Production of $D^{(*)}\bar{D}^{(*)}$ near the thresholds in $e^{+}e^{-}$ annihilation

It is shown that the nontrivial energy dependencies of $D\bar{D}$, $D\bar{D}^{*}$, and $D^{*}\bar{D}^{*}$ pair production cross sections in $e^{+}e^{- }$ annihilation are well described within the approach based on account for the final-state interaction of produced particles. This statement is valid for production of charged and neutral particles. Interaction of $D^{(*)}$ and $\bar{D}^{(*)}$ is taken into account using the effective potential method. Its applicability is based on the fact that for near-threshold resonance the characteristic width of peak in the wave function is much larger than the interaction radius. The transition amplitudes between all three channels play an important role in the description of cross sections. These transitions are possible since all channels have the same quantum numbers $J^{PC}=1^{--}$.

Natural explanation of near-threshold resonances is based on account for the interaction of produced particles.In this approach, resonances arise in two cases (see, e.g., [28,29] and references therein).In the first case, there is a bound state with the binding energy much less than the characteristic value of the interaction potential (about several hundreds of MeV).In the second case, there is no loosely bound state but a slight increase in the depth of potential leads to appearance of such state (this is the so-called virtual level).In both cases, at scattering of produced hadrons on each other, the modulus of scattering length significantly exceeds the characteristic potential size (of the order of 1 fm).At the same time, the wave function at small distances calculated with account for the interaction of produced hadrons has characteristic value much larger than that without account for the interaction.The ratio of squares of the modules of corresponding wave functions for the relative angular momentum l = 0 (or their derivatives for l = 1) is the amplification factor, which can be very large.As a result, resonant structures arise in the particle production cross section.Currently, more and more scientists are coming to the conclusion that taking into account the interaction in the final state is of crucial importance for the correct description of cross sections in the near-threshold region (see, e.g., [30] and references therein).
The description of final-state interaction becomes noticeably more complicated, when there are several nearthreshold resonances with the same quantum numbers and thresholds located close to each other.As a result, non-zero transition amplitudes between resonances arise, which leads to to a significant distortion of the resonance shape.In our recent work [29], we have discussed various cases of coupled channels, where each channel is either loosely bound or virtual state.Moreover, it is shown in Ref. [29] that the account for the final-state interaction allows one to successfully describe the B ( * ) B( * ) production near the thresholds in e + e − annihilation.Similar results for the system of B ( * ) B( * ) mesons were obtained in Ref. [31] using the K-matrix approach.
In this work, the processes e + e − → D ( * ) D( * ) near the thresholds are discussed.Our approach is based on account for the final-state interaction in the case of coupled channels.Certainly, our information on the interaction potential between D mesons is very limited.However, it is not necessary to know these potentials very precisely.As already mentioned, the characteristic size of a peak in the wave function of produced D ( * ) D( * ) system near the threshold is much larger than the characteristic size of the potential.Therefore, specific shapes of the potentials are not important.
Pairs D D, D D * , and D * D * are produced in e + e − annihilation in the states with quantum numbers J P C = 1 −− .In this case, the relative angular momentum of produced particles is l = 1.Due to C-parity conservation, the total spin S of D * D * pair can be either S = 0 or S = 2.However, due to the lack of experimental data for individual spin states in the D * D * channel, we will talk on the total cross section for the production of these states with different spins.At small distances r ∼ 1/ √ s, where √ s is the total energy of electron and positron in the center-of-mass frame, a hadronic system is produced as cc pair and, therefore, has isospin I = 0.However, at large distances r ≳ 1/Λ QCD the difference in masses of charged and neutral D mesons (D * mesons), as well as the Coulomb interaction between charged particles, leads to violation of isospin invariance.Thus, we have six states with C = −1: Taking into account violation of isospin invariance, we conclude that it is necessary to solve the six-channel problem.The threshold of Ψ 1 state production is 3730 MeV.We will count the remaining thresholds ∆ i from this value.Therefore, ∆ 1 = 0, ∆ 2 = 9.6 MeV, ∆ 3 = 142 MeV, ∆ 4 = 150 MeV, ∆ 5 = 284 MeV, and ∆ 6 = 291 MeV.
The radial Schrödinger equation, which describes our six-channel system, has the form where −p 2 r is the radial part of the Laplacian, MeV is the D 0 mass, E is the energy of a system counted from the threshold of D 0 D0 production, and l = 1.The wave function T consists of radial parts ψ i (r) of wave functions of states Ψ i , index T denotes transposition.The matrices V ij are symmetric blocks of dimension 2 × 2 having the form where the diagonal potentials correspond to the transitions without change of particle electric charges, and off-diagonal ones describe processes with charge exchange.These potentials contain contributions from isoscalar and isovector exchange, U ij (r) and U (1) ij (r), respectively.All potentials can be parameterized as Here θ(x) is the Heaviside function, u (I) ij and a (I) ij are some constants that are found from comparison of theoretical predictions with experimental data.
Each solution is determined by the asymptotic behavior at r → ∞, where S (m) i are some coefficients.The cross sections σ (m) of pair production in the states Ψ m have the form Here β m = k m /M D , g i are some constants that determine the production of corresponding states at small distances, ψ(m) i (r) = ∂/∂r ψ (m) i (r).Since an isoscalar state is produced at small distances, then g 1 = g 2 , g 3 = g 4 , and g 5 = g 6 .

III. RESULTS
In Refs.[32][33][34][35][36][37][38][39] detailed experimental data on cross sections σ (m) have been obtained for all m.Parameters u ij , and g i of our model are determined by comparing predictions with all experimental data listed above.We analyze data for energies E up to 450 MeV since, on the one hand, we want to cover the range of thresholds of all six channels, and on the other hand, we use a non-relativistic model and cannot consider too high energies.
provide the best agreement with experiment are given in Table I.Constants, that determine production of different states at small distances, have the values g 1 = g 2 = 0.069, g 3 = g 4 = 0.003 + 0.169 i, and g 5 = g 6 = 0.429 − 0.156 i.As a result of fitting, we have obtained χ 2 /N df = 325/301 = 1.08,where N df is the number of degrees of freedom.The latter equals to the difference between the number of experimental points and the number of parameters in the model.
Figs. 1 and 2 show a comparison of our theoretical predictions with experimental data from Refs.[32][33][34][35][36][37][38][39].It is seen that good agreement of predictions with experimental data is obtained over the entire energy range under consideration.In particular, recent data from Ref. [39] is perfectly described by our model.Few experimental points lie outside of our theoretical predictions, but this is related to the fact that these points have large experimental uncertainties and are not consistent with each other.
The cross sections of different D meson pair production have very non-trivial energy dependencies.There are many peaks of various shapes, as well as sharp gaps between them.Note that experimental data obtained for all six charged and neutral channels have high accuracy.Therefore, for simultaneous description of the cross sections of these processes, it is necessary to take into account all six channels and all possible transitions between them.All potentials (diagonal and off-diagonal, with charge exchange and without charge exchange) are important to obtained good agreement between theory and experiment.

IV. CONCLUSION
It is shown that the final-state interaction in the system of D ( * ) mesons explains the nontrivial energy dependence of the cross sections of e + e − → D ( * ) D( * ) annihilation.Interaction between D ( * ) mesons is described using the effective potentials.Their parameters are determined from comparison of experimental data with theoretical predictions in each channel.Good agreement is obtained for the cross sections of charged and neutral pair production.We emphasize again that, to obtain a good description of experimental data for the processes e + e − → D ( * ) D( * ) , it is necessary to take into account all six channels simultaneously and all transitions between them.
Quite recently, a work [40] has appeared, where the cross sections of processes e + e − → D ( * ) D( * ) have been described using K-matrix approach.Although the approach of Ref. [40] differs significantly from ours and the experimental data averaged over isospin are used in the channels D * D and D * D * , the results of Ref. [40] are consistent with ours qualitatively and quantitatively.

TABLE I .
Parameters of interaction potentials defined by Eqs. (