Discussions on the Line-shape of $X(4660)$ Resonance

A careful re-analysis is made on $e^+e^-\to X(4660)\to (\Lambda_c\bar{\Lambda}_c)/(\psi'\pi\pi)$ processes, aiming at resolving the apparent conflicts between Belle and BESIII data above $\Lambda_c\bar{\Lambda}_c$ threshold. We use a model containing a Breit-Wigner resonance and $\Lambda_c\bar{\Lambda}_c$ four-point contact interactions, with which the enhancement right above $\Lambda_c\bar{\Lambda}_c$ threshold is well explained by a virtual pole generated by $\Lambda_c\bar{\Lambda}_c$ attractive final state interaction, located at $M_V=4.566\pm0.007$ GeV. Meanwhile, $X(4660)$ remains to be a typical Breit-Wigner resonance, and is hence of confinement nature. Our analysis strongly suggests the existence of the virtual pole with statistical significance of $4.2$ standard deviation ($\sigma$). Nevertheless, the conclusion crucially depends on the line-shape of cross sections which are of limited statistics, hence we urge new experimental analyses from Belle II, BESIII, and LHCb to settle down the issue.

Since the discovery of X(3872) in 2003 [1], hadronic exotic states, or called "XY Z" states, open a new window for hadron physics researches. Those states do not match the energy level positions predicted by naive quark models (e.g. the the Godfrey-Isgur model in cc sector [2]), and most of them are very narrow despite of locating above open charm (bottom) thresholds, which have intrigued theorists for recent years. Various models are established to understand such states: for example, modified quark models [3][4][5] which treat them as confining states; "non-resonance" interpretations regarding them as branch cut singularities [6]; also, one widely accepted mechanism is so called "hadronic molecules" [7][8][9][10].
More recently, a much precise measurement by BESIII collaboration gives the cross sections at four center-of-mass energies for e + e − → Λ cΛc cross section near Λ cΛc threshold [23]. As shown in Fig. 2 of Ref. [23] (or the left panel of Fig. 3 in this paper), the BESIII data is questioned to have conflicts with the Belle data [15,24]: especially the line-shape of BESIII data appears nearly horizontal while that from Belle shows a significant growth. The results from some earlier works are not compatible with BESIII data, see e.g. Fig. 2 of Ref. [17].
This work aims at disentangling this problem: it is suggested that a virtual pole via Λ cΛc contact interactions, in addition to the X(4660) Breit-Wigner resonance, could well explain the odd line-shape which are failed to be described by previous studies. To begin with, we assume that X(4630) and X(4660) are the same particle and denote it as X, with quantum number 2s+1 L J = 3 S 1 2 . For Λ cΛc channel, the coupling among XΛ cΛc and the QED transition from a photon to X are introduced as following: where X µν and F νµ are the strength tensors of the X and the photon, respectively. The contact interaction between Λ cΛc is which simulates vector and axial-vector meson exchanges 3 . For ψ ππ channel, the interactions may be complicated with various Lorentz structures, but only the final two pions with total isospin and angular momentum IJ = 00 and their final state interaction (FSI) are considered, just like Ref. [26]. Therefore the momentum dependence from derivative couplings of pions can be absorbed into their FSI, leaving the effective vertices as following where g 2,3 are fit parameters and the A ππ stands for the FSI term between two pions (see e.g. Ref. [26]). Based on Eqs. (1), (2) and (3), a model of K matrix type concerning a mixed mechanism of s channel X state and Λ cΛc contact interaction, can be established as shown in Figs. 1 and 2 4 . Specifically, the K matrix sector satisfying final state theorem is where T X and T ci label the tree diagrams of s channel Λ cΛc → X(4660) → Λ cΛc , and Λ cΛc contact vertex as FSI in 3 S 1 channel, respectively, see Fig. 1; m is the mass of Λ c . Note that Figure 2: The Feynman diagram of e + e − → X(4660) → ψ ππ. A K is the K matrix sector in Eq. (4) (the same as Fig. 1), and V f is the vertex concerning ψ ππ final state.
the ψ ππ state does not show up in the above K matrix, instead it only appears as final state. This greatly simplifies the calculation since it reduces the couple channel problem to a single channel approximation. This simplification is justifiable, as discussed in Ref. [26], because the ψ ππ threshold is distant from the energy region near X(4660), giving a renormalization effect to Λ c . More importantly, the smallness of g 2 in Eq. (3) is fully consistent with experimental observations (see Eq. (5)). Finally, in each channel we adopt a complex number serving as coherent background in the scattering amplitude.
Under the above formulations a combined fit to the data from both Belle [13,14] and BE-SIII [23,27] (also Babar data [12] for ψ ππ final state) is performed, with a quite good fit quality χ 2 /d.o.f. = 26.6/33, indicating that the present model is compatible with both Belle and BESIII data. As shown in Fig. 3, it is evident that the good fit quality originates from an enhancement near the Λ cΛc threshold. Further investigations find that a virtual state lying below but very close to Λ cΛc threshold, located at M V = 4.566 ± 0.007 GeV, causes the enhancement. The main fit results are summarized in Table. 1.

Parameters
Values M pole (GeV) 4.645 ± 0.028 Γ pole (GeV) 0.078 ± 0.029 M V (GeV) 4.566 ± 0.007 M X (GeV) 4.604 ± 0.020 g 1 2.151 ± 0.279 λ (GeV −2 ) −15.899 ± 2.718 Table 1 To proceed, the statistical significance of such virtual pole is studied. As a control, only Breit-Wigner effect is employed to fit all the data, giving the mass and width of X(4660) as M pole = 4.674 ± 0.043 GeV and Γ pole = 0.147 ± 0.110 GeV, but the fit quality becomes χ 2 /d.o.f. = 44.4/34. Comparing with the mixture mechanism (χ 2 /d.o.f. = 26.6/33), the statistical significance of the virtual state is obtained to be 4.2σ, indicating strong evidence in support of the virtual pole as truly existing. Furthermore, to test the stability of the poles, we also use the Breit -Wigner term for X(4360) to replace the constant coherent background, with the mass and width fixed, while the residue varies. Even though the behaviour near ψ ππ threshold changes a little (see Fig. 4), the pole positions are found to be stable against the  It should be emphasized that from a general point of view in quantum scattering theories, virtual states are believed to arise in attractive interactions that are not strong enough, being "precursors" of bound states: when the attractive coupling is strong enough they become bound states. In Table. 1 the parameter λ indicates the attractive force between Λ c andΛ c . Figure. 5 shows the trajectory of the poles against the increase of |λ|: when the Λ cΛc contact interaction grows stronger, the virtual pole moves closer to the threshold; meanwhile, the X(4660) resonance becomes narrower. It is worth noticing that the pole trajectory is model dependent: if the kinematic factor iρ were replaced by the entire two point function B 0 , the virtual pole would go up to the first sheet and become a bound state, given a large enough |λ|. All in all, these analyses exhibits a very clear physical picture: the virtual pole is produced by FSI, while X(4660) state is a typical Breit-Wigner state with a pair of nearby poles. According to the pole counting rule [28] (which has been successfully applied to the studies of "XY Z" physics in Refs. [26,[29][30][31]), our analysis suggests that X(4660) is of confinement nature.
Furthermore, the ratio between the decay widths Γ(X → Λ cΛc ) and Γ(X → ψ ππ) can also be estimated as: which is in agreement with Ref. [19]. In summary, this paper demonstrates the evidence of a virtual state in e + e − → X(4660) → Λ cΛc process with significance of 4.2σ. By employing a model with both s channel X(4660) propagator and Λ cΛc FSI, the data from Belle and BESIII of e + e − → X(4660) → (Λ cΛc ) can be fitted well simultaneously. The virtual state plays a crucial role in respect to that fit since it gives a significant threshold enhancement. This pole is regarded as a Λ cΛc molecular virtual 4 . state and would become a bound state if the Λ cΛc contact coupling were larger, while X(4660) is of confinement nature. Finally, since the statistics of the data is limited, the confirmation of it urgently recalls more experimental measurements with higher precisions, such as Belle II, BESIII, LHCb, etc.