final hadronic states from the ψ(2S) electromagnetic decays and the continuum hadron production + − TABLE VII: Summary of the radiative correction factors, the lowest order cross sections and the R values measured at three energy points, where the interference effects between the ψ(2S) electromagnetic decays and hadron production through non-resonant annihilation of e + e − are taken into account. tort the line shape of the continuum hadron production cross section around the ψ(2S) peak. With the definition of the R given in Eq.
(2), we can estimate the destructive/constructive amount of the cross section due to the interference effects, which is given by The destructive/constructive amounts of the cross sections given in R are estimated to be −0.0581, −0.1026 and +0.0219 at 3.650 GeV, 3.6648 GeV and 3.773 GeV, respectively. After correcting the cross section σ 0 h (s) for the destructive/constructive amounts due to the interference effects, we obtained the lowest order cross section σ 0 crr h (s). The third column of table VII summarizes the σ 0 crr h (s). In the case of considering the interference effects, the correction factor (1+δ(s)) is also changed. The second column of table VII lists the correction factor at the c.m. energies. Dividing the σ 0 crr h (s) by the lowest order cross section for µ + µ − production at the same c.m. energy, we obtain the R values, which are summarized in the fourth column of the table. The errors are statistical, the point-to-point systematic and the common systematic as discussed before.
The weighted average of the R values measured at the first two energy points is R uds = 2.268 ± 0.019 ± 0.091, where the first error is combined from statistical and point-to-point systematic errors, and the second is common systematic.
Following the same procedure as discussed in Sections IV and Section V, we obtained the lowest order cross section for ψ(3770) production to be where the errors are statistical and the systematic arising from some uncanceled systematic uncertainties.