Electronic width of the $\psi(3770)$ resonance interfering with the background

Methods for extracting the $\psi(3770)\to e^+e^-$ decay width from the data on the reaction cross section $e^+e^-\to D\bar D$ are discussed. Attention is drawn to the absence of the generally accepted method for determining $\Gamma_{\psi(3770)e^+e^-}$ in the presence of interference between the contributions of the $\psi(3770)$ resonance and background. It is shown that the model for the experimentally measured $D$ meson form factor, which satisfies the requirement of the Watson theorem and takes into account the contribution of the complex of the mixed $\psi(3770)$ and $\psi(2S)$ resonances, allows uniquely determine the value of $\Gamma_{\psi(3770)e^+e^-}$ by fitting. The $\Gamma_{\psi(3770)e^+ e^-}$ values found from the data processing are compared with the estimates in the potential models.

The charmonium state ψ(3770) was investigated in e + e − collisions by the MARK-I [13,14], DELCO [15], MARK-II [16], BES [17][18][19][20][21][22][23][24][25], CLEO [26][27][28], BABAR [29,30], Belle [31], and KEDR [32] Collaborations. The ψ(3770) production was also observed in the B + → DDK + decays by the Belle [33,34], BABAR [35,36], and LHCb [37] Collaborations. Full compilation of the ψ(3770) production experiments is contained in the review of the Particle Data Group (PDG) [1]. The unusual shape of the ψ(3770) resonance peak, discovered in many experiments [20,21,[23][24][25][29][30][31][32], naturally became the subject of many-sided theoretical analyzes, see, for example, Refs. [38][39][40][41][42][43][44][45][46][47][48][49]. The following circumstance is also of additional interest. According to the CLEO data [26][27][28], the value of the non-DD component in the decay width of the ψ(3770) is negligible. At the same time, the BES analysis [18,19,21,22] does not exclude a noticeable non-DD component. Unfortunately, this contradiction has not yet been resolved. As a result, the PDG [1] gives the following value for the DD component: Theoretical considerations combined with the CLEO data [26][27][28] suggest that the dominance of the ψ(3770) → DD decay can be at the level of 97% − 98%. In what follows, we will consider the ψ(3770) to be an almost elastic resonance coupled to the DD decay channels and apply this assumption to describe its line shape and determine its electronic decay width Γ ψ(3770)e + e − . This paper is organized as follows. Section II gives a brief overview of the commonly used methods for describing the ψ(3770) resonance and the definitions of Γ ψ(3770)e + e − , in particular, selected by PDG [1] for calculations fitted (0.262 ± 0.018) keV and average (0.256 ± 0.016) keV values of Γ ψ(3770)e + e − . Attention is drawn to the fact that some seemingly natural parametrizations of the cross section σ(e + e − → DD), taking into account the interference of the ψ(3770) resonance and background, do not allow to determine the value of Γ ψ(3770)e + e − uniquely. In Section III, we apply to the description of the reaction cross section σ(e + e − → DD) the model for the isoscalar form factor of the D meson, which takes into account the contributions of ψ(3770) and ψ(2S) resonances mixed due to their coupling with the DD decay channels. The model satisfies the requirement of the unitarity condition or the Watson theorem [50] and allows to unambiguously determine the value of Γ ψ(3770)e + e − from the data by fitting. Our analysis substantially develops the approach proposed in Refs. [41,42] by consistently taking into account the finite width corrections in the resonance propagators and clarifying their important role. In Section IV, we compare the values of Γ ψ(3770)e + e − found from phenomenological data processing with theoretical estimates in potential models and briefly state our conclusions.
With increasing accuracy of measurements, there appeared indications on an unusual (anomalous) shape of the ψ(3770) peak in the e + e − → ψ ′′ → hadrons and e + e − → ψ ′′ → DD reaction cross sections, i.e., on possible interference effects that occur directly in the ψ(3770) resonance region [20,21,[23][24][25][29][30][31][32]. In particular, there is a deep dip in the DD production cross section near √ s ≈ 3.81 GeV [20,21,[29][30][31] which strongly distorts the right wing of the ψ ′′ resonance. Such a dip is difficult to describe using Eqs. (1) and (2) for a solitary ψ ′′ resonance contribution. In Ref [41], we showed that the description of the data [20,21,[28][29][30][31] with the use of these formulas turns out to be unsatisfactory for any values of the parameter r. In addition, by performing the analytical continuation of the amplitudes e + e − → ψ ′′ → D 0D0 and e + e − → ψ ′′ → D + D − corresponding to the parameterizations (1) and (2) below the DD thresholds, it is easy to make sure that they have spurious poles and left cuts due to the P -wave Blatt and Weisskopf barrier penetration factors 1/[1 + r 2 p 2 0,+ (s)] [51]. For example, for r ≈ 1 fm ≈ 5 GeV −1 , the indicated singularities appear at about 20 MeV below the DD thresholds. In the next section, we show that taking into account the finite width corrections in the resonance propagators allows us to eliminate these singularities.
If we are not dealing with a solitary resonance, but with a complex of the mixed resonance and background contributions, then a practical question arises about the way of describing it as a whole and the possibilities of adequately determining the individual characteristics of its components. In what follows, we will talk about the process e + e − → DD, in which the isoscalar electromagnetic form factor of the D meson F 0 D (s) is measured. The sum of the e + e − → DD reaction cross sections is expressed in the terms of F 0 D (s) as follows: where α = e 2 /4π = 1/137. Here we neglect the isovector part of the D meson form factor and do not touch on the question about the isospin symmetry breaking. The KEDR Collaboration [32], analyzing their own data on the e + e − → DD cross section, showed that taking into account the interference between the ψ(3770) resonance and background contributions affects the values resonance parameters and therefore the corresponding results cannot be directly compared with those obtained ignoring this effect. In addition, in Ref. [32] within the framework of the accepted parametrization for F 0 D (s), two essentially different solutions were obtained for the production amplitude of the ψ(3770) and its phase relative to the background (see also [48]). These two solutions lead to the same energy dependence of the e + e − → DD cross section and are indistinguishable by the χ 2 criterion. Ambiguities of this type in the interfering resonances parameters determination were found in Ref. [52] (see also [53,54]). The PDG used one of the KEDR solutions [32] [see Eq. (8) below] to determine the value of Γ ψ ′′ e + e − = (0.262 ± 0.018) keV [1], together with the above results from other works [15,16,21,23,26] (in which the interference was not taken into account).
Let us illustrate the ambiguity of the choice of the resonance parameters with a simple example. Consider a model of the reaction amplitude e + e − → hh (where h andh are hadrons) which takes into account the resonance and background contributions Here E is the energy in the hh center-of-mass system, M is the mass and Γ the energy-independent width of the resonance, and A x , ϕ x , and B x are the real parameters. At fixed M and Γ, there are two solutions for A x , ϕ x , and B x [52]: then the values of the electronic decay width of the resonance, Γ e + e − , differ by a factor of two for solutions (I) and (II).
The similar form factor parametrization was used to determine the ψ(3770) resonance parameters in Ref [32]: where Thus, parameterizations of the type (4) and (7) preserving at first glance the usual way of determining the individual characteristics of the ψ ′′ resonance (for example, its electronic width) do not allow to do this unambiguously by fitting. If one of the values of Γ ψ ′′ e + e − from Eqs. (8) and (9) agrees with some theoretical estimate of Γ ψ ′′ e + e − , then it does not yet mean the validity of Eq. (7), which contains the phase φ of unknown origin and does not take into account the transition amplitude between the background and resonance through the common DD intermediate states.
However, just in the case of the ψ ′′ resonance, the above difficulties can be avoided if we take into account the requirement of the unitarity condition. As noted above, the ψ ′′ is the elastic resonance in a good approximation. But in the elastic region (between DD and DD * thresholds) with a width of about 141 MeV, the unitarity condition requires that the phase of the form factor F 0 D (s) coincide with the phase δ 0 1 (s) of the strong P -wave DD elastic scattering amplitude T 0 1 (s) = e δ 0 1 (s) sin δ 0 1 (s) in the channel with isospin I = 0, i.e., where F 0 D (s) and δ 0 1 (s) are the real functions of energy [50]. It is clear that the formulas (4) and (7) contradict the unitarity requirement since the phase of the form factor determined by them depends on the ratio of the background and resonance coupling constants with e + e − , on which δ 0 1 is obviously independent. In the next section, we apply to the description of the data on the reaction e + e − → DD a simple model of the form factor F 0 D (s), which satisfies the requirement of the unitarity condition for the case of the mixed ψ ′′ and ψ(2S) resonances and allows by fitting to uniquely determine the value of Γ ψ ′′ e + e − . Our analysis is an advancement of that suggested earlier in [41,42]. Consider a model that takes into account in the form factor F 0 D (s) and amplitude T 0 1 (s) the contributions of two close to each other resonances ψ ′′ and ψ(2S) strongly coupled only to DD decay channels and mixing with each other due to transitions ψ ′′ → DD → ψ(2S) and vice versa. However, we first write down the contribution of the ψ ′′ to F 0 D in the spirit of the vector dominance model [55][56][57][58], ignoring its mixing with the ψ(2S): where C ψ ′′ is an s-independent constant, D ψ ′′ (s) is the inverse propagator of ψ ′′ , and is the ψ ′′ → DD decay width, where g ψ ′′ DD is the corresponding coupling constant. The function h ψ ′′ (s) describes the contribution of the finite width corrections to the real part of the ψ ′′ propagator. Its explicit form is given in Values C ψ ′′ , m ψ ′′ , g ψ ′′ DD , and r are free parameters of the model. To normalize the form factor F ψ ′′ D (s) at s = m 2 ψ ′′ , we use the relation where Γ ψ ′′ DD ≡ Γ ψ ′′ DD (m 2 ψ ′′ ). Then, taking into account Eqs. (3), (11), and (13) we have (up to a sign) Putting by definition Γ ψ ′′ e + e − = 4πα 2 g 2 ψ ′′ γ /(3m 3 ψ ′′ ), where the constant g ψ ′′ γ describes the ψ ′′ coupling with the virtual γ quantum, we can write C ψ ′′ in the form The effective coupling constant of the ψ ′′ with DD g ef f ψ ′′ DD is related to the constant g ψ ′′ DD from Eq. (12) by the relation From Eqs. (11) and (A1)-(A4) it follows that owing to the finite width corrections in D ψ ′′ (s) the form factor F ψ ′′ D (s) has good analytical properties. In particular, it has no any singularities associated with the poles of the functions 1/[1 + r 2 p 2 0,+ (s)]. In addition, in F ψ ′′ D (s) there are absent spurious bound states in the region 0 < s < 4m 2 D + for r ≥ 0.87 GeV −1 (0.174 fm) (i.e., D ψ ′′ (s) does not vanish anywhere in this region).
C. D meson form factor for the mixed ψ ′′ and ψ(2S) states We now take into account the mixing of ψ ′′ and ψ(2S) resonances due to their common decay channels into D 0D0 and D + D − . The form factor F 0 D (s) corresponding to such a ψ ′′ − ψ(2S) resonance complex can be written as [41,42] F 0 where and Π ψ ′′ ψ(2S) (s) is the non-diagonal polarization operator describing the the transition ψ ′′ → DD → ψ(2S). The polarization operator Π ψ ′′ ψ(2S) (s) is related to the diagonal polarization operator Π ψ ′′ (s) (see Appendix A) by the relation where a and b are unknown constants. In order to the parameters introduced above for the description of solitary ψ ′′ and ψ(2S) resonances (fixed m ψ(2S) , g ψ(2S)γ and free m ψ ′′ , g ψ ′′ γ , g ψ ′′ DD , and g ψ(2S)DD or z) preserve the meaning of individual characteristics for resonances dressed by mixing, we fix the constants a and b by the conditions Note that Eq. (25) keeps the normalization condition (13) for the form factor F 0 D (s) given by formula (20). Using Eqs. (24) and (25), we find Note that the phase of the form factor F 0 D (s), due to the strong resonant interaction of D mesons, is determined by the phase of the denominator in Eq. (20). The numerator in this formula is the first-degree polynomial in s with real coefficients. It is interesting that in the case under consideration we are faced perhaps for the first time with the possibility of the existence of zero in the form factor in the elastic region. As seen from Fig. 1, the data do not contradict the presence of zero in F 0 D (s) at √ s ≈ 3.81 GeV [60].  (26) to the data from BES [20,21], CLEO [28], BABAR [29,30], and Belle [31]. The dashed and dotted curves show the contributions to the cross section from the ψ ′′ and ψ(2S) production amplitudes proportional to the coupling constants C ψ ′′ and C ψ(2S) , respectively; see Eqs. (20). Figures 2 and 3 show the fitting of the data [20,21,[28][29][30][31] in the model of the mixed ψ ′′ and ψ(2S) resonances. The curves in these figures correspond to the following values of the fitted parameters: m ψ ′′ = 3.7884 GeV, g ψ ′′ DD = 60.54, g ψ ′′ γ = −0.2148 GeV 2 , and z = 1.0225. Using these values we get g ef f ψ ′′ DD = 14.72, Γ ψ ′′ DD = 51.88 MeV, and Γ ψ ′′ e + e − = 0.189 keV. The errors in the values of free parameters do not exceed 5%. For this fit, χ 2 = 127.6 which is approximately 3.6 times less than χ 2 for the fit with the solitary ψ ′′ resonance shown in Fig. 1.
The above fit in the model of the mixed ψ ′′ and ψ(2S) resonances has been obtained at the fixed value of the parameter r = 12.5 GeV −1 (≈ 2.5 fm). Let us discuss this parameter in more detail. Its role in the description of the ψ ′′ resonance with the formulas (1) and (2) was discussed in the second section in Ref. [41]. Here a few words about r were said in the two paragraphs after Eq. (16). In Table I, we have collected the conclusions about the parameter r obtained in processing of the data on the ψ(3770) resonance to illustrate the real situation. The parameter r is practically always taken into account when processing resonance data, but, as a rule, it remains not well defined and is often simply fixed by hand. Perhaps, its main role is to suppress of the increasing the P -wave decay width ψ ′′ → DD as √ s increases, see Eq. (12). The suppression occurs faster at high r. But if the fit improves as r increases, then it simultaneously becomes less sensitive to g 2 ψ ′′ DD and r 2 separately, and increasingly depends The curve is the same as the solid curve in Fig. 2, but in comparison only with the data from CLEO [28], BABAR [29,30], and Belle [31]. The inset shows the phase δ 0 1 (s) of the form factor F 0 D (s) and DD elastic scattering amplitude T 0 1 (s) for our fit. Data processing Presented conclusions Rapidis [13] Acceptable fits for all values of r > 1 fm; illustration at r = 3 fm Peruzzi [14] r was varied from 0 to ∞ Schindler [16] r was taken to be 2.5 fm Ablikim [17] r was taken to be 0.5 fm Ablikim [18] r was left free in the fit Ablikim [19] r was taken to be 1 fm Ablikim [21] r was a free parameter in the fit Ablikim [22] r was fixed at 3 fm Ablikim [23] r was of the order of a few fm Ablikim [24] r was fixed at 1.5 fm Dobbs [27] r was taken to be 2.4 fm Anashin [32] r was fixed at 1 fm 1 fm Achasov [41] Analysis of Eqs. (1) and (2) for 0 < r < 4, ... fm on the ratio g 2 ψ ′′ DD /r 2 , see Eq. (12). In such a case the parameter r remains formally unbounded from above [41]. With sequentially increasing of r, one can estimate such its value after which the χ 2 of the fitting actually remains constant. Our fit corresponds to namely such an approximate value of r. If r is decreased, then χ 2 will increase, but not catastrophically. For example, χ 2 turns out to be ≈ 130.4 at r = 5 GeV −1 (≈ 1 fm). In this case Γ ψ ′′ e + e − ≈ 0.14 keV, Γ ψ ′′ DD ≈ 92.2 MeV, and m ψ ′′ ≈ 3.796 GeV. Increasing of the data accuracy would make it possible to determine the value of r more accurately and with it the values of other model parameters too.
One can also express the hope that the model will become more flexible and will improve the data description, if at the next step of the research we take into account the couplings of the ψ ′′ and ψ(2S) resonances with the closed DD * and D * D * decay channels in the region √ s up to 3.872 GeV and the inelastic effects caused by them for √ s > 3.872 GeV. Of course, further accurate measurements of the e + e − → DD cross sections will be decisive for the selection of phenomenological models and understanding the ψ(3770) resonance as a charm factory.
The spread of theoretical estimates for the width Γ ψ ′′ e + e − quite agrees with the spread of its values found in various experiments [1] and also in accompanying phenomenological analyzes [38][39][40][41][42][43][44][45][46][47][48][49] (see discussion in previous sections). Of course, the primary guide is the value of Γ ψ ′′ e + e − = (0.262 ± 0.018) keV given by PDG [1]. However, as noted above, the phenomenological formulas using to obtain this value were rather simplified (or even poorly grounded). If the errors of the data on σ(e + e − → DD) are reduced by approximately two times compared to the existing ones [see Figs. (2) and (3)], then it will be possible to abandon such formulas. When processing new, more accurate data on the cross section σ(e + e − → D 0D0 + D + D − ), it will make sense to take into account the Coulomb interaction in the final state between D + and D − mesons, which amplifies the charged channel by about 8.8% at the peak of the ψ ′′ resonance [64]. Now we summarize. 1) The model of the D meson form factor F 0 D (s) with good unitary and analytic properties is constructed to describe the cross section of the reaction e + e − → DD near the threshold. 2) The model involves the complex of the mixed ψ ′′ and ψ(2S) resonances and satisfactorily describes the data in the √ s region up to 3.9 GeV. 3) A feature of the model is the presence of zero in F 0 D (s) at √ s ≈ 3.818 GeV. 4) The survey of the experimental, phenomenological, and theoretical results for Γ ψ ′′ e + e − is also presented to illustrate the variety of approaches to determining this quantity. 5) The rather small value of Γ ψ ′′ e + e − ≈ 0.19 keV, obtained by us, and the corresponding value of the ratio Γ ψ ′′ e + e − /Γ ψ(2S)e + e − ≈ 0.081 indicate in favor of the D-wave cc nature of the ψ ′′ state.
Improving the data on the shape of the ψ(3770) resonance in the DD decay channels seems to be an extremely important and quite feasible physical problem.

ACKNOWLEDGMENTS
The work was carried out within the framework of the state contract of the Sobolev Institute of Mathematics, Project No. 0314-2019-0021.