Electronic width of the ψ(3770) resonance interfering with the background
aa r X i v : . [ h e p - ph ] F e b Electronic width of the ψ (3770) resonance interfering with the background N.N. Achasov ∗ and G.N. Shestakov † Laboratory of Theoretical Physics, S.L. Sobolev Institute for Mathematics, 630090, Novosibirsk, Russia
Methods for extracting the ψ (3770) → e + e − decay width from the data on the reaction crosssection e + e − → D ¯ D are discussed. Attention is drawn to the absence of the generally acceptedmethod for determining Γ ψ (3770) e + e − in the presence of interference between the contributions of the ψ (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 thecontribution of the complex of the mixed ψ (3770) and ψ (2 S ) resonances, allows uniquely determinethe value of Γ ψ (3770) e + e − by fitting. The Γ ψ (3770) e + e − values found from the data processing arecompared with the estimates in the potential models. I. INTRODUCTION
The charmonium state ψ (3770) [1] predicted in the mid-seventies is considered as the 1 D state of the c ¯ c systemwith small admixtures of n S states [mainly ψ (2 S )] [2–12]. In e + e − collisions, the ψ (3770) is observed in the formof the resonant enhancement, with a width of about 30 MeV, located between the D ¯ D (2 m D ≈ .
739 GeV) and D ¯ D ∗ ( m D + m D ∗ ≈ .
872 GeV) production thresholds. The sizeable width of the ψ (3770) is due to its strong decays into D ¯ D meson pairs. Indeed, the fraction of the radiative decays ψ (3770) → γχ cJ =0 , , , γη c , γη c (2 S ) is less than 1.5 %,and the fraction of the ψ (3770) → J/ψπ + π − , J/ψπ π and J/ψη decays is less than 0 .
5% [1]. The total width of theZweig forbidden decays ψ (3770) → light hadrons must be comparable from the theoretical point of view with thecorresponding decay widths of the J/ψ and ψ (2 S ) resonances located under the D ¯ D threshold. In order of magnitude,it can be about 100 keV, which is less than 0 .
5% of the total decay width of the ψ (3770) meson. For almost ninetydecay channels ψ (3770) → light hadrons are known only upper limits (some of which are rather high) [1]. Only thebranching ratio of the decay ψ (3770) → φη is definitely known, B ( ψ (3770) → φη ) = (3 . ± . × − [1].The charmonium state ψ (3770) was investigated in e + e − collisions by the MARK-I [13, 14], DELCO [15], MARK-II[16], BES [17–25], CLEO [26–28], BABAR [29, 30], Belle [31], and KEDR [32] Collaborations. The ψ (3770) productionwas also observed in the B + → D ¯ DK + 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, 23–25, 29–32], naturallybecame the subject of many-sided theoretical analyzes, see, for example, Refs. [38–49]. The following circumstanceis also of additional interest. According to the CLEO data [26–28], the value of the non- D ¯ D component in the decaywidth of the ψ (3770) is negligible. At the same time, the BES analysis [18, 19, 21, 22] does not exclude a noticeablenon- D ¯ D component. Unfortunately, this contradiction has not yet been resolved. As a result, the PDG [1] gives thefollowing value for the D ¯ D component: B ( ψ (3770) → D ¯ D ) = [ B ( ψ (3770) → D + D − ) = (52 ± )%] + [ B ( ψ (3770) → D ¯ D ) = (41 ± ± )%. Theoretical considerations combined with the CLEO data [26–28] suggest thatthe dominance of the ψ (3770) → D ¯ D decay can be at the level of 97% − ψ (3770) to be an almost elastic resonance coupled to the D ¯ D decay channels and apply this assumption to describeits 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 describingthe ψ (3770) resonance and the definitions of Γ ψ (3770) e + e − , in particular, selected by PDG [1] for calculations fitted(0 . ± . . ± . ψ (3770) e + e − . Attention is drawn to the fact that someseemingly natural parametrizations of the cross section σ ( e + e − → D ¯ D ), 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, weapply to the description of the reaction cross section σ ( e + e − → D ¯ D ) the model for the isoscalar form factor of the D meson, which takes into account the contributions of ψ (3770) and ψ (2 S ) resonances mixed due to their coupling withthe D ¯ D 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 substantiallydevelops the approach proposed in Refs. [41, 42] by consistently taking into account the finite width corrections inthe resonance propagators and clarifying their important role. In Section IV, we compare the values of Γ ψ (3770) e + e − ∗ [email protected] † [email protected] found from phenomenological data processing with theoretical estimates in potential models and briefly state ourconclusions. II. PARAMETERIZATIONS OF THE ψ (3770) RESONANCE STRUCTURE In many experimental works, the cross section of the reaction e + e − → D ¯ D in the ψ (3770) resonance region wasdescribed with minor modification by the following formula [13–22, 26] [below, for short ψ (3770) is also denoted as ψ ′′ ]: σ ψ ′′ ( e + e − → D ¯ D ; s ) = 12 π Γ ψ ′′ e + e − Γ ψ ′′ D ¯ D ( s )( m ψ ′′ − s ) + ( m ψ ′′ Γ totψ ′′ ( s )) , (1)where s is the invariant mass squared of the D ¯ D system, m ψ ′′ , Γ ψ ′′ e + e − , Γ ψ ′′ D ¯ D ( s ), and Γ totψ ′′ ( s ) are the mass,electronic, end total decay widths of ψ ′′ , respectively. The energy-dependent width Γ ψ ′′ D ¯ D ( s ) [dominating in Γ totψ ′′ ( s )]was taken in the form Γ ψ ′′ D ¯ D ( s ) = G ψ ′′ (cid:18) p ( s )1 + r p ( s ) + p ( s )1 + r p ( s ) (cid:19) , (2)where p ( s ) = q s/ − m D and p + ( s ) = q s/ − m D + are the D and D + momenta in the ψ ′′ rest frame, r is the D ¯ D interaction radius [51], and G ψ ′′ is the coupling constant of the ψ ′′ with D ¯ D .For the solitary ψ ′′ resonance, there is no problem with determining Γ ψ ′′ e + e − by fitting the data using Eqs. (1) and(2). Discrepancy between the values found by different Collaborations (Γ ψ ′′ e + e − = 345 ±
85 eV [14], 180 ±
60 eV [15],276 ±
50 eV [16], 279 ± ±
13 eV [21], 220 ±
50 eV [1, 23], 204 ± +41 − eV [26]) is mainly related to the difference inthe collected raw data and uncertainties in the cross section normalization.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 ′′ → D ¯ D reaction cross sections, i.e., on possibleinterference effects that occur directly in the ψ (3770) resonance region [20, 21, 23–25, 29–32]. In particular, there is adeep dip in the D ¯ D production cross section near √ s ≈ .
81 GeV [20, 21, 29–31] which strongly distorts the right wingof 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, 28–31] with the use of these formulas turns out tobe unsatisfactory for any values of the parameter r . In addition, by performing the analytical continuation of theamplitudes e + e − → ψ ′′ → D ¯ D and e + e − → ψ ′′ → D + D − corresponding to the parameterizations (1) and (2)below the D ¯ D thresholds, it is easy to make sure that they have spurious poles and left cuts due to the P -wave Blattand Weisskopf barrier penetration factors 1 / [1 + r p , + ( s )] [51]. For example, for r ≈ ≈ − , the indicatedsingularities appear at about 20 MeV below the D ¯ D thresholds. In the next section, we show that taking into accountthe 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 backgroundcontributions, then a practical question arises about the way of describing it as a whole and the possibilities ofadequately determining the individual characteristics of its components. In what follows, we will talk about theprocess e + e − → D ¯ D , in which the isoscalar electromagnetic form factor of the D meson F D ( s ) is measured. The sumof the e + e − → D ¯ D reaction cross sections is expressed in the terms of F D ( s ) as follows: σ ( e + e − → D ¯ D ; s ) = 8 πα s / (cid:12)(cid:12) F D ( s ) (cid:12)(cid:12) (cid:2) p ( s ) + p ( s ) (cid:3) , (3)where α = e / π = 1/137. Here we neglect the isovector part of the D meson form factor and do not touch on thequestion about the isospin symmetry breaking. The KEDR Collaboration [32], analyzing their own data on the e + e − → D ¯ D cross section, showed that taking into account the interference between the ψ (3770) resonance andbackground contributions affects the values resonance parameters and therefore the corresponding results cannot bedirectly compared with those obtained ignoring this effect. In addition, in Ref. [32] within the framework of theaccepted parametrization for F D ( s ), two essentially different solutions were obtained for the production amplitude ofthe ψ (3770) and its phase relative to the background (see also [48]). These two solutions lead to the same energydependence of the e + e − → D ¯ D cross section and are indistinguishable by the χ criterion. Ambiguities of this typein the interfering resonances parameters determination were found in Ref. [52] (see also [53, 54]). The PDG used oneof the KEDR solutions [32] [see Eq. (8) below] to determine the value of Γ ψ ′′ e + e − = (0 . ± . e + e − → h ¯ h (where h and ¯ h are hadrons) which takes into account the resonance andbackground contributions F ( E ) = A x e iϕ x M − E − i Γ / B x (4)Here E is the energy in the h ¯ h center-of-mass system, M is the mass and Γ the energy-independent width of theresonance, 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]: (I) A x = A, B x = B, ϕ x = ϕ, (5)(II) A x = p A − AB Γ sin ϕ + B Γ , B x = B, tan ϕ x = − tan ϕ + B Γ / ( A cos ϕ ) , (6)which yield the same cross section as a function of energy, σ ( E ) = | F ( E ) | , and different amplitude A x and phase ϕ x .For example, if M = 3.77 GeV, Γ = 0.03 GeV A = 0.045 nb / GeV, ϕ = 0, and B = 1.5 nb / for solution (I), then, forsolution (II), A x = √ A and ϕ x = π/
4. Since A x ∼ √ Γ e + e − Γ, then the values of the electronic decay width of theresonance, 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]: F D ( s ) = F ψ (3770) ( s ) e iφ + F N.R. ( s ) , (7)where F ψ (3770) ( s ) is the Breit-Wigner P -wave resonance amplitude, F N.R. ( s ) the background amplitude, and φ theirrelative phase. F N.R. ( s ) = F ψ (2 S ) ( s )+ F takes into account the contribution of the right wing of the nearest resonance ψ (2 S ) with the mass of 3.686 GeV and the additional constant contribution F . Two solutions indistinguishable in χ are [32] (I) Γ ψ ′′ e + e − = 160 +78 − eV , φ = (170 . ± . ◦ , (8)(II) Γ ψ ′′ e + e − = 420 +72 − eV , φ = (239 . ± . ◦ . (9)Thus, parameterizations of the type (4) and (7) preserving at first glance the usual way of determining the individualcharacteristics 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 doesnot yet mean the validity of Eq. (7), which contains the phase φ of unknown origin and does not take into accountthe transition amplitude between the background and resonance through the common D ¯ D intermediate states.However, just in the case of the ψ ′′ resonance, the above difficulties can be avoided if we take into account therequirement of the unitarity condition. As noted above, the ψ ′′ is the elastic resonance in a good approximation.But in the elastic region (between D ¯ D and D ¯ D ∗ thresholds) with a width of about 141 MeV, the unitarity conditionrequires that the phase of the form factor F D ( s ) coincide with the phase δ ( s ) of the strong P -wave D ¯ D elasticscattering amplitude T ( s ) = e δ ( s ) sin δ ( s ) in the channel with isospin I = 0, i.e., F D ( s ) = e iδ ( s ) F D ( s ) , (10)where F D ( s ) and δ ( s ) are the real functions of energy [50]. It is clear that the formulas (4) and (7) contradict theunitarity requirement since the phase of the form factor determined by them depends on the ratio of the backgroundand resonance coupling constants with e + e − , on which δ is obviously independent.In the next section, we apply to the description of the data on the reaction e + e − → D ¯ D a simple model of theform factor F D ( s ), which satisfies the requirement of the unitarity condition for the case of the mixed ψ ′′ and ψ (2 S )resonances and allows by fitting to uniquely determine the value of Γ ψ ′′ e + e − . Our analysis is an advancement of thatsuggested earlier in [41, 42]. III. THE D MESON ELECTROMAGNETIC FORM FACTOR IN THE ψ (3770) REGIONA. The solitary ψ ′′ resonance Consider a model that takes into account in the form factor F D ( s ) and amplitude T ( s ) the contributions of twoclose to each other resonances ψ ′′ and ψ (2 S ) strongly coupled only to D ¯ D decay channels and mixing with each otherdue to transitions ψ ′′ → D ¯ D → ψ (2 S ) and vice versa. However, we first write down the contribution of the ψ ′′ to F D in the spirit of the vector dominance model [55–58], ignoring its mixing with the ψ (2 S ): F D ( s ) = F ψ ′′ D ( s ) = C ψ ′′ e D ψ ′′ ( s ) = C ψ ′′ m ψ ′′ − s − h ψ ′′ ( s ) − i √ s Γ ψ ′′ D ¯ D ( s ) , (11)where C ψ ′′ is an s -independent constant, e D ψ ′′ ( s ) is the inverse propagator of ψ ′′ , andΓ ψ ′′ D ¯ D ( s ) = g ψ ′′ D ¯ D πs (cid:18) p ( s )1 + r p ( s ) + p ( s )1 + r p ( s ) (cid:19) , (12)is the ψ ′′ → D ¯ D decay width, where g ψ ′′ D ¯ D is the corresponding coupling constant. The function h ψ ′′ ( s ) describesthe contribution of the finite width corrections to the real part of the ψ ′′ propagator. Its explicit form is given inAppendix A. Near s = m ψ ′′ the function h ψ ′′ ( s ) ∼ ( m ψ ′′ − s ) . Values C ψ ′′ , m ψ ′′ , g ψ ′′ D ¯ D , and r are free parametersof the model. To normalize the form factor F ψ ′′ D ( s ) at s = m ψ ′′ , we use the relation σ ψ ′′ ( e + e − → D ¯ D ; s = m ψ ′′ ) = 12 πm ψ ′′ Γ ψ ′′ e + e − Γ ψ ′′ D ¯ D , (13)where Γ ψ ′′ D ¯ D ≡ Γ ψ ′′ D ¯ D ( m ψ ′′ ). Then, taking into account Eqs. (3), (11), and (13) we have (up to a sign) C ψ ′′ = vuut m ψ ′′ Γ ψ ′′ e + e − Γ ψ ′′ D ¯ D α (cid:16) p ( m ψ ′′ ) + p ( m ψ ′′ ) (cid:17) . (14)Putting by definition Γ ψ ′′ e + e − = 4 πα g ψ ′′ γ / (3 m ψ ′′ ), where the constant g ψ ′′ γ describes the ψ ′′ coupling with thevirtual γ quantum, we can write C ψ ′′ in the form C ψ ′′ = g ψ ′′ γ g effψ ′′ D ¯ D . (15)The effective coupling constant of the ψ ′′ with D ¯ D g effψ ′′ D ¯ D is related to the constant g ψ ′′ D ¯ D from Eq. (12) by therelation g effψ ′′ D ¯ D = q πm ψ ′′ Γ ψ ′′ D ¯ D / [ p ( m ψ ′′ ) + p ( m ψ ′′ )] . (16)From Eqs. (11) and (A1)–(A4) it follows that owing to the finite width corrections in e 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 functions1 / [1 + r p , + ( s )]. In addition, in F ψ ′′ D ( s ) there are absent spurious bound states in the region 0 < s < m D + for r ≥ .
87 GeV − (0.174 fm) (i.e., e D ψ ′′ ( s ) does not vanish anywhere in this region).The fit to the data [20, 21, 28–31] with the use of the solitary ψ ′′ resonance model at a fixed value of r = 0 . − is shown in Fig. 1. It corresponds to m ψ ′′ = 3 .
772 GeV, g ψ ′′ D ¯ D = 14 . ψ ′′ D ¯ D ( m ψ ′′ ) ≈ . g ψ ′′ γ = 0 .
245 GeV [i.e., Γ ψ ′′ e + e − ≈ .
25 keV]. Although the obtained values of the ψ ′′ parameters are close tothose given by PDG [1], the fit in itself is unsatisfactory. The corresponding χ = 459 for 84 degrees of freedom. As r increases, the fit becomes even less satisfactory. B. The ψ (2 S ) contribution Let us write the contribution of the state ψ (2 S ) to F D ( s ) by analogy with Eq. (11) in the form F D ( s ) = F ψ (2 S ) D ( s ) = C ψ (2 S ) e D ψ (2 S ) ( s ) = C ψ (2 S ) m ψ (2 S ) − s − h ψ (2 S ) ( s ) − i √ s Γ ψ (2 S ) D ¯ D ( s ) , (17)where m ψ (2 S ) = 3 . F ψ (2 S ) D ( s ) is calculated according Eqs. (12) and (A1)–(A4), where index ψ ′′ shouldbe replaced everywhere by ψ (2 S ). The constant C ψ (2 S ) in Eq. (17) can be represented by analogy with Eq. (15) inthe form C ψ (2 S ) = g ψ (2 S ) γ g effψ (2 S ) D ¯ D . (18) (cid:143)!!!! s H GeV L Σ H e + e - ® DD (cid:143)(cid:143)(cid:143) LH nb L æ BES ò BABARBelle í CLEO
Figure 1: The variant of the solitary ψ ′′ resonance model. The curve is the fit using Eqs. (3), (11)–(15) to the data from BES[20, 21], CLEO [28], BABAR [29, 30], and Belle [31] for σ ( e + e − → D ¯ D ). There are 87 points in the fit. For more details onthe data see Ref. [41] and also a footnote in Ref. [59]. The constant g ψ (2 S ) γ describes the ψ (2 S ) coupling with the virtual γ quantum. From the PDG data [1], Γ ψ (2 S ) e + e − =2 .
33 keV, and the relation Γ ψ (2 S ) e + e − = 4 πα g ψ (2 S ) γ / (3 m ψ (2 S ) ), we get g ψ (2 S ) γ = ± .
723 GeV . As a free parameterfor the ψ (2 S ) contribution, it is convenient to use the proportionality coefficient z between the coupling constants ofthe ψ (2 S ) and ψ ′′ with D ¯ D : g ψ (2 S ) D ¯ D = z g ψ ′′ D ¯ D and g effψ (2 S ) D ¯ D = z g effψ ′′ D ¯ D (19)[the relation between g ψ ′′ D ¯ D and g effψ ′′ D ¯ D is definite by Eq. (16)]. C. D meson form factor for the mixed ψ ′′ and ψ (2 S ) states We now take into account the mixing of ψ ′′ and ψ (2 S ) resonances due to their common decay channels into D ¯ D and D + D − . The form factor F D ( s ) corresponding to such a ψ ′′ − ψ (2 S ) resonance complex can be written as [41, 42] F D ( s ) = C ψ ′′ ∆ ψ (2 S ) ( s ) + C ψ (2 S ) ∆ ψ ′′ ( s ) e D ψ ′′ ( s ) e D ψ (2 S ) ( s ) − e Π ψ ′′ ψ (2 S ) ( s ) , (20)where ∆ ψ (2 S ) ( s ) = e D ψ (2 S ) ( s ) + z e Π ψ ′′ ψ (2 S ) ( s ) , (21)∆ ψ ′′ ( s ) = e D ψ ′′ ( s ) + z − e Π ψ ′′ ψ (2 S ) ( s ) , (22)and e Π ψ ′′ ψ (2 S ) ( s ) is the non-diagonal polarization operator describing the the transition ψ ′′ → D ¯ D → ψ (2 S ). Thepolarization operator e Π ψ ′′ ψ (2 S ) ( s ) is related to the diagonal polarization operator Π ψ ′′ ( s ) (see Appendix A) by therelation e Π ψ ′′ ψ (2 S ) ( s ) = z Π ψ ′′ ( s ) + a + s b, (23)where a and b are unknown constants. In order to the parameters introduced above for the description of solitary ψ ′′ and ψ (2 S ) resonances (fixed m ψ (2 S ) , g ψ (2 S ) γ and free m ψ ′′ , g ψ ′′ γ , g ψ ′′ D ¯ D , and g ψ (2 S ) D ¯ D or z ) preserve the meaningof individual characteristics for resonances dressed by mixing, we fix the constants a and b by the conditionsRe e Π ψ ′′ ψ (2 S ) ( m ψ (2 S ) ) = 0 , (24)Re e Π ψ ′′ ψ (2 S ) ( m ψ ′′ ) = 0 . (25)Note that Eq. (25) keeps the normalization condition (13) for the form factor F D ( s ) given by formula (20). UsingEqs. (24) and (25), we find e Π ψ ′′ ψ (2 S ) ( s ) = z " Π ψ ′′ ( s ) − Re Π ψ ′′ ( m ψ ′′ ) + s − m ψ ′′ m ψ ′′ − m ψ (2 S ) Re (cid:16) Π ψ ′′ ( m ψ (2 S ) ) − Π ψ ′′ ( m ψ ′′ ) (cid:17) . (26)Note that the phase of the form factor F D ( s ), due to the strong resonant interaction of D mesons, is determinedby the phase of the denominator in Eq. (20). The numerator in this formula is the first-degree polynomial in s withreal coefficients. It is interesting that in the case under consideration we are faced perhaps for the first time withthe possibility of the existence of zero in the form factor in the elastic region. As seen from Fig. 1, the data do notcontradict the presence of zero in F D ( s ) at √ s ≈ .
81 GeV [60]. (cid:143)!!!! s H GeV L Σ H e + e - ® DD (cid:143)(cid:143)(cid:143) LH nb L æ BES ò BABARBelle í CLEO
Figure 2: The model of the mixed ψ ′′ and ψ (2 S ) resonances. The solid curve is the fit using Eqs. (3) and (20)–(26) to thedata from BES [20, 21], CLEO [28], BABAR [29, 30], and Belle [31]. The dashed and dotted curves show the contributionsto the cross section from the ψ ′′ and ψ (2 S ) production amplitudes proportional to the coupling constants C ψ ′′ and C ψ (2 S ) ,respectively; see Eqs. (20). Figures 2 and 3 show the fitting of the data [20, 21, 28–31] in the model of the mixed ψ ′′ and ψ (2 S ) resonances. Thecurves in these figures correspond to the following values of the fitted parameters: m ψ ′′ = 3 . g ψ ′′ D ¯ D = 60 . g ψ ′′ γ = − . , and z = 1 . g effψ ′′ D ¯ D = 14 .
72, Γ ψ ′′ D ¯ D = 51 .
88 MeV, andΓ ψ ′′ e + e − = 0 .
189 keV. The errors in the values of free parameters do not exceed 5%. For this fit, χ = 127 . χ for the fit with the solitary ψ ′′ resonance shown in Fig. 1.The above fit in the model of the mixed ψ ′′ and ψ (2 S ) resonances has been obtained at the fixed value of theparameter r = 12 . − ( ≈ . ψ ′′ resonance with the formulas (1) and (2) was discussed in the second section in Ref. [41]. Here a fewwords about r were said in the two paragraphs after Eq. (16). In Table I, we have collected the conclusions aboutthe parameter r obtained in processing of the data on the ψ (3770) resonance to illustrate the real situation. Theparameter r is practically always taken into account when processing resonance data, but, as a rule, it remains notwell defined and is often simply fixed by hand. Perhaps, its main role is to suppress of the increasing the P -wavedecay width ψ ′′ → D ¯ D as √ s increases, see Eq. (12). The suppression occurs faster at high r . But if the fit improvesas r increases, then it simultaneously becomes less sensitive to g ψ ′′ D ¯ D and r separately, and increasingly depends (cid:143)!!!! s H GeV L Σ H e + e - ® DD (cid:143)(cid:143)(cid:143) LH nb L ò BABARBelle í CLEO (cid:143)!!!! s H GeV L ∆ H s LH deg r ee L ∆ H s L Figure 3: The model of the mixed ψ ′′ and ψ (2 S ) resonances. The curve is the same as the solid curve in Fig. 2, but incomparison only with the data from CLEO [28], BABAR [29, 30], and Belle [31]. The inset shows the phase δ ( s ) of the formfactor F D ( s ) and D ¯ D elastic scattering amplitude T ( s ) for our fit.Table I: Information about the parameter r from the ψ (3770) → D ¯ D decay descriptions (1 fm ≈ − ).Data processing Presented conclusionsRapidis [13] Acceptable fits for all values of r > r = 3 fmPeruzzi [14] r was varied from 0 to ∞ Schindler [16] r was taken to be 2.5 fmAblikim [17] r was taken to be 0.5 fmAblikim [18] r was left free in the fitAblikim [19] r was taken to be 1 fmAblikim [21] r was a free parameter in the fitAblikim [22] r was fixed at 3 fmAblikim [23] r was of the order of a few fmAblikim [24] r was fixed at 1.5 fmDobbs [27] r was taken to be 2.4 fmAnashin [32] r was fixed at 1 fm 1 fmAchasov [41] Analysis of Eqs. (1) and (2) for0 < r < , ... fm on the ratio g ψ ′′ D ¯ D /r , 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 of the fitting actually remainsconstant. Our fit corresponds to namely such an approximate value of r . If r is decreased, then χ will increase, butnot catastrophically. For example, χ turns out to be ≈ . r = 5 GeV − ( ≈ ψ ′′ e + e − ≈ . ψ ′′ D ¯ D ≈ . m ψ ′′ ≈ .
796 GeV. Increasing of the data accuracy would make it possible to determinethe 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 atthe next step of the research we take into account the couplings of the ψ ′′ and ψ (2 S ) resonances with the closed D ¯ D ∗ and D ∗ ¯ D ∗ decay channels in the region √ s up to 3.872 GeV and the inelastic effects caused by them for √ s > . e + e − → D ¯ D cross sections will be decisive for the selection ofphenomenological models and understanding the ψ (3770) resonance as a charm factory. IV. COMPARISON WITH THEORETICAL ESTIMATES AND CONCLUSIONS
Theoretical estimates of the electronic width of the ψ ′′ resonance, that is considered mainly as the 1 D charmo-nium state, show that it is very sensitive to the relativistic corrections, QCD corrections, and mixing of S − D c ¯ c configurations due to tensor forces and transitions via D ¯ D coupled-channels [2–12, 61–63]. The literature cited herepresents a rather wide range of theoretical values for Γ ψ ′′ e + e − . For example, in the nonrelativistic limit, Γ ψ (3770) e + e − turns out to be ≈ .
070 keV due to the 2 S − D mixing in the coupled-channel scheme [6]. Γ ψ ′′ e + e − can increaseto ≈ .
160 keV [6], if one takes into account the relativistic corrections (i.e., the inequality to zero of the secondderivative of the radial wave function at the origin [5]), and further to ≈ .
230 keV with connection of the the S − D mixing due to tensor forces [6]. The relativistic corrections (without mixing) give for Γ ψ ′′ e + e − , for example, ≈ . ≈ .
060 keV [8]. The recent theoretical schemes did not give more definite predictions for the width:Γ ψ ′′ e + e − ≈ .
091 keV [61], ≈ .
270 keV [62], ≈ .
113 keV [63].The spread of theoretical estimates for the width Γ ψ ′′ e + e − quite agrees with the spread of its values found in variousexperiments [1] and also in accompanying phenomenological analyzes [38–49] (see discussion in previous sections).Of course, the primary guide is the value of Γ ψ ′′ e + e − = (0 . ± . σ ( e + e − → D ¯ D ) are reduced by approximately two times compared to the existing ones [seeFigs. (2) and (3)], then it will be possible to abandon such formulas. When processing new, more accurate data onthe cross section σ ( e + e − → D ¯ D + D + D − ), it will make sense to take into account the Coulomb interaction in thefinal 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 D ( s ) with good unitary and analytic properties isconstructed to describe the cross section of the reaction e + e − → D ¯ D near the threshold. 2) The model involves thecomplex of the mixed ψ ′′ and ψ (2 S ) 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 D ( s ) at √ s ≈ .
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 todetermining this quantity. 5) The rather small value of Γ ψ ′′ e + e − ≈ .
19 keV, obtained by us, and the correspondingvalue of the ratio Γ ψ ′′ e + e − / Γ ψ (2 S ) e + e − ≈ .
081 indicate in favor of the D -wave c ¯ c nature of the ψ ′′ state.Improving the data on the shape of the ψ (3770) resonance in the D ¯ D decay channels seems to be an extremelyimportant 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.
Appendix A
The twice subtracted dispersion integral corresponding to the one-loop P -wave Feynman diagram has the form f , + ( s ) = s π ∞ Z m D , + p , + ( s ′ ) ds ′ √ s ′ s ′ ( s ′ − s − iε )= s − m D , + π − sρ , + ( s ) π ln ρ , + ( s ) + 1 ρ , + ( s ) − , for s < , = s − m D , + π + s | ρ , + ( s ) | π ( π − | ρ , + ( s ) | ) , for 0 < s < m D , + , = s − m D , + π + sρ , + ( s )8 π (cid:18) iπ − ln 1 + ρ , + ( s )1 − ρ , + ( s ) (cid:19) , for s > m D , + , (A1)where ρ , + ( s ) = 2 p , + ( s ) / √ s = q − m D , + /s . The polarization operators of the ψ ′′ resonance Π ψ ′′ ( s ) and Π + ψ ′′ ( s )corresponding to the contributions of the D ¯ D and D + D − intermediate states are expressed in terms of the functions f ( s ) and f + ( s ) as follows:Π , + ψ ′′ ( s ) = g ψ ′′ D ¯ D π s π ∞ Z m D , + p , + ( s ′ ) ds ′ √ s ′ (1 + r p , + ( s ′ )) s ′ ( s ′ − s − iε )= g ψ ′′ D ¯ D π
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