Strong decays of the explicitly exotic doubly charmed $DDK$ bound state

Nowadays, it is generally accepted that the $DK$ interaction in isospin zero is strongly attractive and the $D_{s0}^*(2317)$ can be described as a $DK$ molecular state. Recent studies show that the three-body $DDK$ system binds as well with a binding energy about 60$\sim$70 MeV. The $DDK$ bound state has isospin $1/2$ and spin-parity $0^-$. If discovered either experimentally or in lattice QCD, it will not only provide further support on the molecular nature of the $D_{s0}^*(2317)$, but also provide a way to understand other exotic hadrons expected to be of molecular nature. In the present work, we study its two-body strong decay widths via triangle diagrams. We find that the partial decay width into $DD_s\pi$ is at the order of $2\sim3$ MeV, which seems to be within the reach of the current experiments such as BelleII. As a result, we strongly recommend this decay channel of the $DDK$ bound state to be searched for experimentally.


I. INTRODUCTION
In 2003, the BaBar Collaboration observed a narrow state in the inclusive D + s π 0 invariant mass distribution of the e + e − collision at energies near 10.6 GeV [1], i.e., the D * s0 (2317) (D s0 for short in the present work), which was later confirmed by the CLEO Collaboration [2] and the Belle Collaboration [3]. Because the low mass, small width, and decay mode of the D s0 are quite different from those of a conventional J P = 0 + cs state in the naive quark model, its nature has remained a topic of tremendous theoretical interests ever since its discovery . In recent years, the importance of the DK interaction in forming the D s0 has been confirmed by lattice QCD simulations [38][39][40][41][42]. See Ref. [43] for a short summary of the theoretical, experimental, and lattice QCD supports for the molecular interpretation of the D s0 as a DK bound state.
If the D s0 is indeed (dominantly) a DK bound state, a natural question to ask is whether the DDK three-body system is still bound. In Ref. [44], by describing the D s0 D interaction via one kaon exchange (OKE), it was shown that the OKE interaction is strong enough to form a D s0 D molecular state * Electronic address: lisheng.geng@buaa.edu.cn † Electronic address: amartine@if.usp.br ‡ Electronic address: kanchan.khemchandani@unifesp.br with a binding energy of 50 ∼ 60 MeV, regardless whether the D s0 DK coupling is determined by treating the D s0 as a cs state or a DK molecule. In Ref. [45], a study was done by explicitly considering the three-body D(DK − D s π − D s η) system and by solving the Faddeev equations using the twobody inputs provided by the unitarized chiral perturbation theory and the local hidden symmetry approach. A three-body bound state was found in this latter work, with a total mass around 4140 MeV, which is an isospin doublet containing two states (R ++ , R + ). In a more recent work [46], using the Gaussian expansion method, the existence of this state has been further confirmed though with a lightly smaller binding energy of ∼ 60 − 70 MeV)and it has been found that even the DDDK or DDD s0 system is bound. It is interesting to note that the DD * K [47][48][49], the DKK and DKK [50], as well as the DKN [51] systems bind as well, because of the strong attraction between D and K.
As pointed out in Ref. [45], the three-body DDK bound state can decay strongly via diagrams such as those shown in Fig. 1. In the present work, we calculate explicitly the partial decay widths from such processes, aiming to provide further motivation for the experimental search for this state. The present manuscript 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 Sec. IV.

II. THEORETICAL FORMLISM
In the following, we focus on the doubly charged state R ++ . Due to isospin symmetry, the decay width of its isospin partner R + can be calculated analogously and only small differences are expected because of the slightly different masses of its molecular components. As mentioned in Ref. [45], though the R ++ is a bound state of the DDK system or D s0 D system, it is possible for such a state to decay strongly. Keeping in mind that the D + s0 is observed in the inclusive D + s π 0 invariant mass distribution, which violates isospin, the R ++ can decay via R ++ ≡ D + s0 D + → (D + s π 0 )D + . An alternative process, without involving isospin breaking, is via triangle diagrams such as those shown in Fig. 1. These processes conserve isospin and therefore should be the dominant ones, as compared to the ones that violate isospin. In the following, we explain how to calculate the four diagrams shown in Fig. 1. In order to calculate the Feynman diagrams shown in Fig. 1, we need to determine the relevant vertices. For the vertex of R ++ D + s0 D + , since the R ++ can be treated as a bound state of D + s0 D + [44], this coupling can be determined by the Weinberg compositeness condition. In the present work, we adopt the method developed in Refs. [52][53][54][55][56][57][58][59][60][61][62][63][64][65][66][67][68]. In this framework, the interacting Lagrangian between R, D s0 , and D can be written as [52,53] R ++ (k0) FIG. 2: Self-energy of the R ++ and R + states.
where ω i = m i /(m i + m j ) is a kinematical parameter with m i and m j being the masses of the involved mesons. In the Lagrangian of Eq. (1), an effective correlation function Φ(y 2 ) is introduced to reflect the distribution of the two constituents, D + s0 (2317) and D + (D 0 ), in the hadronic molecular R ++ (R + ) state. The introduced correlation function also serves the purpose of making the Feynman diagrams ultraviolate finite.
The coupling constant g RD s0 D in Eq. (1) could be determined by the compositeness condition [52,53], where the renormalization constant of the composite particle should be zero, i.e., with Σ R (m 2 R ) being the derivative of the mass operator of the R. The concrete form of the mass operator of the DDK bound state R corresponding to the diagram in Fig. 2 is with k 0 , m R denoting the four-momenta and mass of the R, respectively. Here, we set m R = m D s0 + m D − E b with E b being the binding energy of R, k 1 , and m D s0 are the four-momenta and mass of the D s0 , and m D is the mass of the D-meson, respectively.
In the present work, we calculate the two-body decay width of the R via the triangle diagrams shown in Fig. 1. To evaluate the diagrams, in addition to the Lagrangian of Eq. (1), the following effective Lagrangian terms, responsible for the interactions between heavy-light pseudoscalar and vector mesons, are needed as well [33] L PφP * = ig P * µ u µ P † − Pu µ P * † µ , where P = (D 0 , D + , D + s ) and P * = (D * 0 , D * + , D * + s ), u µ is the axial vector combination of the pseudoscalar-meson fields and their derivatives, where u 2 = U = exp(i φ f 0 ), f 0 =92.4 MeV, and the pseudoscalar-meson octet φ is represented by the 3 × 3 matrix From Eqs. (5)- (7), one can easily obtain the interaction vertices ηDD * ,KDD * s , and πD * D. The coupling constant g can be determined from the strong decay width Γ(D * + → D 0 π + ) = 56.46 ± 1.22 keV, together with the branching ratio BR(D * + → D 0 π + ) = (67.7 ± 0.5)% [69]. With the help of Eq. (5), the two body decay width Γ(D * + → π + D 0 ) is related to g via where p π is the three-momentum of π + in the rest frame of the decaying vector meson D * + . Using the corresponding experimental strong decay width and the masses of the relevant particles given in Table I [69], we obtain g = 1.097 ± 0.012 GeV. In the chiral unitary approaches [20,21,33,70], the D s0 is found to be dynamically generated from the DK and D s η Swave interactions. As a result, the vertices D s0 DK and D s0 ηD s can be easily written as where the coupling of the D s0 to DK and D s η states, g D s0 DK and g D s0 D s η , can be obtained from the coupling constant of the D s0 to the DK and ηD s channels in isospin zero, which are found to be g D s0 DK = 10.21 GeV(10.203 GeV) and g D s0 D s η = 6.40 GeV(5.876 GeV) in Ref. [21]( [20]), multiplied by the appropriate Clebsch-Gordan (CG) coefficients, namely, g D + s0 D + K 0 = g D + s0 D 0 K + = −g D s0 DK / √ 2 and g D + s0 D + s η = g D s0 D s η . With the above vertices, the amplitudes of the triangle diagrams of Fig. 1, evaluated in the center of mass frame of final states, are The corresponding partial decay width then reads where J = 0 is the total angular momentum of the initial R state, the overline indicates the sum over the polarization vectors of final hadrons, and | p 1 | is the 3-momenta of the decay products in the rest frame of the (R ++ , R + ) states. Then the total decay widths of the (R ++ , R + ) states are

III. RESULTS AND DISCUSSIONS
To estimate the partial decay widths of the R, we first need to determine the coupling constants related to the molecular state and its components.
In Refs. [44][45][46], the R ++ state is found to have a binding energy about 15 ∼ 45 MeV, with respect to the D + s0 D + threshold. In this mass range, the coupling constant is dependent on the mass of the bound state R as shown in Fig. 3. One finds that the coupling constant g R ++ D + D + s0 decreases with m R ++ . With a value of the mass m R ++ = 4140 MeV, the corresponding coupling constants is g R ++ D + D + s0 = 9.02 GeV. We show the dependence of the total decay width on the masses of the bound state R ++ in Fig. 4. One can see that the total decay width increases slightly with the mass of the bound state R ++ from 4.13 to 4.17 GeV. The predicted total decay width is small and found to be Γ R ++ = 2.5 − 2.6 MeV.   In Fig. 4, we also show the partial decay widths of the R ++ → D + D * + s and D + s D * + as a function of the mass of the bound state R. The corresponding partial decay widths for several masses of the bound state are listed in Table.II. We note that the transition R ++ → D + D * + s is the main decay channel, almost saturating the total width. The corresponding partial decay widths are Γ R + →D + D * + s = 2.30 − 2.50 MeV and Γ R ++ →D + s D * + = 0.26 − 0.29 MeV, which yields a total decay width of 2.6 ∼ 2.8 MeV. We note that the results depend only moderately on the cutoff. For instance, varying the cutoff from 0.5 to 1.5 GeV, the total decay width changes from 1.6 to 3.4 MeV within the R ++ mass range of 4.13 to 4.17 GeV.
We find that the contribution from the η meson exchange is very small, because the ηD 0 D * 0 vertex, which involves the creation or annihilation of an additional ss quark pair, is strongly suppressed. Moreover, the main component of the D s0 (2317) is DK [20,21,33,70] and the coupling constant related to this vertex is larger than the others. These two factors make the contribution from the K meson exchange the most important one.
In Fig. 4, we also show the ratio of the partial decay widths into D + D * + s and D + s D * + . The ratio of branching fractions is found to be of the order of 8 ∼ 9, and is almost independent of the mass of the bound state R.

IV. SUMMARY
In this work, inspired by the recent series of studies that showed the likely existence of a DDK bound state, we have studied its partial decay widths into D s D * and DD * s . Such a decay involves the treatment of the R state as a quasi-bound state of D * s0 (2317)D and utilizing the Weinberg compositeness condition to determine the corresponding coupling. Our studies find a relative small total decay width, of the order of 2 ∼ 3 MeV, mainly to DD * s , and the results depends only moderately on the single parameter of the method, the cutoff Λ.
The predicted decay width seems to suggest that it is possible to observe such a state at Belle or BelleII, e.g., via the inclusive invariant mass distribution D + D + s π 0 , which is quite similar to the experimental discovery of the D * s0 (2317) by BaBar, Belle, and CLEO. On the other hand, its production yields at these experimental setups remain to be studied.
Recent lattice QCD studies of compact tetraquark states, see, e.g., Refs. [71,72], suggest that a study of the DDK bound state in terms of its minimal quark content ccqs might be within the reach of the state of the art of lattice QCD simulations, even taking explicitly into account its three-meson molecular nature [73].