Prediction of an exotic state around 4240 MeV with $J^{PC}=1^{-+}$ as C-parity partner of Y(4260) in molecular picture

The possibility of the Y(4260) being the molecular state of $D\bar D_1(2420)+\mathrm{c.c.}$ is investigated through one boson exchange model. It turns out that the potential of $J^{PC}=1^{--}$ state formed by $D\bar D_1(2420)+\mathrm{c.c.}$ is attractive and it is strong enough to bind them together when the momentum cutoff $\Lambda \gtrsim 1.5$ GeV. To produce the Y(4260) with correct binding energy, we need $\Lambda\approx 2.3$ GeV. Besides, $D\bar D_1(2420)+\mathrm{c.c.}$ can also form a state with exotic quantum numbers, $J^{PC}=1^{-+}$, and its potential is more attractive than that of the $J^{PC}=1^{--}$ state. Therefore, an exotic state with mass around 4240 MeV is expected to be exist. Our estimation of the mass of the $J^{PC}=1^{-+}$ state in charmonium region is in agreement with that prodicted by the chiral quark model and the lattice QCD.

The possibility of the Y(4260) being the molecular state of DD1(2420) + c.c. is investigated through one boson exchange model. It turns out that the potential of J P C = 1 −− state formed by DD1(2420) + c.c. is attractive and it is strong enough to bind them together when the momentum cutoff Λ 1.5 GeV. To produce the Y(4260) with correct binding energy, we need Λ ≈ 2.3 GeV. Besides, DD1(2420) + c.c. can also form a state with exotic quantum numbers, J P C = 1 −+ , and its potential is more attractive than that of the J P C = 1 −− state. Therefore, an exotic state with mass around 4240 MeV is expected to be exist. Our estimation of the mass of the J P C = 1 −+ state in charmonium region is in agreement with that prodicted by the chiral quark model and the lattice QCD.

I. INTRODUCTION
In 2005 a vector charmoniumlike state, the Y (4260), was reported by BaBar Collaboration [1] in the initialstate radiation process e + e − → γ ISR J/ψπ + π − , which was confirmed by CLEO Collaboration [2] and Belle Collaboration [3] later. It is clear that the Y (4260) contains cc quarks and is above the threshholds for DD, DD * +c.c. and D * D * . However, no signals of the Y (4260) appear in these channels [4][5][6], which indicates that it is not a conventional charmonium. Besides, there seems no room for the Y (4260) in the 1 −− cc spectrum [7]. As a candidate for the exotic meson, its nature still remains controversial and has been attracting much attention.
Several models were proposed to account for the peculiar behaviors of the Y (4260) (see Ref. [8] for a detailed discussion), among which a molecular state of DD 1 (2420) + c.c. seems to be a good choice since the Y (4260) is just below the threshold of DD 1 (2420) + c.c. and they can couple in S-wave. The mechanisms of the formation of the molecular Y (4260) was discussed in Refs. [9,10]. Although it was argued in Ref. [11] that the production of DD 1 (2420) + c.c. in the electron-positron collisions is forbidden in the heavy quark limit due to the heavy quark spin symmetry (HQSS) and in turn suppressed in the real world, Ref. [12] showed that the HQSS breaking is strong enough so that the molecule interpretation of the Y (4260) does not contradict the current experimental data. From the light-quark perspective, it is claimed that the Y (4260) has a sizeable DD 1 (2420)+c.c. component, which is, however, not completely dominant [13]. By assuming the Y (4260) being the DD 1 (2420) + c.c. molecule, its properties have been discussed in Refs. [12,[14][15][16][17]34]. Furthermore, such interpretation is supported by the new expermental data: an enhancement at the DD 1 (2420) + c.c. threshold in the J/ψππ channel [19] and the observations of Z c (3900)π [20,21] and X(3872)γ [22] in the mass region of the Y (4260). We refer to Ref. [23] for more details of this molecule picture.
The DD 1 (2420) + c.c. can also form a system with positive C-partity, which is definitely an exotic state, if exists, since J P C = 1 −+ is not allowed for tradiational qq mesons. Within the chiral quark model, Ref. [10] showed that the DD 1 (2420) + c.c. with J P C = 1 −+ can form a bound state with a mass of 4253 ∼ 4285 MeV. Besides, it is predicted by using the lattice QCD [24] that the J P C = 1 −+ state in the charmonium region has a mass of m(1 −+ ) = m ηc + 1233 ± 16 MeV = 4217 ± 16 MeV, just below the Y(4260), which gives us more confidence in the existence of the DD 1 (2420)+c.c. bound state with J P C = 1 −+ . On the other hand, the production and the decay of such exotic state were discussed in Ref. [25] under the assumption of the Y(4260) being a molecule of DD 1 (2420) + c.c. where some guidance for the experiments was given.
In this paper we use the vector meson exchange interaction between DD 1 (2420) + c.c. to investigate whether it is possible for them to form the J P C = 1 −− and J P C = 1 −+ molecules. In addition, we also discuss the influence of σ exchange on the potential. Note that there are two D 1 states with similar masses while quite different decay widths, the narraw D 1 (2420) and the wide D 1 (2430). We only use the narraw one (denoted by D 1 throughout the rest of the paper) since D 1 (2430) is too wide to form a molecular state. We assume that the potential between the components of the DD 1 + c.c. molecule is dominant by the vector meson exchange interactions since the psudoscalar meson exchange between DD 1 + c.c. is forbidden by parity conservation [26]. It is different from the assumption in Ref. [9] where the arXiv:1910.14455v1 [hep-ph] 31 Oct 2019 Y(4260) was considered as the molecule of DD 1 or D 0 D * through psudoscalar mesons exchange (off-diagonal potential) and σ exchange (diagonal potential). The vector mesons exchange was not included because some of the related coupling constants were not available. We emphasize that it is not advisable because D 0 is too wide to be the component of the Y(4260).
This paper is organized as follows: In section II, the vector meson exchange potential between DD 1 + c.c. is derived; Numerical results and discussions are shown in Section III.

II. FRAMEWORK
The potential between D(D) andD 1 (D 1 ) is related to the corresponding scattering amplitude. For the state with J P C = 1 −− , the element of S matrix reads Note that we have adapted the following charge conjugation conventions, and in turn the flavor wave functions of positive and negative C-parity states now read There are four feynman diagrams for DD 1 +c.c. elastic scattering by one boson (vector mesons and σ) exchange, shown in Fig.(1). Note that the scattering amplitudes of u-channel processes in the positive and negative C-parity cases carry opposite signs and in turn yield opposite potentials.
A. The vector exchange potential

The Langrangian
The coupling of heavy mesons and light vector meson nonet can be described by the effective Lagrangians, which satisfies the hidden gauge symmetry [27]. For D and D 1 mesons, the Lagrangians read explicitly [9] where and The angular momentum can be L = 0, 2 and we only consider L = 0.

Estimation of coupling constants
There are several parameters in the effective Lagrangians introduced in the last subsection. The already known ones are collected in the following, see Refs. [28], [29] and [27], respectively. These lead to g DDV ≈ 3.7. The rest constants β 2 , µ 1 and ζ 1 are not available now. The L D1D1V contains two types of interaction, which are denoted by g D1D1V and g D1D1V in Eq. (8). The second type vanishes in the nonrelativistic limit since ∂ µ V ν ∼ q µ V ν and the exchanged four-momentum q µ = (0, q) vanishes. Therefore, we only consider the first interaction, which has nothing to do with the angular momentum of D 1 . As a rough estimation, we take g D1D1V ≈ g DDV since they all describe the P-wave coupling of heavy mesons and the light vector meson. D and D 1 have the same behaviors in such case where the spin of D 1 does not participate in.
The L DD1V also contains two types of interaction, denoted by g DD1V and g DD1V in Eq.(9). The first one describes the S-wave coupling, which dominates the interaction and hence the second one is neglected. We assume that the coupling of KK 1 V is approximately the same as that of DD 1 V because s quarks in K and K 1 and c quarks in D and D 1 are all spectators during the interactions. We use the decay of K 1 into Kρ to estimate g KK1V and in turn g DD1V .
The partial wave amplitude [30] for L = 0 can be expressed as where 3/2 accounts for the isospin factor. Sum of the polarizations of vector mesons leads to where the K µ 1 is the polarization of K 1 and ρ µ is the polarization of ρ. The decay width reads In PDG [31], there are two different K 1 states, K 1 (1270) and K 1 (1400). The decay widths are respectively, which lead to In our calculation, g DD1V will vary from 0.6 GeV to 3.9 GeV.

The potential in position space
For the vector exchange in the first two diagrams in Fig.(1), the scattering amplitude reads where 1 and 2 are the polarizations of initial and final D 1 's, respectively. Note that 1 · * 2 = 1 for the S-wave to S-wave scattering.
The corresponding potential in momentum space reads After Fourier transformation we obtain the potential in position space where is the Yukawa potential. For the vector exchange in the last two diagrams, and the potential (see e.g. Ref. [32] for more details of such Fourier transformation) reads with S 12 (r) = 3 1 ·r 2 ·r − 1 · 2 . We have used the facts that 1 · 2 = 1 and S 12 = 0 for the S-wave to S-wave scattering [33]. The delta function has been ignored since the components have finite sizes. After taking the isospin factor into account, we obtain For the J P C = 1 −± state, the total vector exchange potential reads where I = 0, 1. Note that m ρ ≈ m ω , the potentials for isovector (I=1) are very weak and we only consider the isoscalar states here. A form factor is introduced to the potential at each vertex in order to take into account the actual size of the mesons. The potentials in position space, Eqs. (31,36), become B. The σ exchange potential The σ exchange potential has been calculated in Ref. [9]. For the t-channel process, and for the u-channel process, In our calculation, the constants in the above potentials are taken to be (52) as in Ref. [9]. Note that the isospin factor is trivial in this case. For the J P C = 1 −± state, the total σ exchange potential reads (53)

III. NUMERICAL RESULTS AND DISCUSSIONS
We use the following values from PDG [31], m D1 = 2.420 GeV, (55) m ρ = 0.775 GeV, (56) m ω = 0.783 GeV, (57) m σ ≈ 0.600 GeV. (58) Besides, we take g D1D1V ≈ g DDV ≈ 3.7, as analyzed above. Using the decay of K 1 we estimate that g D1D1V is in the range from 0.6 GeV to 3.9 GeV. The vector and σ exchange potentials are shown in Fig. 3 and Fig.4, respectively. From Fig. 3 we see that Meanwhile, the σ exchange potentials are much smaller than the vector ones and therefore, it is expected that the The coupling constant g DD1V is estimated in the range from 0.6 GeV to 3.9 GeV and it is adjustable in our calculation. For each value of g DD1V in this range, we use the fact that Y(4260), as a bound state of DD 1 with J P C = 1 −− , has a binding energy of around 59 MeV to determine the momentum cutoff Λ 0 . With the same g DD1V and Λ 0 , we obtain the binding energy of the bound state of DD 1 with J P C = 1 −+ , around 60 ∼ 120 MeV. We also include the σ exchange potential and its influence turns out to be insignificant. The results for different g DD1V and g DD1σ are listed in Table I. If we assume that the Y(4260) is a pure DD 1 + c.c. bound state with Λ 0 ≈ 2.3 GeV, its 1 −+ partner should has a mass around 4200 MeV. Since the Y(4260) may be a mixture of DD 1 + c.c. molecule and ψ(nD) [34], then a more commonly used Λ 0 ≈ 1.5 GeV leads to a 1 −+ DD 1 + c.c. molecule with a mass around 4280 MeV. Therefore, we expected this exotic 1 −+ DD 1 + c.c. molecule to be around 4240 MeV.
In summary, we have used the one boson exchange potential between the DD 1 + c.c., for both J P C = 1 −− and J P C = 1 −+ systems, to investigate if it is possible for them to form bound states. We use the effective Lagrangians, which satisfy the heavy quark symmetry, to describe the interaction between D and D 1 . First, we only consider the vector exchange. The coupling constabt g DD1V is taken to be in the rang from 0.6 GeV to 3.9 GeV, which is estimated from the decay width of K 1 → Kρ. It turns out that with a momentum cutoff Λ = 2.20 ∼ 2.44 GeV, the attractive force between the DD 1 + c.c. with J P C = 1 −− is strong enough to form a bound state, corresponding to the Y (4260). The Cparity partner of the Y(4260), i.e. the exotic DD 1 + c.c. bound state with J P C = 1 −+ , is predicted to be exist. Its mass is expected to be less than the Y(4260), which is consistent with the prediction by lattice QCD and chiral quark model. The σ exchange potential is then included and it turns out to have little influence on the binding energies. The possible decay modes of the predicted exotic 1 −+ state include η c η [25] and χ c1 η [35].
We GeV for the right one. The "t-channel" reprensents the potential for the first two diagrams in Fig.(1) and the "u-channel" for the second two diagrams. "C = +" and "C = −" reprensent the total potentials for positive and negetive C-parity states, respectively.  I. The binding energy of DD1 + c.c. bound state. We choose gDD 1 σ = 0 or ± 0.58, gDD 1 V = 0.6 or 3.9 and Λ = 1.5, 2.0 or 2.5 GeV to investigate how the binding energy of the DD1 bound state with J P C = 1 −+ or 1 −− changes. In the last two columns, Λ0 means the cutoff, which, together with other specified couplings, yields the experimental binding energy of the Y(4260), 59 MeV and E0 means the expected corresponding bindig energy of the J P C = 1 −+ state.