Production of $D^*_{s0}(2317)$ and $D_{s1}(2460)$ in $B$ decays as $D^{(*)}K$ and $D^{(*)}_s\eta$ molecules

The molecular nature of $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ have been extensively studied from the perspective of their masses, decay properties, and production rates. In this work, we study the weak decays of $B \to \bar{D}^{(\ast)}D_{s0}^{*}(2317)$ and $B \to \bar{D}^{(\ast)}D_{s1}(2460)$ by invoking triangle diagrams where the $B$ meson first decays weakly into $\bar{D}^{(\ast)}D_{s}^{(\ast)}$ and $J/\psi K$($\eta_{c}K$), and then the $D_{s0}^{\ast}(2317)$ and $D_{s1}(2460)$ are dynamically generated by the final-state interactions of $D_{s}^{(\ast)}\eta$ and $D^{(\ast)}K$ via exchanges of $\eta$ and $D^{(\ast)}$ mesons. The obtained absolute branching fractions of Br$[B \to \bar{D}^{(\ast)}D_{s0}^{*}(2317)]$ are in reasonable agreement with the experimental data, while the branching fractions of Br$[B \to \bar{D}^{(\ast)}D_{s1}(2460)]$ are smaller than the experimental central values by almost a factor of two to three. We tentatively attribute such a discrepancy to either reaction mechanisms missing in the present work or the likely existence of a relatively larger $c\bar{s}$ component in the $D_{s1}(2460)$ wave function.


I. INTRODUCTION
In 2003, the BaBar Collaboration discovered a quite narrow state near 2.32 GeV in the inclusive D + s π 0 invariant mass distribution [1], named as D * s0 (2317), which was subsequently confirmed by the CLEO [2] and Belle Collaborations [3].Taken as a cs state with the quantum number of I(J P ) = 0(0 + ), its mass is lower by 160 MeV than the prediction of the Godfrey-Isgur (GI) quark model [4].Such a large deviation has also appeared within the lattice QCD simulations [5,6].To explain the discrepancy, many different interpretations of the D * s0 (2317) have been proposed, such as a P -wave cs excited state [7][8][9], a compact tetraquark state [10], or a hadronic molecule [11][12][13][14][15].Among them, the hadronic molecular interpretation has attracted considerable attention.
In Refs.[16,17], the authors interpreted D * s0 (2317) as a hadronic molecule generated by the DK and D s η coupled-channel interactions in the chiral unitary approach, which is also supported by many other studies [18][19][20][21].The DK coupled-channel interactions [22][23][24] have been simulated on the lattice, and a bound state below the DK mass threshold is found, which can be identified as D * s0 (2317).In addition, a D * K molecule as the partner of D * s0 (2317) is predicted via the heavy quark spin symmetry (HQSS), and it can be identified as D s1 (2460) [15,18,25,26], discovered by the CLEO Collaboration in the D * s π mass distribution [2] and confirmed by the Belle Collaboration [3].Up to now, only the upper limits for the widths of D * s0 (2317) and D s1 (2460) are known, i.e., Γ D * s0 (2317) < 3.7 MeV and Γ D s1 (2460) < 3.5 MeV [27].In the molecular picture, Faessler et al. took the effective Lagrangian approach to estimate the the dominant partial decay widths of D * s0 (2317) → D s π and D s1 (2460) → D * s π to be 80 keV and 50∼79 keV [28,29].Very recently, an effective field theory study estimated their partial decay widths to be 120 keV and 102 keV [30], respectively.
Recently, we proposed a novel approach to verify the molecular nature of exotic states from the existence of relevant three-hadron molecules (see Refs. [31,32] for reviews).The molecular nature of D * s0 (2317) can be verified by searching for the three-body molecule DDK, where the DK interaction is determined by reproducing the mass of D * s0 (2317) and plays a dominant role in forming the DDK molecule [33,34].In Ref. [35], assuming D * s0 (2317) as a DK molecule, we employed the one-kaon-exchange potential and predicted the existence of a DD * s0 (2317) molecule, whose mass and quantum numbers are consistent with those of the DDK molecule.Moreover, we have investigated the DDK system [36], and it was found that the DD * s0 (2317) configuration accounts for about 87% of the DDK configuration, which indicates that the DK interaction plays the most important role in forming the DDK molecule as well [37].If the DDK molecule is discovered by experiments, it will also verify the molecular nature of D * s0 (2317).It should be noted that although the DK molecular interpretation is the most favorable, the cs component is found to play a non-negligible role in describing the mass of D * s0 (2317) in the unquenched quark models [38][39][40][41][42].In a recent work [43], by fitting to the lattice QCD finite volume spectra, Yang et al. found that the cs component accounts for about 32% of the wave function of D * s0 (2317), while the cs component accounts for more than half of the D s1 (2460) wave function, which is consistent with a number of earlier studies [44][45][46].
The production of D * s0 (2317) in the molecular picture has also been extensively investigated.In Ref. [47]  s η.We note that a similar approach has earlier been employed to study a 0 (980) generated by the coupled-channels πη and K K in the process D s → ππη [56], where the theoretical results are found in good agreement with the experimental data.This work is organized as follows.We briefly introduce the triangle mechanism for the decays of B → D( * ) D * s0 (2317) and B → D( * ) D s1 (2460) and the effective Lagrangian approach in Sec.II.Results and discussions are given in Sec.III, followed by a short summary in the last section.

II. THEORETICAL FORMALISM
The mesonic weak transition form factors and decay constants are the two main ingredients in the study of hadronic weak decays of mesons, which are less certain for P -wave charmed mesons than for S-wave
[27]   and B +(0) → J/ψ(η c )K +(0) has the following form where G F is the Fermi constant, V bc and V cs are the Cabibbo-Kobayashi-Maskawa(CKM) matrix elements, c ef f 1,2 are the effective Wilson coefficients, and O 1 and O 2 are the four-fermion operators of (sc The effective Lagrangians accounting for the interactions between the charmonium states (J/ψ, η c ) and a pair of charmed mesons read [63,64] where g ψDD , g ψD * D , g ψD * D * , g ηcD * D , and g ηcD * D * are the couplings of the charmonium mesons to the charmed mesons.The coupling constants are determined as follows: The effective Lagrangian describing the interaction between the charmed mesons (D and D * ) and η are written as [67] L  are determined from the residues of D * s0 (2317) on the complex plane, where it is treated as a molecule dynamically generated by the DK and D s η coupled-channel interactions.In this work, we take g D * s0 DK = 9.4 GeV and g D * s0 Dsη = 7.4 GeV given in the effective field theory approach [30], in agreement with those obtained in Ref. [17].The D s1 (2460) is regarded as the HQSS partner of D * s0 (2317), which is dynamically generated by the D * K and D * s η coupled-channel interactions.The couplings of g D s1 D * K = 10.1 GeV and g DsD * s η = 7.9 GeV are also taken from Ref. [30].Taking into account isospin symmetry, the relevant couplings are obtained as

C. Decay amplitudes and partial decay widths
The decay amplitudes of B +(0) → D ( * )+ s D( * )0 (D ( * )− ) and B +(0) → J/ψ(η c )K +(0) can be written as the products of two hadronic matrix elements [68,69] where with N c the number of colors.It should be noted that a 1 and a 2 can be obtained in the factorization approach [70].
The current matrix elements between a pseudoscalar meson or vector meson and the vacuum have the following form: where The hadronic matrix elements can be parameterised in terms of form factors [71] where q, q and q represent the momentum transfer of p B + −p D * 0 , p B + −p D0 , and p B + −p K + , respectively, and The form factors of F 1,0D (t), F 1,0K (t), A 0 (t), A 1 (t), A 2 (t), and V (t) with t ≡ q ( )2 can be parameterized as [71] X For these form factors, we adopt those of the covariant light-front quark model, i.With the above relevant Lagrangians, one can easily compute the corresponding decay amplitudes of Fig. 2, where q 1 , q 2 , and q 3 denote the momenta of D * , D s , and η for Fig. 2  Similarly, the corresponding decay amplitudes of Fig. 3 are written as and the corresponding amplitudes of Fig. 4 are written as where the representation of momenta are the same as Eqs.(16)(17)(18)(19).
The vertices representing the D( * ) mesons scattering into D( * ) and η mesons and J/ψ(η c ) mesons scattering into D( * ) and D ( * ) mesons are written as The vertices describing the D * s0 (2317) and D s1 (2460) molecules generated by D ( * ) K and D ( * ) s η coupled channels are expressed as With the above amplitudes determined as specified above, the corresponding partial decay widths can be finally written as where the overline indicates the sum over the polarization vectors of final states, and | p | is the momentum of either final state in the rest frame of the B meson.

III. NUMERICAL RESULTS AND DISCUSSION
With the above preparation and the masses of relevant particles given in Table II, we can obtain the decay widths of B → D( * ) D * s0 (2317) shown in Table III.We note that the branching ratios of g D * Dη and g ηc/J/ψ D( * ) D .We assume that the breaking of SU (3)-flavor symmetry is at the order of 20% and that of heavy quark spin symmetry is at the level of 20% [75].Adding them in quadrature, we obtain the theoretical uncertainty of 28% given in Table III.For the D s1 (2460) state, our predictions for all the four processes studied are smaller than the PDG averages by about a factor of 3 and than the BaBar results by roughly a factor of 2. On the other hand, the results of Ref. [55] are in better agreement with the data.In Ref. [55], the authors estimated such branching ratios via a naive factorisation approach, where the determination of the couplings f D * s0 and f D s1 depends on the choice of cutoff parameter and relies on the SU(4) symmetry which relates the weak vertices D * → KW and D → K * W . Furthermore, in Ref. [55], D * s0 (2317) and D s1 (2460) are treated as pure DK and D * K molecules, while in our approach it is shown that the coupled channel D ( * ) η plays an important role as well.
The discrepancy between our results and the experimental data can be attributed to either missing reaction mechanisms or the neglect of the likely existence of a relatively large cs component in the wave function of D s1 (2460).In most of the unquenched quark models, both D s1 (2460) and D * s0 (2317) contain sizable cs components, while the former contains a larger cs component.In addition, in the molecular picture, other reaction mechanisms than the triangle mechanism studied here can also contribute to the production of D * s0 (2317) and D s1 (2460) in B decays, such as those studied in Refs.[48,76].We decompose the contributions of the η and D ( * ) exchanges in Table IV.Note that the processes mediated by the η meson contain stronger weak-interaction vertices but weaker strong-interaction scattering vertices with respect to those mediated by the D ( * ) meson, while the couplings of the D * s0 (2317) and D s1 (2460) molecules to their constituents D ( * ) K and D From Table IV, one can see that among the eight branching ratios studied, the contribution of the η exchange is comparable to that of the D ( * ) exchange except for the processes B → DD * s0 (2317), where the D ( * ) contribution is accidentally one order of magnitude smaller that of the η exchange.

IV. SUMMARY AND DISCUSSION
To distinguish the nature of D * s0 (2317) as either a DK molecule, a cs state, or a combination of both has motivated a lot of experimental and theoretical studies.In this work, we utilized the triangle mecha- We note that the degree of agreement between our predictions and the experimental data indeed provides further support for the molecular nature of D * s0 (2317) and D s1 (2460).However, more precise data and further theoretical studies are needed in order to pin down the precise percentage of the cs and D * K/D ( * ) s η components in their wave functions.
and D ( * ) mesons.The obtained absolute branching fractions of Br[B → D( * ) D * s0 (2317)] are in reasonable agreement with the experimental data, while the branching fractions of Br[B → D( * ) D s1 (2460)] are smaller than the experimental central values by almost a factor of two to three.We tentatively attribute such a discrepancy to either reaction mechanisms missing in the present work or the likely existence of a relatively larger cs component in the D s1 (2460) wave function.
, assuming D * s0 (2317) as either a conventional cs state, a compact multiquark state or a hadronic molecule, Cho et al. adopted the coalescence model and statistical model to estimate the corresponding yield of D * s0 (2317) in heavy ion collisions, which would help probe its nature in future experiments.On the other hand, the production of D * s0 (2317) in the weak decays of B and B s mesons also provides a very good platform to study the meson-meson interactions and the nature of D * s0 (2317).In Ref. [48], Miguel et al. investigated the nature of D * s0 (2317) by extracting the DK interaction via the DK invariant mass distributions of the processes

TABLE III .
[55]ching ratios (10 −3 ) of B → D( * ) D * s0 (2317) and B → D( * ) D s1 (2460).asapureDKmolecule.Their branching ratios are shown in Table III.We note that the branching ratios of B → DD * s0 (2317) and B → D * D * s0 (2317) are consistent with ours, but those of B + → D * 0 D * s0 (2317) and B 0 → D * − D * s0(2317) are smaller than ours and in worse agreement with the experimental data.We note that many recent works claim that D * s0 (2317) contains a cs component of 30%, which is not explicitly taken into account in both our work and Ref.[55].Considering such an uncertainty, both our results and those of Ref.[55]are consistent with the experimental data.