Derived Born cross sections of $e^+e^-$ annihilation into open charm mesons from CLEO-c measurements

The exclusive Born cross sections of the production of $D^0$, $D^+$ and $D_s^+$ mesons in $e^+e^-$ annihilation at 13 energy points between 3.970 and 4.260 GeV are obtained by applying corrections for initial state radiation and vacuum polarization to the observed cross sections measured by CLEO-c experiment. Both the statistical and the systematic uncertainties for the obtained Born cross sections are properly estimated.


Introduction
In the energy range between the mass of ψ(3770) (just above the smallest open charm production threshold of D 0D0 ) and 4.7 GeV which contains quite a few open charm production thresholds, several vector charmonium-like structures (known as Y states, e.g. Y (4260), Y (4360), Y (4660) ...) were discovered in the e + e − annihilation into the hidden-charm final states [1][2][3][4][5][6] over the past decade. The strong coupling of them to the hidden-charm final states and the difficulty in fitting all of the Y states to the potential model prediction [7] indicate that they could be exotic particle candidates, like hybrids, tetraquarks, meson molecules and so on [8]. The exclusive cross sections of e + e − annihilation into open charm final states could give the detailed coupling strength of these charmonium-like states to the open charm final states and could help understanding nature of these charmonium-like states. Such exclusive open charm cross sections have been measured by BaBar [9][10][11] and Belle [12][13][14][15][16][17] experiments through initial states radiation (ISR) and by CLEO-c [18] experiment with a set of scan data. The errors of CLEO-c's results are relatively small but the observed exclusive cross sections are not corrected for the radiative correction. As the event selection efficiency in this CLEO-c measurement does not depend on the radiative correction, it is possible to apply the radiative correction directly to the observed cross sections to get the Born cross sections which have clear definition and are more convenient in results comparison and theoretical applications.
In this paper, we calculate the ISR correction to the observed exclusive cross sections of the open charm production in e + e − annihilation measured by CLEOc [18], including the channels e + e − → D 0D0 , D * 0D0 + c.c., D * 0D * 0 , D + D − , D * + D − + c.c., D * + D * − , D + s D − s , D * + s D − s +c.c., D * + s D * − s , D * D π +c.c., and D * D * π. With the calculated ISR correction and the vacuum polarization (VP) correction from Ref. [19], we obtain the exclusive Born cross sections of the open charm production in e + e − annihilation. The uncertainties of the final Born cross sections from both the statistical errors of the original measurements and the method to calculate the ISR correction are estimated properly. By summing up the radiatively corrected exclusive open charm cross sections, we also obtain the Born-level inclusive open charm cross section.
This paper is organized as follows: the theoretical formulae for the radiative correction is described first, followed by the calculation procedure, and at last the final results are presented.

Radiative correction
At a center-of-mass energy √ s, the experimentally observed cross section (σ obs ) of e + e − annihilation is the integral of the radiative function F (x, s) and the dressed cross section x))| 2 [20], where σ B (s) is the Born cross section and 1/|1 − Π(s)| 2 is the VP factor. The integral variable x = s /s, where s is the e + e − center of mass energy squared after emitting ISR photons. The upper limit of the integral x m = 1−s m /s, where √ s m is the production threshold of a specific channel. The radiative function F (x, s) is expressed as where α is the fine-structure constant and m e is the mass of the electron.
To extract the Born cross section from the observed one, we define 1 + δ(s) = σ dre (s)/σ obs (s) (6) as the ISR correction factor and factorize the Born cross section as 3 Calculation of ISR correction factors

Methods to connect the data
To calculate the ISR correction factor, we need full information of the observed cross section σ obs depending on the energy square s, not only the discrete experimental data. Five methods to connect the observed cross sections at discrete energy points (the cross section at the threshold is fixed to zero) are considered and listed below.
1. Linear interpolation, connecting the experimental data points with straight lines.
2. Cubic spline interpolation with natural boundary condition [21]. The second derivatives at both ends are zero.
(Only the linear interpolation is considered for e + e − → D * + s D * − s and D * D * π.) In addition, BaBar and Belle have published the exclusive cross sections of some channels of charm production in e + e − annihilation at the energy region overlapping with that of CLEO-c. In order to make our results more reliable, for a certain channel we combine the data of CLEO-c with those of BaBar or Belle and connect them with linear interpolation as an additional connection method, as illustrated in Fig. ??. To be specific, we consider the data of BaBar [10] for the channel e + e − → D * 0D0 + c.c, Belle [12] for D 0D0 , D + D − , Belle [13] for D * + D − + c.c., D * + D * − , and Belle [16] for Since both BaBar and Belle obtain their results using ISR method, we do not apply the ISR correction to their data. where the Belle data Ref. [13] and CLEO-c data are combined (only statistical errors shown here). This curve serves as the initial input of the ISR factor calculation for the additioanl connection method and then is to be updated within the iterative procedure described later in the text.
The comparison of the six curves for the channel e + e − → D * + D − + c.c. is shown in Fig. 2 For the channel e + e − → D 0D0 and e + e − → D + D − , the energy points, from √ s = 3.970 to 4.260 GeV at which CLEO-c [18] experiment is performed, are far from the production thresholds around 3.74 GeV. We use the experimental data from Belle [12] to fill the energy gap.

Iteration
The ISR correction factor is obtained in an iterative method via with σ dre 0 (s) = σ obs (s). The iteration is continued until the difference between the two consecutive results is smaller than the given upper limit, 1% of the statistical error of the observation. The results from the last iteration, denoted by σ dre f (s) and 1 + δ f (s), are regarded as the final dressed cross section and ISR correction factor, respectively.
For example, following the above iteration procedure, we get the ISR correction factors for e + e − → D * + D − +c.c. with the linear interpolation and the results converge fast as shown in Fig. 3

Dressed cross sections
We can get the dressed cross sections simultaneously as the ISR correction factors from the observed cross sections via Eq. (10). Besides the connection methods, the errors of the observed cross sections also have impacts on the obtained dressed cross sections. The systematic errors of CLEO-c experiment are postulated to be the same percentage of the central values of the observed cross sections (TABLE III in Ref. [18]), which have no modification to the line shape and in turn the ISR correction factors. (For e + e − → D 0D0 and D + D − , the systematic errors should be treated in the same way as the statistical errors since Belle data are involved and their systematic errors are independent of those of CLEO-c data.) While considering the statistical errors of the observed cross sections which are independent between energy points, the line shape and the obtained dressed cross sections could be different. To investigate the effects of the statistical errors of the observation on the dressed cross sections, we perform samplings of the σ dre 0 (s) according to a Gaussian distribution at each energy point, where the mean value and the standard deviation of the Gaussian distribution are the central value and the statistical error of the observed cross section, respectively. (The negative cross sections from the samplings are discarded.) Following the iteration procedure with a certain connection method, we obtain the dressed cross sections for each sampling.
After 30,000 samplings we get a distribution of the dressed cross section at each energy point. We find that most results satisfy Gaussian distribution so the fitted mean and standard deviation are taken as the central value and the statistical error of the dressed cross section. However, at some energy point the observed cross section is so small that it is possible to get some sampled values which yield very small (going to zero during iterations) ISR factors or equivalently very small dressed cross sections. It also indicates that at these energy points all the observed events could be from ISR. For these results, if there is a small amount of σ dre f (s) accumulating at zero, we ignore it and still try to fit the sampling results with Gaussian distributions (Fig. 4(a)), and if the accumulation at zero dominates, we try to estimate the upper limit for the dressed cross section (Fig. 4(b)).
where max({x i }) yields the maximum one among x i . Note that ISR± sys is just an estimation of the uncertainty from connection methods. For the simulation results described by the 90% upper confidence limit, we just take the biggest upper limit among these different connection methods as the result considering the systematic uncertainties from the correction methods.

VP factors and Born cross sections
To obtain the Born cross sections, we divide the dressed cross sections by the VP factors 1/|1 − Π(s)| 2 , which are calculated in Ref.
[19] and listed in Appendix A. The obtained exclusive Born cross sections are shown in Fig. 6 and listed in Appendix B.  Summing up all the exclusive cross sections in Fig. 6 we can obtained the inclusive Born cross section. At most of the energy points the exclusive results have symmetric statistical errors so the summing up is straight forward. For some energy points, part of the exclusive results have asymmetric statistical errors. In such cases, we try to obtain the summed results also through the toy MC method. We sum the exclusive cross sections up after a certain sampling for each channel and get distributions for the summed cross section after the 30,000 samplings (see section 4). These distributions are quite symmetric and can be fitted by the Gaussian distribution to yield the central value σ B inc and the statistical error ISR sta,inc . The systematic errors from ISR for the summation are all estimated by assuming the systematic errors of different exclusive channels are independent, where i is the index of exclusive channels. As mentioned in Section 4, the systematic errors of the original measurement are assumed to be the same percentage of the central values within a certain channel. For the Born cross section, the same percentage of the central values are taken directly as part of the systematic errors for this channel, denoted by CLEOc sys,i . Assuming that this part of systematic errors for different channels are independent, they are summed up in the same way as Eq. (13) to yield this part of the systematic error of the inclusive Born cross section, denoted by The comparison between the inclusive Born cross section obtained by this study and by CLEO-c is illustrated in Fig. 7. The top figure shows the cross sections and the bottom one shows the differences between the nominal overall radiative correction factors (1 + δ) · |1 − Π| 2 = σ B /σ obs . No significant difference is observed. In principle, it is more reasonable to obtain the inclusive cross section by summing up the exclusive ones because by treating each channel separately the individual open-charm production thresholds and different exclusive cross section line shapes can be implemented in the ISR correction.

Summary and discussion
In summary, we have obtained the Born cross sections for charm production in e + e − annihilation after correcting the observed cross sections measured by CLEO-c experiment for initial states radiation and vacuum polarization. A Monte Carlo sampling method is used to estimate the uncertainty of the corrected cross sections caused by the original statistical errors. The uncertainty from the connection of the observations at discrete energy points is estimated with different connection methods when we determine the errors of the Born cross sections and regarded as a part of the systematic error. It turns out that such systematic errors are of the same order as the statistical ones. In addition, we make our results more reliable by taking the corresponding data obtained by Belle and BaBar collaboration into consideration.
The Born cross sections for the exclusive open charm meson productions in e + e − annihilation, which are from the observation at CLEO-c corrected for ISR and VP effects, can provide more conventional information to determine the relative decay strength into different charm mesons for the charmonium(-like) states in the open charm region and can be easily compared or combined with the results from other experiments. Besides, the summation of the radiatively corrected exclusive cross sections, which is more reliable than the one obtain by correcting the observed inclusive cross section directly, can be used in the calculation of the anomalous magnetic moment of muon g−2 and the vacuum polarization factor 1/|1 − Π(s)| 2 . The Born cross sections for final states consisting of two charged nonstrange mesons. The first error of each cross section is statistical and the second is systematic ( ISR sys ). The upper limits are at 90% confidence level.  Table 5. The Born cross sections for final states consisting of two charm mesons and an extra pion. The first error of each cross section is statistical and the second is systematic ( ISR sys ). The systematic error ( ISR sys ) for e + e − → D * D * π is not estimated so only the statistical one is presented.