$Y(4260)$ as four-quark state

We treat the $Y(4260)$ resonance as a four-quark state in the framework of the covariant confining quark model. We study two choices of the interpolating current, either the molecular-type current which effectively corresponds to the product of $D$ and $\bar D_1$ quark currents or tetraquark one. In both cases we calculate the widths of decays $Y(4260)\to Z_c(3900)+\pi$ and $Y(4260)\to D^{(\ast)}+\bar D^{(\ast)}$. It is found that in both approches the mode $Y\to Z^+_c + \pi^-$ is enhanced compared with the open charm modes. However the absolute value of the $Y\to Z^+_c + \pi^-$ decay width obtained in molecular picture is arguably too large. On the other hand the value obtained in tetraquark picture is reasonable.


I. INTRODUCTION
In 2005 BABAR Collaboration observed a broad resonance around 4.26 GeV in analyzing the mass spectrum of π + π − J/ψ in initial-state-radiation (ISR) production e + e − → γ ISR π + π − J/ψ [1]. Since this resonance was found in the e + e − annihilation through ISR, its spin-parity is J P C = 1 −− . However, its mass does not fit any mass of charmonium states in the same mass region, such as the ψ(4040), ψ(4160), and ψ(4415). Moreover, the Y (4260) has strong coupling to the π + π − J/ψ final state, but no evidence was found for coupling to any open charm decay modes as D ( * )D( * ) , D s where D ( * ) = D or D * [2][3][4][5][6]. These properties perhaps indicate that the Y (4260) state is not a conventional state of charmonium [7].
In addition to the Y (4260), the the BESIII Collaboration reported on the observation of another exotic state named as Z c (3900) in the reaction e + e − → π + π − J/ψ [8]. It carries an electric charge and couples to charmonium. A fit to the π ± J/ψ invariant mass spectrum results in a mass of M Zc = 3899.0 ± 3.6(stat) ± 4.9(syst) MeV and a width of Γ Zc = 46 ± 10(stat) ± 20(syst) MeV. This state was confirmed by Belle [9] and CLEO [10] Collaborations. Then the BESIII Collaboration observed a distinct charged structure in the (DD * ) ∓ invariant mass distribution of the process e + e − → π ± (DD * ) ∓ [11]. Assuming this structure and the Z c (3900) → πJ/ψ signal are from the same source, the ratio of partial widths is Γ(Z c → DD * )/Γ(Z c → πJ/ψ) = 6.2 ± 2.7 . That means that the Z c (3900) state has a much stronger coupling to DD * than to πJ/ψ [12]. Now we go back to the Y (4260) and shortly review some theoretical efforts to understand the underlying structure of this state. We refer to Ref. [7] for more complete review of this subject. Probably, one of the first attempts to analyze the possible interpretations of the Y (4260) was undertaken in Ref. [13]. The conclusion has been done that only the hybrid charmonium picture is not in conflict with available experimental data from BABAR measurement. The interpretation of the Y (4260) as a charmonium hybrid has been also explored in Refs. [14,15].
The three-body J/ψππ and J/ψKK systems have been treated as coupled channels in Ref. [16]. It was found by solving the Faddeev equations that the resonance Y (4260) can be generated due to the interaction between these three mesons. The Y (4260) has been identified as the low member of the pair ψ(4S) − ψ(3D) charmonium by using simple quark model [17].
In the paper [18] it was suggested that the Y (4260) is a χ c1 − ρ 0 molecule. In that picture one can show that the width of decay Y (4260) → π + π − J/ψ is larger than Y (4260) → DD which has not been observed.
It was proposed in Ref. [19] to interpret the Y (4260) as the first orbital excitation of a diquark-antidiquark state ([cs][cs]). In this case the Y (4260) should decay predominantly to D sDs .
Masses of heavy tetraquarks have been calculated in the relativistic quark model [20]. It However, as mentioned above, no evidence was found for the decays Y (4260) → s [2][3][4][5][6]. In the Ref. [21] it was assumed that the Y (4260) is DD 1 molecular state where D = D(1870) is the psedoscalar meson with the quantum numbers I(J P ) = However, in Ref. [22] it was argued that the production of an S-wave DD 1 pair in ℓ + ℓ − annihilation is forbidden by the heavy quark spin symmetry. This argument is certainly not in the favor of considering the Y (4260) as S-wave DD 1 state. Despite of this, there are many studies of the Y (4260) as DD 1 molecular state. We briefly mention some of them. By assuming that the Y (4260) is a DD 1 molecular state, some hidden-charm and charmed pair decay channels of the Y (4260) via intermediate DD 1 meson loops within an effective Lagrangian approach have been investigated in Ref. [23]. By treating the Y (4260) as a DD 1 weakly bound state and also the Z c (3900) as a DD * molecule [24], the two-body decay Y (4260) → Z c (3900)+π has been studied. Moreover the decay mode Y (4260) → J/ψ+π + π − was also computed.
The approach we propose is based on the covariant confining quark model (CCQM) [25][26][27] which represents an effective quantum field treatment of hadronic effects. The model is derived from Lorentz invariant non-local Lagrangian in which a hadron is coupled to its constituent quarks. Hadrons are characterized by size parameters Λ H from which the strength of the quark-hadron coupling can derived. It is done by using so-called compositeness condition [28,29], this condition requires the wavefunction renormalization constant of the hadron to be zero Z H = 0. Besides reducing the number of free parameters (i.e. couplings), it also guarantuees a correct description of bound states as dressed (with no overlap with bare states) and solves the double counting problem. The vertices are described by a Gaussian-type vertex functions which are supposed to effectively include contributions from gluons (which are not present). Thanks to the built-in confinement, based on a cutoff in the integration space of Schwinger parameters (stemming from representation of quark proparators), the model can be used for description of arbitrary heavy hadrons. The model should be understood as a practical tool for computing hadronic form factors from assumed quark currents, which is, in this text, applied to Y (4260) and Z c (3900) states.
In our earlier papers devoted to description of the multi-quark states Refs. [30,31], first, we have explored the consequences of treating the X(3872) meson as a tetraquark, i.e. diquark-antidiquark bound state. We have calculated the decay widths of the observed channels and concluded that for reasonable values of the size parameter of the X(3872) one finds consistency with the available experimental data. Then we have critically checked in Ref. [32] the tetraquark picture for the Z c (3900) state by analyzing its strong decays. We found that Z c (3900) has a much more stronger coupling to DD * than to J/ψπ which is in discord with experiment. As an alternative we have employed a molecular-type four-quark current to describe the decays of the Z c (3900) state. We found that a molecular-type current gives the values of the above decays in accordance with the experimental observation. By using molecular-type four-quark currents for the recently observed resonances Z b (10610) and Z b (10650), we have calculated in Ref. [33] their two-body decay rates into a bottomonium state plus a light meson as well as into B-meson pairs. A brief sketch of our findings may be found in Ref. [34].
In the present paper we treat the Y (4260) resonance as a four-quark state. We study two choices of the interpolating currents either the molecular-type current which effectively corresponds to the product of D andD 1 quark currents or tetraquark one. In both cases we calculate the widths of decays Y (4260) → Z c (3900) + π and Y (4260) → D ( * ) +D ( * ) .
The paper is organized as follows: Two subsequent sections II and III are dedicated to the general formalism for describing Y (4260) as four quark molecular state and tertaquark state respectively, full expressions of studied quark currents and related amplitudes are provided.
In the next, last section the decay width formulas are written down and used to reach our numerical results which are presented together with our conclusion.

II. Y(4260) AS FOUR-QUARK STATE WITH MOLECULAR-TYPE CURRENT
We start with an assumption that both the Y (4260) and Z + c (3900) resonances are fourquark states with the molecular-type currents given in Table I.
Their nonlocal generalizations are given by The reduced quark masses are specified as where we assume no isospin-violation in the u − d sector, i.e. m u = m d . The Fouriertransform of the vertex function Φ may be written as read as Here, Γ 1 ⊗ Γ 2 = γ 5 ⊗ γ 5 for DD pair, ǫ * ν 1 γ ν 1 ⊗ γ 5 for D * D pair, and ǫ * ν 1 γ ν 1 ⊗ ǫ * ν 2 γ ν 2 for D * D * pair. The momenta are defined as The calculation of the matrix element of the decay Y → Z c + π is more involved because it is described by three-loop diagram as shown in Fig 1 b. One has Here The momenta are defined as

III. Y(4260) AS FOUR-QUARK STATE WITH TETRAQUARK CURRENT
Now we treat the Y (4260) as four-quark state with the tetraquark current: where the charge conjugate matrix is chosen in the form C = γ 0 γ 2 so that C T = −C, C † = C and C 2 = C. Its nonlocal generalization is given by The matrix elements of the decays Y u → D 1 +D 2 read as The momenta are defined as The matrix element of the decay Y → Z c + π is written down The momenta are defined as ,

IV. NUMERICAL RESULTS AND CONCLUSION
We remind the formulas for the two-body decay widths expressed via Lorentz form factors.
We calculate the decay widths and put their numerical values in Table II. We have taken the value of Z c size parameter to be equal Λ Zc = 3.3 GeV as was obtained in our paper [32]. We vary the value of Y size parameter in some vicinity of this average value Λ Y = 3.3 ± 0.1 GeV. One can see that in both approches the mode Y → Z + c + π − is enhanced compared with the open charm modes. The two approaches differ in the decay width values Γ(Y → Z + c π − ). Comparison with the total decay width of the Y (4260) particle from experiment 55 ± 19MeV [35] disqualifies the molecular picture. As a result, one can conclude that the CCQM model calculations favor the tetraquark picture of the Y (4260) state since it leads to reasonable number of the decay width into Z + c π − . Y → D * 0 +D 0 (0.39 ± 0.14) · 10 −2 0.32 ± 0.09 Y → D * 0 +D * 0 0 (0.19 ± 0.08) · 10 −3