Excited $K$ meson, $K_c(4180)$, with hidden charm as a $D\bar{D}K$ bound state

Motivated by the recent discovery of two new states in the $B^+\rightarrow D^+D^-K^+$ decay by the LHCb Collaboration, we study the $D\bar{D}K$ three-body system by solving the Schr\"odinger equation with the Gaussian Expansion Method. We show that the $D\bar{D}K$ system can bind with quantum numbers $I(J^P)=\frac{1}{2}(0^-)$ and a binding energy of $B_3(D\bar{D}K)=48.9^{+1.4}_{-2.4}$ MeV. It can decay into $J/\psi K$ and $D_s\bar{D}^*$ via triangle diagrams, yielding a partial decay width of about 1 MeV. As a result, if discovered, it will serve as a highly nontrivial check on the nature of the many exotic hadrons discovered so far and on non-perturbative QCD as well. Assuming heavy quark spin symmetry, the same formalism is applied to study the $D\bar{D}^*K$ system, which is shown to also bind with quantum numbers $I(J^P)=\frac{1}{2}(1^-)$ and a binding energy of $B_3(D\bar{D}^*K)\simeq 77.3^{+3.1}_{-6.6}$ MeV, consistent with the results of previous works.

Introduction: Starting from 2003 [1,2], many exotic hadronic states have been discovered experimentally. Most of them cannot easily fit into the conventional quark model picture, i.e., baryons consisting of qqq and mesons of qq. As a result, they have attracted a lot of attention and inspired intensive discussions about their true nature. Among the many possible interpretations, the molecule picture has been widely employed to interpret some of these exotic states (for recent comprehensive reviews, see, e.g., Refs. [3][4][5][6][7][8][9][10]). Nonetheless, it is difficult if not impossible to distinguish different scenarios because there are always some free parameters in each model to fit the experimental data. In this letter, we propose to study the existence of a three-body DDK bound state to decisively confirm or repute the molecular picture, with the assumption that D * s0 (2317) and D s1 (2460) are predominantly DK and D * K bound states.
Being about 160 and 70 MeV lower than the corresponding cs states predicted by the naive quark model, D * s0 (2317) and D s1 (2460) can be naturally interpreted as DK and D * K bound states [11][12][13][14][15]. If this is the case, in the sense that the deuteron is a bound state of proton and neutron, a naive but straight forward question to ask is whether DDK (and/or DD * K) can form 3-body bound states? A number of recent studies showed that they indeed bind [16][17][18][19][20]. Lately, the Belle Collaboration has performed the first experimental search for the existence of the DDK bound state and reported an upper limit for its production yield [21].
Leading order chiral perturbation theory dictates that in the isospin zero channel, theDK andD * K interactions are only half those of the DK and D * K interactions [22]. In addition, both X(3872) and Z c (3900) can be explained as DD * molecules 1 , which implies that the DD * interaction is attractive as well, but not as strong as the DK and D * K interactions. A recent lattice QCD study shows that the DD interaction is attractive such that a shallow DD bound state exists, consistent with the early theoretical results [30] and more recent analysis of the γγ → DD reaction [31]. Motivated by these facts, the existence of a three-body DD * K bound state has been studied. In Ref. [32], the authors studied the DD * K system using the Born-Oppenheimer approximation via delocalized π bond. A bound state with the quantum numbers of K * and a mass of 4317.92 +6.13 −6.55 MeV was found. In Ref. [33], using the so-called fixed-center approximation in coupled channels, the authors solved the Faddeev equation and found a heavy hidden charm K * meson with a mass about 4307 ± 2 MeV, consistent with Ref. [32].
Lately, the LHCb Collaboration found two new states in the B + → D + D − K + decay [34,35], namely X 0 (2866) and X 1 (2900). The former has been interpreted as aD * K * bound state of spin zero [36][37][38][39][40][41][42][43]. Although a DDK bound state can not be found in the D + D − K + invariant mass spectrum, the recent experimental discovery indicates that a three-body DDK bound state, if it exists, could have been formed already and remain to be discovered at the current facilities. Motivated by these theoretical and experiment works, we study the strange hidden charm DDK system using the Gaussian Expansion Method (GEM) [44]. We indeed find a DDK bound state as well as its heavy quark spin symmetry partner, a DD * K bound state.
Theoretical framework: There exit several widely used approaches to solve three-body problems, such as the Faddeev equation [45], the Gaussian expansion method [44], the stochastic variational method [46], and the hyperspherical harmonic expansion method [47]. With the same inputs, the results of all these methods agree very well with each other as shown in the benchmark study [48]. In this work, we utilize the Gaussian expansion method to study the DD ( * ) K three-body systems, which has been widely used to solve three-, four-and even five-body problems [49], because of its high precision and rapid convergence. Namely, we study the three-body DD ( * ) K systems by solving the following Schödinger equationĤ where the HamiltonianĤ includes the kinetic term and three two-body interaction termŝ In order to solve the Schödinger equation, we have to first specify the two-body interactions.
For the DK interaction, we refer to chiral perturbation theory, in which the most important contribution is the leading order Weinberg-Tomozawa (WT) term [22] where the pion mass decay constant f π = 130 MeV and C W (I) represents the strength of the WT interaction with C W (0) = 2 for the isospin 0 and C W (1) = 0 for the isospin 1 configurations, respectively. This DK potential can be rewritten in coordinate space by Fourier transformation and we use the same form of the DK interaction as that adopted in Ref. [18], which explicitly reads Here R C is a coordinate space cutoff representing the effective interaction range. In this work, we choose R c ranging from 0.5 to 2.0 fm to study the related uncertainties. The C(R c ) is a running constant related to R c , which can be determined by reproducing the D * s0 (2317) state. TheDK interaction can be related to the DK interaction in chiral perturbation theory, where the leading order DK potential is half of the DK interaction in the isospin zero channel [22]. Thus we take theDK potential to have the same form as that of Eq. (4) but multiplied with 1/2. Assuming heavy quark spin symmetry, theD * K interaction is the same as theDK interaction.
For the DD interaction, there is no concrete experimental data and we have to resort to phenomenological models, e.g., the one boson exchange model of Ref. [50]. In Ref. [18], the DD OBE potential has been derived with the exchange of σ, ρ and ω mesons. According to G-parity, the only difference between the DD potential and the DD potential in the OBE model is the sign of the ω exchange potential. The explicit form of the DD interaction can be found in Ref. [18]. For the DD * interaction, one can also exchange a π in addition to the σ, ρ and ω exchanges. We use the DD * OBE potential of Ref. [50] which reproduces the well-known X(3872) state as a molecular state. We choose a cutoff Λ = 1.01 GeV to reproduce the binding energy 4.0 MeV of X(3872) with respect to the DD * threshold. It should be mentioned that whether the DD system can form a bound state is still under discussion [30,31,51]. In Ref. [30], the authors found a very narrow heavy scalar with mass around 3700 . In Ref. [51], the authors found a shallow DD bound state with a binding energy −4.0 +3.7 MeV in lattice QCD. In Ref. [31], the authors claimed that the S-wave DD final state interaction can produce a bound state around 3720 MeV with I = 0 by investigating the γγ → DD reaction. In our model, with a cutoff Λ = 1.01 GeV, the DD system can not form a bound state with the OBE potential, but will do so if a larger cutoff is adopted. 2 In the following, consistent with the DD * case, we choose Λ = 1.01 GeV for the DD OBE potential. As all the two-body interactions have been specified, we use the GEM to solve the Schödinger equation. The three-body wave functions can be constructed in Jacobi coordinates as where c = 1 − 3 is the label of the Jacobi channels shown in Fig. 1. In each Jacobi channel the wave function Ψ(r c , R c ) reads where C c,α is the expansion coefficient and the α = {nN, tT, lLλ} labels the basis number with the configuration sets of the Jacobi channels. H c t,T is the three-body isospin wave function where t is the isospin of the subsystem in Jacobi channel c and T is the total isospin. Considering that the isospin 1 DK interaction is 0 in our model, the isospin t DK of the DK subsystem should be 0 and thus the total isospin T of the DDK(DD * K) system is 1/2.
The three-body spatial wave function Φ(r c , R c ) is constructed by two two-body wave functions as Here N nl (N N L ) is the normalization constant of the Gaussian basis, n(N ) is the number of Gaussian basis used and l(L) is the orbital angular momentum corresponding to the Jacobi coordinates r(R). Since only S-wave interactions are considered, the total orbital angular momentum is λ = 0 and thus the quantum numbers I(J P ) of this DDK three-body system are 1 2 (0 − ), while those for DD * K are 1 2 (1 − ). With the constructed wave functions, the Schödinger equation can be transformed into a generalized matrix eigenvalue problem with the Gaussian basis functions where T ab αα is the matrix element of kinetic energy, V ab αα is the matrix element of potential energy, and N ab αα is the normalization matrix element.
Predictions and discussions: We first study whether the three-body DDK and DD * K systems bind in the theoretical framework and with the two-body interactions specified above.
The DDK system is found to bind with quantum numbers I(J P ) = 1 2 (0 − ) and a three-body binding energy The results are weakly cutoff dependent, and therefore we vary the cutoff R c from 0.5 to 2.0 fm to estimate the uncertainties originating from the DK andDK interactions. More concretely, the central value of the binding energy is obtained with R c = 1.0 fm while the uncertainties are taken from R c = 0.5 and 2.0 fm in the specific numerical calculations.
Assuming heavy quark spin symmetry, the same formalism is applied to study the DD * K system, where a bound state is found as well, with quantum numbers I(J P ) = 1 2 (1 − ) and a binding energy The binding energy of the DD * K bound state is larger than the one of DD * K mainly due to the more attractive DD * interaction. As mentioned earlier, the DD * K state has been studied using other methods. In Table I, we compare our predictions with those of two earlier studies. It is clear that although there are some differences in detail, the existence of a DD * K bound state seems to be a robust prediction.
In Fig. 2, we show the root mean square (RMS) radii of the three subsystems in the DDK and DD * K bound states. With a cutoff of R c = 1.0 fm, the RMS radius of the DK subsystem in the DDK bound state is predicted to be 1.26 fm, while those of theDK and DD subsystems are much larger due to the less attractive interactions with respect to the DK one, yielding 2.27 and 2.10 fm, respectively. For the DD * K bound state, the RMS radii for DK, D * K, and DD * are 1.15, 1.27, and 0.93 fm, respectively. The later two are much smaller compared to the ones of DK and DD in the DDK bound state. It can be easily understood because the DD * K bound state has a much larger binding energy. In Fig. 2, we also show the RMS radii of the DDK bound state taken from Ref. [18], of which the DK one is about 1.32 fm and the DD one is about 1.36 fm. Following Refs. [19,52] where the decays of the DDK and D * D K bound states via triangle diagrams have been estimated, in the following we compute the decay of the DDK state, which for convenience will be denoted by K c (4180). The decay of K c can proceed via its coupling to D * s0 andD, and then decays to J/ψK and D sD * via triangle diagrams as shown in Fig. 3. We employ the effective Lagrangian approach to calculate the amplitudes of these hadronic loops.
The interaction between K c (4180) and its components is described by the following effective Lagrangian where ω i = m i /(m i + m j ) is a kinematical parameter with m i and m j being the masses of the components of K c , Φ(y) is a form factor which we take a Gaussian form and g KcDs0D is the coupling constant between K c and its components. We can determine the value of g KcDs0D by the compositeness condition, see, e.g., Refs. [19,53,54]. In studies of hadronic molecules, the cutoff in the form factor is often chosen to be about 1 GeV, and the corresponding coupling is found to be 5.38 GeV. The Lagrangians describing the interaction between D * s0 (2317) and DK/D s η have the following form where g Ds0DK and g Ds0Dsη denote the D * s0 (2317) coupling to DK and D s η with g Ds0DK = 10.21 GeV and g Ds0Dsη = 6.40 GeV, respectively [30]. The effective Lagrangian between J/ψ andDD, and that between η andDD * can be written as where g ψDD = m ψ /f ψ with f ψ = 0.426 GeV and gDD * η = g/( √ 3f π ) with g = 1.097 GeV and f π = 0.0924 GeV [54]. With the above Lagrangians, the amplitudes of the two triangle diagrams of Fig. 3 can be straightforwardly calculated where k 2 and k 1 are the momenta ofD and D * s0 (2317), q the intermediate momenta, and ψ µ and D * µ are the polarization vectors of ψ andD * .
With these amplitudes, the decay width can be easily obtained where p 1 is the 3-momenta of either final state in the rest frame of K c and m Kc is the mass of the DDK bound state. With a cutoff of 1 GeV for the form factor Φ Kc (y 2 ), the decay width of K c (4180) to J/ψK and D sD * are found to be 0.5 MeV and 0.2 MeV, respectively, which indicates that this state is very narrow. Compared to the DDK bound state R(4140), for which Γ 2 − 3 MeV, the relative smallness of the decay width of K c (4180) is mainly due to the suppression of the D exchange contribution in comparison with the corresponding K exchange contribution.
Summary: We employed the Gaussian expansion method to study the DDK system with the leading order DK andDK potentials obtained in chiral perturbation theory and the DD potential from the OBE model. We found the existence of a DDK bound state with a binding energy about 49 MeV. It is interesting to note that the predicted DDK three-body bound state is only 4 MeV below the respectiveDD * s0 (2317) threshold, though its binding could increase by 20 MeV if the DD interaction is strong enough to generate a bound state as claimed by the recent lattice QCD study. We also studied its heavy quark spin partner, the DD * K system, in the same framework, and we found a DD * K bound state with a binding energy about of 77 MeV, consistent with the results of earlier works.
We studied the decays of the DDK bound state via triangle diagrams and found that its partial decay widths to J/ψK and D sD * are about 0.5 MeV and 0.2 MeV, respectively. Different from the DDK bound state, these two exotic states are more likely to be discovered at the current facilities because of their hidden charm nature. It is interesting to note that a recent study in QCD sum rules does not find aDD * s0 (2317) bound state [55], consistent with the current picture that the K c (4180) state is a three-body molecule. As a result, we strongly encourage our experimental colleagues to search for it.