Strong decays of $\bar{D}^{*}K^{*}$ molecules and the newly observed $X_{0,1}$ states

Lately, the LHCb Collaboration reported the discovery of two new states in the $B^+\rightarrow D^+D^- K^+$ decay, i.e., $X_0(2866)$ and $X_1(2904)$. In the present work, we study whether these states can be understood as $D^*\bar{K}^*$ molecules from the perspective of their two-body strong decays into $D^-K^+$ via triangle diagrams and three-body decays into $D^*\bar{K}\pi$. The coupling of the two states to $D^*\bar{K}^*$ are determined from the Weinberg compositeness condition, while the other relevant couplings are well known. The obtained strong decay width for the $X_0(2866)$, in marginal agreement with the experimental value within the uncertainty of the model, hints at a large $D^*\bar{K}^*$ component in its wave function. On the other hand, the strong decay width for the $X_1(2904)$, much smaller than its experimental counterpart, effectively rules out its assignment as a $D^*\bar{K}^*$ molecule.


I. INTRODUCTION
Ever since the experimental discovery of the X(3872) and D * s0 (2317), many hadrons that cannot be simply classified into conventional mesons of qq and baryons of qqq have been discovered, with the latest addition being the cccc states discovered by the LHCb Collaboration [1]. See, e.g., Refs. [2][3][4][5] for recent reviews. It should be noted that most of the so-called exotic hadrons mix with conventional hadrons or can be understood as hadron-hadron molecules or threshold effects such that they are not that "exotic". Curiously, two of the truly exotic candidates, the θ + (1540) [6] and the X(5568) [7] seem to fade away with time. In such a context, the latest LHCb announcement of two structures observed in the D − K + invariant mass of the B + → D + D − K + decay points to the likely existence of genuinely exotic mesonic states with a minimum quark content of csud [8]. Their masses and widths are, in units of MeV, respectively X 0 (2866) : M = 2866 ± 7 and Γ = 57.2 ± 12.9, X 1 (2900) : M = 2904 ± 5 and Γ = 110.3 ± 11.5.
The spin-parities of these two states are determined to be 0 + and 1 − .
It is interesting to note that these two states are just below (X 0 ) and close to (X 1 ) the D * K * threshold. Although the existence of compact tetraquark states in this energy region has been predicted, in either quark models [9][10][11][12], or QCD sum rules [13,14] 1 . In the present work, we examine the possibility whether they can be understood as D * K * molecules. For such a purpose, we first assume that they are bound states of D * K * , and then employ the weinberg compositeness rule to determine their couplings to D * K * . The two body strong decays then follow from the exchange of a pseudoscalar meson between the D * K * pair, which then transforms into D − K + .
Such a process is depicted in Fig. 1. In addition, the D * K * molecules can also decay into a three-body finale state D * K π, as shown in Fig. 2. 2 If within the uncertainties of the model, the so-obtained strong decay widths are consistent with data, then it is possible to assign the state under study as a molecular state, otherwise, the possibility is excluded. Such an approach has been widely applied to study newly observed (exotic) hadrons, see, e.g., Refs. [16][17][18][19][20][21][22][23] for a partial list. 1 It is interesting to note that a state of the art lattice QCD study found no compact tetraquark state ofcsud with I = 0 and spin-parity 0 + and 1 + [15]. 2 As the D * is very narrow, we treat it as a stable particle.
It is interesting to note that the DDK bound state of isospin 1/2 and spin-parity 0 − with a mass around 4140 MeV [24][25][26] is different from those observed by the LHCb Collaboration in the D − K + spectrum. Though the former is built from the DDK interaction, it decays into DD s π [16] instead of D − K + because of parity conservation. It would be interesting if in the future the LHCb collaboration can search for the existence of such a state.
This work is organized as follows. In Section II, we explain the theoretical formalism. Results and discussions are provided in Section III, followed by a short summary in Section IV.

II. THEORETICAL FRAMEWORK
In the following, we explain how the strong decays into DK, Fig. 1, and D * Kπ, Fig. 2, are computed. We take advantage of the fact that the D * is very narrow (with a width of less than 100 keV) and therefore can be treated as a stable particle for our purpose. We shall construct the amplitudes using the isospin formalism, where theDK * isospin doublet reads |K * D * , |K * D * , I = 1 = 1 Considering quantum numbers and phase space, the two body strong decay modes of X J are X J → D − K + and X J →D 0 K 0 . In this work, we only explicitly compute the partial decay width of X J → D − K + , and that of X J →D 0 K 0 can be obtained by isospin symmetry Γ X J →D − K + = Γ X J →D 0 K 0 . The sum of the two parts is the total decay width of the X J →DK.
In order to calculate the Feynman diagrams shown in Fig. 1, we need to determine the relevant vertices. For the vertex of X JD * K * , since the X J is considered as a bound state ofD * K * , this coupling can be determined by the Weinberg compositeness condition. In the present work, we adopt the method developed in Refs. [16][17][18][19][20][21][22][23]. In this framework, the relevant Lagrangians for the X 0 (2866) can be written as [17] X J (k 0 )D * 0 (k 1 ) while for the X 1 (2904) the Lagrangian has the form [27] where is a kinematical parameter with m i and m j being the masses of the involved mesons. In the Lagrangians, an effective correlation function Φ(y 2 ) is introduced to describe the distribution of the two constituents,D * and K * , in the hadronic molecular X J states.
The introduced correlation function also serves the purpose of making the Feynman diagrams ultraviolate finite. Here we choose the Fourier transformation of the correlation function to have a Gaussian form, where β being the size parameter which characterizes the distribution of the constituents inside the molecule. The value of α has to be determined by fitting to data. It is found that the experimental total decay widths of some states that can be considered as molecules (see, e.g., Refs. [16][17][18][19][20][21][22][23] and references therein) can be well explained with α ≈ 1.0 GeV. Therefore we take α = 1.0±0.1 GeV in this work to study whether the X J states can be interpreted as molecules composed ofD * K * .
The coupling constant g X JD * K * is determined by the compositeness condition [16][17][18][19][20][21][22][23]. It implies that the renormalization constant of the hadron wave function is set to zero, i.e., The Σ T 1 is the transverse part of the self-energy operator Σ µν The concrete forms of the mass operator of the X J corresponding to Fig. 3 are where z = 2 + α + β, ∆ = −4ωD * k 0 − 2βk 0 , and k 2 0 = m 2 X with k 0 , m X denoting the fourmomenta and mass of the X J , respectively. Here, we set m X J = mD * + m K * − E b with E b the binding energy of X J , k 1 , and mD * are the four-momenta and mass of theD * , and m K * is the mass of K * , respectively. I is isospin and isospin symmetry implies that To evaluate the diagrams of Fig. 1 and Fig. 2, in addition to the Lagrangians in Eqs.( 5,6), the following effective Lagrangians, responsible for the interaction between a vector meson and a pseudoscalar meson, are needed as well [28] where P and V µ represents the vector fields of the 16-plet of the ρ and the SU (4) and Then we obtain The coupling g h is fixed from the strong decay width of K * → Kπ. With the help of Eq. (17), the two-body decay width Γ(K * + → K 0 π + ) is related to g h 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 listed in Table I [29], we obtain g h = 9.11. With the above formalism, the decay amplitudes of the triangle diagrams of Fig. 1, evaluated in the final state center of mass frame, are where the expressions in the curly brackets, {1, i(k α 2 − k α 1 )ǫ X α }, are for X 0 and X 1 , respectively.

B. Three-body decay width
Similarily, the decay amplitudes of the triangle diagrams of Fig. 2, evaluated in the initial state center of mass frame, are where the expressions in the curly brackets, {1, i(q − p 2 ) α ǫ α (p)}, are for X 0 and X 1 , respectively.
Once the amplitudes are determined, the corresponding partial decay widths can be easily obtained, which read as, where J is the total angular momentum of the X J , | p 1 | is the three-momenta of the decay products in the center of mass frame, and the overline indicates the sum over the polarization vectors of the final hadrons. The ( p * 3 , Ω * p 3 ) is the momentum and angle of the particle K in the rest frame of K and π, and Ω p 2 is the angle of theD * in the rest frame of the decaying particle. The m πK is the invariant mass for π and K and m π + m K ≤ m πK ≤ M − mD * . The total decay width of the X J is the sum of Γ(X J →DK) and Γ(X J → πKD * ).

III. RESULTS AND DISCUSSIONS
In order to obtain the allowed two body decay widths through the triangle diagrams shown in Fig. 1 and three body decay widths in Fig. 2, we first compute the coupling constant g X JD * K * (≡ g X J ). With a value of the cutoff α = 0.9 − 1.1 GeV, these coupling constants are shown in Fig 4. We note that they decrease slowly with the increase of the cutoff, and the coupling constant is almost independent of α. The different α dependences reflect the different distribution of the two constituents,D * and K * , in the hadronic molecular X J states. We show the dependence of the total decay width on the cutoff α in Fig. 5. In the present study, we vary Λ from 0.9 to 1.1 GeV. In this α range, the total decay width increases for the case of J P = 0 + , while it decreases for the J P = 1 − case. The three-body decay widths for both J P = 0 + , 1 − and I = 0, 1 are in the range of 2 to 3 MeV, while the two-body decay width for J P = 0 + are at the order of a few tens of MeV, but that for the J P = 1 − are less than 1 MeV (see also Table II). A possible explanation for this is that the width of a P -wave molecule is heavily dependent on the spatial distributions of its constituents, as one can see from Eqs. (22)(23)(24)(25).
From Fig. 5, we find that the calculated total decay width for the case of I(J P ) = 1(0 + ) is comparable with that of the experimental total width in the range of α = 1.06 − 1.1 GeV, while an even larger α is needed for I(J P ) = 0(0 + ). Although a value of α = 1.0 is preferred based on previous studies [16][17][18][19][20][21][22][23], considering that the fact our results should be considered as the lower limits because it is possible that other decay modes exist, our study did indicate a sizeable D * K * component in the X 0 wave function. The corresponding partial decay widths of X J →DK, D * πK, and the total decay widths for different spin-parity and isospin assignments of X J are listed in Tab. II. For comparison, we show the results from the LHCb Collaboration as well [8].
The results show that the X 0 (2866) might have a sizeable D * K * component while the X 1 (2904) cannot be explained as a D * K * molecule. We note that in Ref. [30], the X 0 (2866) is found to be compatible with a compact tetraquark state. Exp. [8] 57.2 ± 12.9 110.3 ± 11.5

IV. SUMMARY
We studied the two-body and three-body strong decays of the two states X 0 (2866) and X 1 (2904) assuming that they are bound states of D * K * . The couplings of these states to their and the total decay width (black solid lines) with different spin-parity and isospin assignments for the X J as a function of the parameter α. The oycn error bands correspond to the experimental total decay width [8].
components are fixed by the Weinberg compositeness condition. The two-body decays are via triangle diagrams with exchanges of a pseudoscalar meson π, η, or D s , where the three-body decays happen at tree level. With the other couplings fixed from relevant experimental data, the only remaining parameter is the cutoff α. We showed that with the well accepted range of 0.9 ∼ 1.1 GeV, the so-obtained decay width for the X 0 (2866) is in marginal agreement with the LHCb measurement but that for the X 1 (2904) is much smaller. As a result, we conclude that the X 0 (2866) may have a large D * K * component (also a non-negligible compact tetraquark component) but the X 1 (2904) cannot be of molecular nature.
Such a conclusion is consistent with the OBE model of Ref. [31]. We note that a recent study by Karliner and Rosner favors the explanation of the X 0 as a compact tetraquark state [30], while the lattice QCD study of Ref. [15] found no tetraquark candidate in this channel. As a result, more works are urgently needed to clarify the nature of these latest additions to the family of exotic mesons.