Possible $P$- wave $D_s\bar{D}_{s0}(2317)$ molecular state $Y^{'}(4274)$

Stimulated by the measurement of the $J\psi\phi$ decay model of $Y(4274)$ by the LHCb Collaboration, we consider a possible interpretation of this state as a hadron molecular-a bound state of $D_s$ and $\bar{D}_{s0}(2317)$ mesons. Using effective Lagrangian approach, we calculate the two-body strong decay channels $Y(4274)\to{}J/\psi\phi,\chi_{c0}\eta,\chi_{c0}\eta,D^{*}_s\bar{D}_s$, $D\bar{D}^{*}$, $K\bar{K}^{*}$, and $\phi\phi$ through hadronic loops and three-body decays into $\pi^0{}D_s\bar{D}_s$. In comparison with the LHCb data, our results show that $Y(4274)$ cannot be assigned to be a $D_s\bar{D}_{s0}(2317)$ molecular state. The calculated partial decay widths with $J^P=1^{+}$ $D_s\bar{D}_{s0}$ molecular state picture indicates that allowed decay modes, $\chi_{c0}\eta$ and $\chi_{c1}\eta$, may have the smallest branching ratio and are of the order of 0.0 MeV. Future experimental measurements of such two processes can be quite useful to test the different interpretations of the $Y(4274)$. If $P-$wave $D_s\bar{D}_{s0}$ molecular exist [we marked as $Y^{'}(4274)$], the total decay is at the order of 1.06-1.84 MeV, which seems to be within the reach of the current experiments such as Belle II. In addition, the calculated partial decay widths indicate that allowed decay mode, $D\bar{D}^{*}$, may have the biggest branching ratio. The experimental measurements for this strong decay process could be a crucial to observe such a new state $Y^{'}(4274)$.


I. INTRODUCTION
Thanks to the great progress of the experiment in the past several decades, many hadrons that cannot be ascribed into the simpleqq configuration for mesons or qqq configuration for baryons have been reported [1]. For example, various hiddencharm pentaquark were observed in the J/ψp(Λ) invariant mass from the heavy baryon decay Λ + b → K − J/ψp (Ξ − b → K − J/ψΛ) at LHCb Collaboration, P c (4312, 4440, 4450) and P cs (4459) [2,3]. Their confirmation, and determination of their quantum numbers, would allow new insights into the binding mechanisms present in multiquark systems, and help improve understanding of QCD in the nonperturbative regime.
In 2017, a charmonium-like meson named Y(4274) was observed again by the LHCb collaboration in the analysis of the B + → J/ψφK + reaction [4,5]. The observed resonance masses, widths, and favorable quantum numbers are MeV, respectively, which are consistent with the early CDF Collaboration [6] report Since the statistic is not enough, its spin-parity quantum number was not confirmed by the CDF Collaboration. However, * corresponding author † Electronic address: yin.huang@apctp.org the isospin of this state is zero and it contains at least four valence quarks from the observed J/ψφ decay mode. Following the discovery of the Y(4274), several theoretical studies have been performed. In the QCD sum rules approach, based on the analysis of the mass spectrum, the Y(4274) can be interpreted as the S -wave cscs state with spinparity J P = 1 + [7,8]. The compact tetraquark model, implemented by Stancu, can also describe the Y(4274) [9], while only one J P = 1 + state exists. While in Ref. [10] the mass of the Y(4274) was studied in the relativized quark model and it was shown that the Y(4274) cannot be explained as a tetraquark state, however, it can be a good candidate of the conventional χ c1 (3 3 P 1 ) state. And the χ c1 (3 3 P 1 ) explain for Y(4274) was also proposed in Ref. [11]. In the context of the QCD two-point sum rule method by taking into account the quark, gluon, and mixed vacuum condensates, Ref. [12] assigned the Y(4274) as a sextet [cs][cs] diquark-antidiquark state with spin-parity J P = 1 + . Based on the spin-spin interaction, Maiani et al. suggest that the Y(4274) may have quantum number 0 ++ or 2 ++ [13], which contradicts the experimental observation [4,5]. Moreover, a detailed calculation is performed by Zhu [14] where the Y(4274) may be described simultaneously by adding the up and down quark components.
Although the studies of Refs. [7][8][9][10][11][12][13][14] seem to indicate that this state is a compact tetraquark state or diquark-antidiquark state, Y(4274) might still be a D sDs0 (2317) hadronic molecule state. Since the mass of Y(4274) is about 12 MeV below the threshold of D sDs0 (m D s0 = 2317.8 ± 0.6 MeV and m D s = 1968.34±0.07 MeV [1]), it is reasonable to regard it is a bound state of D sDs0 . The idea comes from molecular state interpretation of deuteron due to deuteron mass is a little below the corresponding threshold and exhibit sizable spatial extension. And because the quantum numbers ofD s0 and D s are J P = 0 + and J P = 0 − , respectively, to form a bound state with quantum number J P = 1 + , the coupling between Y(4274) and its constituents should be a P wave. Indeed, it is shown in Ref. [15] that the interaction between a D s meson and aD s0 meson is strong enough to form a bound state with a mass about 4274 MeV.
From Ref [15], Y(4274) may be a molecular state. However, currently, we cannot fully exclude other possible explanations such as a compact pentaquark state [7][8][9][10][11][12][13][14]. Further research is required to distinguish whether it is a molecular or compact multi-quark state. One way to distinguish the two scenarios is to study the allowing strong decay widths of the Y(4274) baryon due to the strong decay almost saturates the total strong decay width. In the present paper we consider possible strong decay modes using an effective Lagrangian approach by assuming that Y(4274) is a hadronic molecule state of D s andD s0 .
This work is organized as follows. The theoretical formalism is explained in Sec. II. The predicted partial decay widths are presented in Sec. III, followed by a short summary in the last section.

II. FORMALISM AND INGREDIENTS
Besides the J/ψφ decay model, which other decay is allowed. We first find that the transition from Y(4274) to final states composed of purely neutral state AĀ are strictly forbidden by the conservation of the c parity. Thus, the decay of Y(4274) into η c η, χ c0 η, χ c1 η, φφ, ηη, D * sD s , DD * , KK * and π 0D s D s are allowed by considering appropriate phase space [1]. However, the transitions Y(4274) → η c η and ηη are also strictly forbidden by the conservation of angular momentum. In this work, we will calculate J/ψφ, χ c0 η, χ c1 η, D * sDs , DD * , KK * and φφ strong decay patterns of p-wave D sDs0 molecular state within the effective Lagrangians approach, and find the relation between D sDs0 molecular state and Y(4274) by comparing with the experiment observation.
Before introducing the theoretical framework, we need construct the flavor functions for D sDs0 system with definite I(J PC ). Since the Y(4274) carry quantum numbers I(J PC ) = 0(1 ++ ), the flavor function for a definite charge parity C = 1 can be easy obtained [16] Considering the quantum number J P = 1 + and the flavor function, Y(4274) should couple to its components dominantly via P− wave, and the corresponding effective Lagrangian is in the form [17] where ω D s = m D s /(m D s + m D s0 ) and ω D s0 = m D s0 /(m D s + m D s0 ).
The coupling constant g YD sDs0 in Eq. (4) can be computed by the compositeness condition [37,38], which indicates that the renormalization constants of a composite particle wave function should be zero, i.e., where the Σ T Y is the transverse part of the mass operator and relates to its mass operator via the relation.
The concrete forms of the mass operators of the Y(4274) corresponding to the diagrams in Fig. 1 is where k 2 0 = m 2 Y with k 0 , m Y denoting the four momenta and mass of the Y(4274), respectively, k 1 , m D s , and m D s0 are the four-momenta, the mass of the D s meson, and the mass of theD s0 meson, respectively. With above preparations, we can obtain the coupling constant of the D sDs0 molecule to its components Since the Y(4274) was observed in the J/ψφ invariant mass, we first calculate the J/ψφ two-body decay width of the Y(4274) via the triangle diagrams show in Fig. 2. The
We also show the definitions of the kinematics(k 0 , k 1 , k 2 , p 1 , p 2 , q,and q ′ ) used in the calculation.
hadronic decay of the D sDs0 molecular state into J/ψφ mediated by the exchange of the D * s meson. To evaluate the diagrams, in addition to the Lagrangian in Eq. 4, the following effective Lagrangians, responsible for vector meson The coupling constant g J/ψD s0 D * s = 0.225 GeV −1 and g φD s0 D * s = 0.135 GeV −1 are got from Ref. [17].
To compute the D s D * s V vertices, we also need the following effective Lagrangian [39,40] [39,40], and m ρ = 0.775 GeV. V µ and P are standard S U(4) matrices constructed with 16-plet of the vector meson contain ρ and the 16-plet of pseudoscalar mesons containing the π, respectively.
(13) With above effective Lagrangians, we can get the decay amplitudes corresponding to the diagrams in Fig. 2 B. The decay Y(4274) → χ c0 η and χ c1 η In this section, we compute the other possible decays of Y(4274) with D sDs0 molecular. Fig. 4 shows the hadronic decay of the D sDs0 molecular state into χ c0 η and χ c1 η mediated by the exchange of D s meson. The ingredients need are the The coupling constants g χ c1 D + s D − s and g χ c0 D + s D − s can be fixed from the heavy quark field theory [42].
Moreover, the effective Lagrangian, responsible for the coupling of D s0 to ηD s , is needed as well [43] L D s0 D s η = g D s0 D s η D s0 D s η, where the coupling constant is found to be g D s0 D s η = 6.40 [44]. Thus, we can obtain the following amplitudes for the de- and The minus sign in Eqs. (27) and (24) The coupling g is fixed from the strong decay width of K * → Kπ. With the help of Eq. (25), the two-body decay width Γ(K * + → K 0 π + ) is related to g as where P πK * is the three-momentum of the π in the rest frame of the K * . Using the experimental strong decay width (Γ K * + = 50.3 ± 0.8 MeV) and the masses of the particles needed in the present work [1] we obtain g = 4.61. Thus, we can obtain the following amplitudes for the decay D. The decay Y(4274) → D sDs π 0 Now we turn to D sDs π 0 three-body decay channel of Y(4274). Under the D sDs0 molecular state assignment, Y(4274) first dissociates into D + s D − s0 or D − s D + s0 . Then, the decay Y(4274) → D + s D − s π 0 occur via the transitions D ± s0 → D ± s π 0 , where D ± s0 decay into D ± s π 0 by considering the η − π 0 mixing mechanism [45,46]. The relevant Feynman diagrams ars shown in Fig. 5.
FIG. 5: Feynman diagrams for Y(4274) → D sDs π 0 decay processes. We also show the definitions of the kinematics(k 0 , p 1 , p 2 , p 3 ) used in the calculation.
The Lagrangian including η − π 0 mixing for π 0 D s0 D s have been constructed in Ref. [47] and in the form where the coupling constant g π 0 D s0 D s can be extracted by the relation In above, P π 0 is the three-momentum of π in the rest frame of D s0 . At present, the experimental partial decay width on D ± s0 → D ± s π 0 is absent. We only find the absolute branching fraction B(D ± s0 → D ± s π 0 ) is measured as 1.00 +0.00 −0.14 (stat) +0.00 −0.14 (syst) [48] and the upper limit on D s0 width is 3.8 MeV at the 95% confidence level (C.L.) [49]. In this work, we focus on whether Y(4274) can be a P-wave D sDs0 molecular state and hope that the theoretical maximum decay width cannot be compared with the experimental data. Thus, we use upper limit Γ(D s0 ) = 3.8 MeV to determine the coupling constant g π 0 D s0 D s = 1.3124 GeV and in further calculation.
The general expression of the decay amplitude Y(4274) → D sDs π 0 is where q = p 2 + p 3 .
E. The decay Y(4274) → DD * , KK * , and φφ In this section, we calculate the two-body Okubo-Zweig-Iizuka (OZI) allowed strong decays Y(4274) → DD * , KK * , and φφ. The processes is described as a quark-antiquark pair cc or ss annihilation in the initial state. Meanwhile, a light quark-antiquark pair is created and then regroups into two outgoing hadrons by a quark rearrangement process. The decay of Y(4274) into other channels, such as Y(4274) → ππ, are ignored because in these processes, which involve the creation or annihilation of twoqq (q = u, d, s, c) quark pairs, are usually strongly suppressed. The quark-level diagrams are depicted in Fig. 6 FIG. 6: The Y(4274) decay process via the OZI mechanism. q = u, d quarks for diagram (a) and q = u, d, s quarks for diagram (b). are in Fig. 7. Besides the Lagrangians above, the effective Lagrangian of KDD s0 vertex is also needed [43] where the coupling constant g KDD s0 = √ 2g K − D 0 D − s0 = 10.21 is obtained from the coupling of the D s0 to the DK channel in isospin I = 0 [44]. Putting all pieces together, we obtain the amplitudes for Y(4274) → DD * , KK * , and φφ which correspond to the diagrams in Fig. 7, where J=1 and −1 are for D 0 (D 0 ) and D + (D − ) exchange, respectively.
Once the amplitudes are determined, the corresponding partial decay widths can be obtained, which reads, where J is the total angular momentum of the Y(4274) state, the | p 1 | is the three-momenta of the decay products in the center of mass frame, the overline indicates the sum over the polarization vectors of the final hadrons, and MB denotes the decay channel of MB, i.e., J/ψφ, χ c0 η, χ c1 η, D * sDs , DD * , KK * and φφ. In Eq. 42, the p * 3 and Ω * p 3 are the momentum and angle of the particle D ± s in the rest frame of D ± s and π 0 , respectively, and Ω p 1 is the angle of D ± s in the rest frame of the decaying particle Y(4274). m π 0 D ± s is the invariant mass for D ± s and π 0 and must meet

III. RESULTS AND DISCUSSIONS
In this work, we study the strong decays of the Y(4274) to the two-body final states J/ψφ, χ c0 η, χ c1 η, D * sDs , DD * , KK * , φφ and three-body decay into π 0 D sDs assuming that Y(4274) is a D sDs0 (2317) molecular state. In order to obtain the decay width shown in Figs. 2, 4, and 5, the coupling constant g YD sDs0 should be computed first. According to the compositeness condition that we introduced in Eq. 6, Λ dependence of the coupling constant g YD sDs0 is computed. With a value of cutoff Λ = 0.9 − 1.1 GeV, the corresponding coupling constants shown in Fig 8. We note that they decrease slowly with the increase of cut-off, and the coupling constant is almost independent of Λ, where Y(4274) is P−wave D sDs0 molecular state. According to the studies in Refs [18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36], a typical value of Λ = 1.0 GeV is often employed. Thus, in this work we take Λ = 1.0 GeV and the corresponding coupling constants are g YD sDs0 = 13.34 +1.11 −0.89 GeV, which the error reflects variation in Λ from 0.9 to 1.1 GeV. Once the coupling constant g YD sDs0 = 13.34 +1. 11 −0.89 GeV are determined, the decay widths of Y(4274) can be calculated straightforwardly. In Fig. 9, we show the partial decay widths of Y(4274) → J/ψφ, χ c0 η, χ c1 η, D * sD s , π 0 D sDs , DD * , KK * , and φφ as a function of cutoff parameter Λ. We find that the estimated two-body decay widths increases with increase of cut-off and are all insensitive to cut-off parameter Λ. While the Y(4274) → π 0 D sDs three-body decay decreases, but very slowly. We also find that the partial decay width is the largest for transition Y(4274) → DD * . Thanks the flavor symmetry of the wave function that shown in Eq. (3), the Y(4274) → χ c0 η and χ c1 η two-body decay widths are of the order of about 0.0 MeV. We also note that the three-body transition strength is quite small, and the decay width is of the order of 0.158-0.190 MeV. The small three-body decay width can be easy understand due to the decay Y(4274) → D sDs π 0 is an isospin violation process.
Two reasons can help us to understand why transition Y(4274) → DD * provides the dominant contribution. First is that transition Y(4274) → DD * is s-wave decay, and the lowest angular momentum gives the dominant contribution. Larger Y(4274) → DD * decay can also be understood due to the main component of D s0 is DK, and the coupling constant related to this vertex is larger than the others. The same D s0 DK coupling also exist in the Y(4274) → KK * reaction. However, its partial decay width is small. A possible explanation is that a light quark-antiquark pair creation or annihilation in Fig. 6 is easier than that of a heavy quark-antiquark pair. We also show the dependence of the total decay width on cutoff Λ in Fig. 10. In present calculation, we vary Λ from 0.9 to 1.1 GeV. In this Λ range, the total decay width increases, and predicted decay width Γ Y =1.25-2.0 MeV is much smaller than the experimental width, which disfavors Y(4274) in a D sDs0 molecular picture. If we increase Λ to higher values, the total widths of Y(4274) cannot be reproduced until a much larger Λ value of about 9.6 is adopted. Unfortunately, there are no such studies on taking Λ = 9.6 or higher, which is reasonable. Hence, the assignment as a P−wave D sDs0 molecular state is impossible for Y(4274) based on the total decay width experimentally measured. This is quite different from the con-clusion in Ref. [15] that the interaction between a D s meson and aD s0 meson is strong enough to form a bound state with a mass about 4274 MeV, which can be associated to Y(4274). Comparing our results with those in Ref. [15], it seems that a study of the spectroscopy alone does not give a complete picture of its nature.
Combination our results in Fig. 10 with conclusion that in Ref. [15], we predict a P−wave D sDs0 molecular that we marked as Y ′ (4274) may exist. Taking a typical value Γ(D ± s0 → πD ± s ) = 79.3 KeV [19], where D s0 is assumed to be a DK bound state, the corresponding partial decay widths 018 MeV, which yields a total decay width of 1.06-1.84 MeV. And find that the transition Y ′ (4274) → DD * is the main decay channel, almost saturating the total width. The experimental measurements for this strong decay process could be a crucial to observe such a state.

IV. SUMMARY
In this work, inspired by the studies in Ref. [15] that showed the likely existence of a D sDs0 bound state, we have studied its partial decay widths into Y(4274) → J/ψφ, χ c0 η, χ c1 η, φφ, ηη, D * sDs , DD * , KK * and π 0D s D s . These decays involve the treatment of Y(4274) state as a quasi-bound state of D sDs0 and utilizing the Weinberg compositeness condition to determine the corresponding coupling. Our studies find the P−wave D sDs0 assignments for Y(4274) is not allowed, it may be a compact tetraquark state or diquark-antidiquark state [7][8][9][10][11][12][13][14]. If Y(4274) is a compact tetraquark state or diquark-antidiquark state [7][8][9][10][11][12][13][14], we suggest experimental to search for Y(4274) in the Y(4274) → χ c0 η and Y(4274) → χ c1 η reactions that the partial decay widths are of the order of 0.0 MeV by assuming Y(4274) is a D sDs0 molecular state. Theoretical investigations on decay modes and further experimental information on partial decay widths will be helpful to distinguish which the inner structure of the Y(4274) state is possible.
However, a P−wave D sDs0 molecular with the total decay width is at the order of 1.06 to 1.84 MeV [we marked as Y ′ (4274)] is found. The predicted decay width seems to suggest that it is possible to observe such a state at Belle or Belle II, e.g., via the inclusive invariant mass distribution DD * , which is the largest transition. On the other hand, its production yields at these experimental setups remain to be studied.
If the Y(4274) could be a P−wave D sDs0 molecular. It naturally leads us to think about whether there are contributions come from the light pseudoscalar meson exchanges (π, K, η, η ′ ) and the vector meson exchanges (ρ, ω, K * )..., or there exist other decay mechanisms for the few-body systems. However, only J/ψ, φ, η, K, D, D s , and D * s exchange are included because there is no information on studies about other meson exchange in the vertices of charm-strange meson. Moreover, we always think the two-body decay modes of the multi-quark states are usually the dominant ones. Hence, the currently calculation is enough to explain the Y(4274) cannot be a P−wave D sDs0 molecular.