Disentangling the role of the $Y(4260)$ in $e^+e^-\to D^*\bar{D}^*$ and $D_s^*\bar{D}_s^*$ via line shape studies

Whether the $Y(4260)$ can couple to open charm channels has been a crucial issue for understanding its nature. The available experimental data suggest that the cross section line shapes of exclusive processes in $e^+e^-$ annihilations have nontrivial structures around the mass region of the $Y(4260)$. As part of a series of studies of the $Y(4260)$ as mainly a $\bar{D}D_1(2420)+c.c.$ molecular state, we show that the partial widths of the $Y(4260)$ to the two-body open charm channels of $e^+e^-\to D^*\bar{D}^*$ and $D_s^*\bar{D}_s^*$ are much smaller than that to $\bar{D}D^*\pi+c.c.$. The line shapes measured by the Belle Collaboration for these two channels can be well described by the vector charmonium states $\psi(4040)$, $\psi(4160)$ and $\psi(4415)$ together with the $Y(4260)$. It turns out that the interference of the $Y(4260)$ with the other charmonia produces a dip around 4.22~GeV in the $e^+e^-\to D^*\bar{D}^*$ cross section line shape. The data also show an evidence for the strong coupling of the $Y(4260)$ to the $D\bar D_1(2420)$, in line with the expectation in the hadronic molecular scenario for the $Y(4260)$.


I. INTRODUCTION
The mysterious state Y (4260) has attracted a lot of attention since its observation in 2005 by the BaBar Collaboration [1]. Although many different models were proposed as solutions in the literature, it is unfortunate that not all of these scenarios have been systematically studied and compared with the existing experimental data (see e.g. several recent reviews [2][3][4][5] for summaries of some theoretical interpretations proposed in the literature). Following a series of recent studies by treating the Y (4260) as mainly aDD 1 (2420)+ c.c. hadronic molecule, we are motivated to examine as many as possible exclusive processes where the Y (4260) can contribute. Such systematic studies with more experimental constraints would either support or invalidate the picture of the Y (4260) being a hadronic molecule ofDD 1 (2420) + c.c. and should provide more insights into its intrinsic structure. Therefore, we investigate the cross section line shapes of the e + e − → D * D * and D * will dominantly decay intoDD * π + c.c. via the decays of its constituent hadrons [6][7][8]. Moreover, due to the strong S-wave coupling to the nearbyDD 1 (2420) channel, the cross section line shape for the e + e − →DD * π + c.c. process should not be described by a Breit-Wigner parametrization. This is generally true for any states that strongly couple to nearby thresholds via an S-wave interaction.
Namely, it is natural to expect a nontrivial cross section line shape for e + e − →DD * π + c.c. around the mass of the Y (4260). This phenomenon has been investigated in detail in Refs. [7,8] which are closely correlated with the study of the nature of the charged charmonium states Z c (3900).
One interesting question arising from the above mentioned analysis is whether the Y (4260) should have significantly large decay widths into other open charm channels apart fromDD * π+c.c..
Given that the total width of Y (4260) is dominated by theDD * π + c.c. channel [15], which has a partial width of about 65 MeV in Ref. [8], while its decays into the hidden charm channels, i.e.
J/ψππ, h c ππ, and χ c0 ω, turn out to be relatively small, the Y (4260) decays into other open charm channels should also have small widths in order to match the total width extracted in the combined analysis of e + e − → J/ψππ, h c ππ, andDD * π + c.c. In this sense, to accommodate the experimental data for e + e − → D * D * and D * sD * s in the same framework is a challenge for the molecular picture, and should provide more information about its structure.
In this work, we analyze the cross section line shapes of the e + e − → D * D * and D * sD * s processes from threshold to about 4.6 GeV. These two processes have been measured by the Belle Collaboration using the initial state radiation (ISR) in e + e − annihilations [16,17]. One can see that the cross sections for e + e − → D * D * have been measured with a high precision [16], but there are still large uncertainties in the data for e + e − → D * sD * s [17]. The former process has been studied in [18] which considers the P -wave coupled-channel effects due to a pair of ground state charmed mesons and the ψ(4040) but not the Y (4260). In our analysis, in addition to the Y (4260) which is included as aDD 1 (2420) hadronic molecule, we also include several conventional vector charmonium states established in this mass region including the ψ(4040), the ψ(4160) and the ψ(4415).
We try to understand the behavior of the molecular state Y (4260) in this energy region and its interference with other charmonium states in the description of the cross section line shapes. We note in advance that our focus is mainly in the vicinity of the Y (4260), i.e. around the threshold of DD 1 (2420). Although there are additional exotic candidates above theDD 1 (2420) threshold, such as the Y (4360), to be neglected in this analysis, we find that we can still draw a clear conclusion on the Y (4260) contribution due to the relatively isolatedDD 1 (2420) threshold.
In this paper, we first estimate the partial decay width of Y (4260) → D * D * in the molecular picture in Sec. II, and then we study the cross section line shapes of e + e − → D * D * and D * sD * s considering the Y (4260) and three charmonium states mentioned above in Sec. III. A brief summary will be given in Sec. IV.
In our scenario, the Y (4260) is treated as mainly an S-wave molecule ofDD 1 (2420) + c.c. with a small mixture of a compact cc core [8]. This treatment recognizes the HQSS breaking in the production of Y (4260) via e + e − annihilations. Namely, its production in e + e − annihilations is mainly via the direct coupling to its compact cc core which contains the 3 S 1 (cc) configuration.
Then, the HQSS breaking allows the mixture of the 3 S 1 (cc) core with the long-distance component ofDD 1 (2420) + c.c. which can couple to 3 D 1 (cc) via an S-wave interaction. The wave function renormalization will dress the nonvanishing γ * -3 S 1 (cc) coupling and the coupling of Y (4260) tō DD 1 (2420) + c.c. as investigated in Ref. [8]. As a result of this scenario, it allows for the decay of Y (4260) → D * D * to occur not only via the dominantDD 1 (2420) + c.c. component but also through the direct coupling of the cc core to D * D * as illustrated in Fig. 1.
In the framework of non-relativistic effective field theory (NREFT) the Lagrangians for the coupling vertices in Fig. 1 can be written as [7,8,19,20] where f π = 132 MeV and the effective coupling for Y (4260) andDD 1 (2420) is y eff = (3.94 ± 0.04)GeV −1/2 which has been determined by the combined analysis of e + e− → J/ψππ, h c ππ and DD * π + c.c. [7,8]; the effective coupling constants h ′ and g π can be determined by the processes of D 0 1 → D * + π − and D * − → D 0 π − , respectively. The direct coupling for Y (4260) → D * The decay amplitude for the loop diagrams in Fig. 1 (b) and (c) can then be expressed as where p 1 , p 2 and l are the four momenta of the D * ,D * and π, respectively. In the last step, we have used p 2 1 = p 2 2 = m 2 D * and p i 2 = −p i 1 in the center-of-mass (c.m.) frame. The factor of 3/2 comes from the isospin symmetry and function C ijk and I ijk are defined as follows: It is interesting to compare the transition of Fig. 1 with the hidden charm decay channels such as Y (4260) → Z c (3900)π [8,19] and χ c0 ω [21]. Following the NREFT power counting scheme of Refs. [5,22,23], it can be seen that the loop amplitude for the Y (4260) → Z c (3900)π is ultraviolet where the factor of v 3 comes from the vertices, which is significantly suppressed in respect of the contact interaction. Because of the D-and P -wave pionic couplings given by Eqs. (2) and (3) , respectively, the loop decay amplitude can be split into P -wave and F -wave parts as The first term contributes to the decay into the D * D * in a P -wave, while the second contributes to that in a F -wave. While the F -wave part is UV convergent, the P -wave part diverges and needs to be regularized and renormalized. The UV divergence can be absorbed by introducing a counterterm. However, the tree-level term of Fig. 1 (a) cannot serve as the counterterm for the loop amplitude of Fig. 1 (b) and (c) since diagram (a) is introduced to incorporate the 3 S 1 (cc) coupling to D * D * while in diagrams (b) and (c) the S-waveDD 1 (2420), which leads to the transitions to D * D * , couples to the 3 D 1 (cc) in the heavy quark limit [24]. This means that the UV divergence here needs to be absorbed into a different counterterm. Here we will regularize the UV divergence practically using a form factor with a cutoff, see below. The cutoff will be treated as a free parameter, which effectively takes the place of the counterterm at a given scale.
In order to regularize the UV contributions in the loop integral, we introduce a monopole form factor for each propagator to take into account the off-shell effects in the loop integral: where Λ 1 ≡ m D 1 + αΛ QCD and Λ 2 ≡ m D + αΛ QCD , with Λ QCD = 220 MeV and α a parameter of order unity, are defined for the heavy charmed mesons. For the light pion exchange the cut-off Λ π is within a range of 0.5 ∼ 1 GeV as usually adopted. Then the loop amplitude I ijk can be expressed as: with Here the functions A and B are defined as where P ≡ p 1 + p 2 is the initial momentum, x and y are the Feynman parameters, and ∆ =   Fig. 3 where the tree-level diagram represents the charmonium transitions and the loop diagram illustrates the Y (4260) contribution via its molecular component.
As mentioned earlier, the tree diagram also contains the contribution from the short-distance core of the Y (4260).
The effective Lagrangian for the vector charmonium couplings to the virtual photon is described by the vector meson dominance (VMD) model: while the strong couplings for ψ i (i = 1, 2, 3) to the D * (s)D * (s) meson pairs are as follows [20,25]: where the coupling constants g ψ i D * where q 1 and q 2 are the incoming four-momenta of the electron and positron, respectively, p 1 is the Eqs. (16) and (17), respectively. The Gaussian form factor suppresses the resonance contributions when they become far off-shell, and the parameter β controls the suppression. As a reasonable assumption to reduce the number of parameters, we assume that these two processes share the same value for β which means that the strong couplings for ψ i to D * D * and D * sD * s have the same suppression behavior when the resonances become off-shell. The ψ i states and the Y (4260) can interfere through many possible intermediate hadron loops which can introduce energy-dependent complex phases. In order to parameterize such effects, we also introduce a few constant phases, TABLE I. The masses, total widths and leptonic partial widths adopted for the charmonium states from PDG [28]. In Eq.
is the product of the bare coupling y/f Y and the Y (4260) propagator defined in the molecular picture [8] which has the following expression: where the subtracted self-energy [26] with Σ 1 (E cm ) the Y (4260) self-energy due to theDD 1 (2420) loop. In the MS subtraction scheme, the self-energy is given by [8]. We use m Y = (4.217 ± 0.002) GeV and Γ non−DD 1 = (0.056 ± 0.003) GeV are determined in the combined analysis of e + e − → J/ψππ and h c ππ [7], and the wave function renormalization constant Z ≃ 0.13 is determined in Ref. [8]. The values of m ψ i , Γ ψ i →e + e − and Γ ψ i (i = 1, 2, 3) are taken from those given by the Particle Data Group (PDG) [28], which are listed in Table I. The leptonic decay coupling constants of the charmonium states defined by the VMD model in Eq. (14) can thus be determined.
To further reduce the number of parameters we assume the SU(3) flavor symmetry for the strong couplings of the same charmonium states to D * D * and D * sD * s so that they take the same value, i.e. g ψ 2 D * D * = g ψ 2 D * sD * s and g ψ 3 D * D * = g ψ 3 D * sD * s . In total there are 11 parameters to be fitted from the cross section data: four cutoff parameters (α, Λ π , Λ K and β), four coupling constants (g ψ i D * D * and g eff Y D * D * which is the coupling of the short-distance core of the Y (4260) to the D * D * ), and three phases. We note in advance that due to the lack of precise experimental measurements for the e + e − → D * sD * s some of the parameters cannot be well constrained in the numerical fitting, and we anticipate that the main contributions to the χ 2 value will be from the D * D * channel.
In Table II the values of the fitted parameters are listed. The value of β, which bears a large uncertainty, is consistent with the reasonable order of 1 GeV. The cutoff parameter α is consistent    Table III. With the fitted couplings for the charmonium states to D * D * and D * sD * s , we can also obtain their corresponding partial decay widths which are also listed in Table III. With the fitted parameters in Table II, we find that the cross section of e + e − → D * D * can be well described. The line shape from the best fit is plotted in Fig. 4 and compared with the Belle data [16]. An apparent feature is that the threshold enhancement in the measured e + e − → D * D * line shape can be largely accounted for by the ψ(4040) while the contributions from Y (4260) and other charmonium states are rather small below 4.2 GeV. The bump between 4.1 and 4.2 GeV can be described well by the contributions from ψ(4040) and ψ(4160) and their interference. Notice that the relative phase between these two states leads to destructive interference in the energy regions of below ψ(4040) or above ψ(4160) while in the region between their masses the interference is constructive. As a result, the rise of the cross section in the near threshold region is enhanced, although the ψ(4040) coupling to D * D * is in a P wave.
Comparing the dotted curve (denoted by "ψ i " in the figure), which is the sum of the contributions from all considered conventional charmonium states, with the solid curve, which is the best fit result, or with the experimental data, one sees that the dip around 4. We also tried a fit without including the Y (4260), and found that the cross section at energies above the dip, which correspond to the region around theDD 1 (2420) + c.c. threshold, cannot be well described. In fact, a negligibly small contribution from the Y (4260) is consistent with the molecular picture.
From Fig. 4 we also see that the cross sections in the region of 4.4 ∼ 4.6 GeV can be well described by the interference between the ψ(4415) and the Y (4260). Despite this, we need to mention that the S-wave open thresholds of D * D 1 (2420) + c.c. and D * D 2 (2460) + c.c. have not been taken into account, and they could play a role in the region between 4.4 and 4.5 GeV. We leave their contributions to be investigated more elaborately in future studies when more data are available.
As already mentioned the present experimental data for e + e − → D * sD * s [17] do not allow a reliable determination of the parameters in this channel. As shown in Fig. 5, with the present fitted parameters only the ψ(4415) can produce a resonance structure in the cross section line shape. The exclusive contributions from these three states are also presented. The theoretical curve shows a flattened line shape near threshold which is different from that in e + e − → D * D * .
Although the D * + s D * − s threshold, 4.22 GeV, is very close to the mass of the Y (4260), we do not see a near-threshold enhancement due to the Y (4260). We expect that the contribution of the Y (4260) in this process should be smaller than that in the e + e − → D * + D * − since the intermediate kaon in Fig. 3 (b) cannot go on shell, contrary to the case of the pion. However, one notices that the poor data quality do not allow a more quantitative restriction on the Y (4260) contribution to this process. This situation is also reflected by the poor determination of the cutoff energy Λ K shown in Table II. A more precise measurement of the e + e − → D * sD * s cross section line shape is highly recommended.

IV. SUMMARY
In this work we have studied the cross section line shapes of the e + e − → D * D * and D * threshold. The current data present a clear evidence for such a structure. Yet, more precise data are necessary to make the conclusion more solid. For the e + e − → D * sD * s channel, the present experimental data from Belle [17] have poor quality. However, although the data do not allow any conclusion on the role played by charmonium states, we do not expect sizeable contributions from the Y (4260). The future precise data from BESIII for these two channels will be able to clarify the role played by the Y (4260) and provide valuable insights into its internal structure.