Faddeev fixed-center approximation to the $D\bar{D}K$ system and the hidden charm $K_{c\bar{c}}(4180)$ state

We perform a theoretical study on the $D\bar{D}K$ three body system, using the fixed center approximation to the Faddeev equations, considering the interaction between $D$ and $K$, $D$ and $\bar{D}$ from the chiral unitary approach. We assume the scattering of $K$ meson on a clusterized system $D\bar{D}$, where a scalar meson $X(3720)$ could be formed. Thanks to the strong $DK$ interaction, where the scalar $D^*_{s0}(2317)$ meson is dynamically generated, a resonance structure shows up in the modulus squared of the three body $K$-$(D\bar{D})_{X(3720)}$ scattering amplitude and supports that a $D\bar{D}K$ bound state can be formed. The result is in agreement with previous theoretical studies, which claim a new excited hidden charm $K$ meson, $K_{c\bar{c}}(4180)$ with quantum numbers $I(J^P) = \frac{1}{2}(0^-)$ and mass about $4180$ MeV. It is expected that these theoretical results motivate its search in experimental measurements.


I. INTRODUCTION
The study of hadron structure and spectrum is one of the most important issues in hadronic physics and is attracting much attention. The traditional quark model classifies hadrons into the mesons which are composed of a pair of quark and anti-quark (qq), and the baryons which are composed of three quarks (qqq). That well explains the basic properties of most of the experimentally discovered mesons and baryons [1][2][3]. However, the existence of exotic states and the investigation of their properties will extend our knowledge of the strong interaction dynamics [4][5][6][7]. Recently, the discovery of the Z cs (3985) in the D * − s D 0 and D − s D * 0 invariant mass distribution of the e + e − → K + (D * − s D 0 + D − s D * 0 ) reaction by the BESIII collaboration in Ref. [8] was a turning point in hadron spectroscopy since it provided a clear example of an exotic meson state. However, its quantum numbers need to be determined by more precise experimental data. The Z cs (3985) has the same valence quarks with the K meson, and it could be the first candidate for a charged hidden-charm tetraquark with strangeness. On the theoretical side, there are much work that take the Z cs (3985) as a molecular state by using directly dynamics for meson-meson interactions [9][10][11][12][13][14][15][16][17][18][19][20][21][22].
Meanwhile, there is also evidence that some excited hadronic states are predicted in terms of bound state of resonances of three particles. For example, the DDK three body system, with hidden charm, was investigated in Ref. [23] by solving the Schrödinger equation with the Gaussian Expansion Method. Thanks to the strong interaction of DK, it was found that the DDK system can form a bound state, the excited K meson, with quantum numbers I(J P ) = 1 2 (0 − ) and with mass about 4180 MeV. Indeed, the effec- * Electronic address: weixiang@ucas.ac.cn † Electronic address: shenqinghua21@ucas.ac.cn ‡ Electronic address: xiejujun@impcas.ac.cn tive DK interaction is strong. Within the chiral dynamics the scalar D * s0 (2317) meson could be explained as a bound DK state [24][25][26][27][28][29][30][31].
For the DD interaction, a scalar DD bound state was predicted in Ref. [32] by solving the Schrödinger equation with one vector meson exchange potential. Within the chiral unitary approach, the scalar DD bound state X(3720) was obtained with mass about 3720 MeV [26,33]. This DD state was studied in the B 0 → D 0D0 K 0 and B + → D 0D0 K + reactions in Refs. [34], ψ(3770) → γD 0D0 reaction [35], and Λ b → ΛDD reaction [36]. Based on the experimental measurements, the analyses of e + e − → J/ψDD and γγ → DD reactions were done in Refs. [37,38], where the evidence of the existence of a S-wave DD bound state was also claimed. Recently, a scalar DD bound state was also found according to the lattice calculation in Ref. [39]. Furthermore, in Refs. [40,41], the contributions from S-wave D + D − interactions were also investigated in the B + → D + D − K + decay by the LHCb collaboration.
Motivated by the work of Ref. [23], we reinvestigate the three body DDK system by considering the strong interactions of DD and DK. Making the DD bound state, X(3720), as a cluster, and in terms of the two body DK andDK scattering amplitudes obtained within the chiral unitary approach, we solve the Faddeev equations by using the fixed center approximation (FCA). The main purpose of this work is to test the validity of the FCA to study the DDK system which was done in Ref. [23] with the Gaussian expansion method.
The FCA has been employed before, in particular in the similar systems to the DDK. For example: the DKK and DKK systems [42]; the DD * K system [43], the BDD and BDD systems [44]. In the baryon sector, within the FCA, the N DK and N DD systems were studied in Ref. [45], while the N DD * system was studied in Ref. [46]. Besides, in Ref. [47], the Schrödinger equation for the DD * K system was solved, and very similar results were obtained in Ref. [43]. In Ref. [48], by solving the full Faddeev equations of the DDK system considering the DD s η and DD s π coupled channels, the three-body scattering amplitudes were obtained. It was found that an isospin I = 1/2 state is formed at 4140 MeV, which is compatible with the one found in Ref. [49] where the D-D * s0 (2317) system was studied without considering explicit three-body dynamics.
This may indicate that the FCA is a good approximation for the problem where one has a cluster of two particles and allows the third particle to undergo multiple scattering with this cluster. And the cluster is assumed not to be changed by the interaction with the third particle. Intuitively this would mostly happen when the third particle has a smaller mass than the particles in the cluster [50][51][52]. This is just the situation of the present DDK system, where we keep the strong correlations of the DD system that generate the scalar meson X(3720).
Along this line, in this work, we mainly study the DDK three body system by using the FCA approach, and prove it can produce a bound state. On the other hand, we take the advantage that the DDK system was already studied with other approach in Ref. [23], such that comparison with the result of the above work can give us a feeling of the accuracy of the FCA, which is, technically, much easier than most of other methods to solve the full Faddeev equations.
The paper is organized as follows. In Sec. II, we present the theoretical formalism for studying the DDK system with the FCA, and in Sec. III, we show our numerical results. Finally, a short summary is given in the last section.

II. FORMALISM
We are going to use the FCA of the Faddeev equations in order to obtain the scattering amplitude of the three body DDK system, where DD is considered as a bound state of X(3720), which allows us to use the FCA to solve the Faddeev equations. From the analysis of the KX(3720) → KX(3720) scattering amplitude one could study the dynamically generated hidden charm K cc states.
The important ingredients in the calculation of the total scattering amplitude for the DDK system using the FCA are the two-body DD and DK unitarized s-wave interactions from the chiral unitary approach. Since the form of these two body interactions have been reported in many previous works, we direct the reader for details to Refs. [26][27][28], thus we will directly start with the form factor of X(3720) that is a bound state of DD.
A. Form factor for the X(3720) We firstly redo the work of Ref. [26] including the channels D + s D − s , DD, ηη c , ηη, KK, and ππ. Using a cutoff regularization method for the two body loop function and the transition potentials derived as in Ref. [26], the Bethe-Salpeter equations can be solved, and then one gets the two body scattering amplitude t DD→DD in the isospin I = 0 sector. In Fig. 1 we show the modulus squared of the scattering amplitude |t DD→DD | 2 as a function of the invariant mass M DD of the DD system, where the peak of X(3720) state is clearly seen. Note that the numerical results are obtained with a cutoff Λ = 850 MeV to regularize the loop function for the integral of intermediate two meson propagators. It is found that, by adjusting the value of Λ, there is always a clear peak. For example, with Λ = 950 MeV, one could get a peak around 3700 MeV. In addition, the masses of the particles considered in this work are shown in table I, which are taken from the review of particle physics [53].  Next, following Refs. [54,55], one can easily obtain the expression of the form factor F R (q) for the bound state X(3720), which is given by [55][56][57], with the normalization factor N , which is where M is the mass of X(3720), E 1 and E 2 are the energies of D andD, respectively. The cutoff parameter Λ is needed to regularize the loop functions in the chiral unitary approach. In fig. 2 we show the results of the form factor for X(3720) as a function of q, where we take M = 3720 MeV and Λ = 850 MeV. In Eq. (1) the condition | p − q | < Λ implies that the form factor is exactly zero for q > 2Λ. Therefore the integration in Eq. (1) has an upper limit of 2Λ.

B. Faddeev equations under fixed center approximation
In the framework of FCA, we consider the DD bound state X(3720) as a cluster, and the K meson interacts with the components of the cluster. The total three-body scattering amplitude T can be simplified as the sum of two partition functions T 1 and T 2 , by summing all diagrams in Fig. 3, starting from the interaction of particle 3 with particle 1(2) of the cluster. Thus, the FCA equations can be written in terms of T 1 and T 2 , which read [58,59] where T 1 represents the amplitudes of all multiple scattering processes in which particle 3 first scatters with particle 1 in-side the cluster, similarly, T 2 represents the amplitudes of all multiple scattering processes in which particle 3 first scatters with particle 2 inside the cluster. While t 1 and t 2 represent, in the present work, the two body scattering amplitudes of DK → DK andDK →DK, respectively. We will take the results in the work of Ref. [26] for t 1 and t 2 , which will be discussed in following. In Eqs. (3) and (4), the G 0 is the K meson propagator between the D andD in the cluster, which is, with E 2 K (q) = | q | 2 + m 2 K , and q 0 is the energy of K meson in the rest frame of the cluster where the form factor F R (q) is calculated, and its expression is: where s is the invariant mass squared of the DDK three body system. In Fig. 4, we show the real (red line) and imaginary (blue line) parts of G 0 as a function of the invariant mass of the DDK system.

C. Scattering amplitude of the three body DDK system
In this work we study the K(DD) X(3720) configuration of the KDD system, which means that we need to study the DDK system where DD is treated as X(3720). Since D and D are isospin 1/2 states, then the DD isospin I = 0 state can be written as where the last numbers in the kets indicate the I z components of D andD mesons, (|I z D , I z D ). We take D = (D + , D 0 ) and D = (D 0 , D − ).
Then the three body scattering amplitude DDK|t|DDK can be easily obtained in terms of the two-body potentials V KD and V KD derived in Refs. [26,33], which is written as: This leads to the following amplitudes for the single-scattering contribution, It is worth noting that the argument of the total scattering amplitude T depends on the total invariant mass squared s, while the argument in t 1 is s DK and in t 2 is sD K , where s DK and sD K are the invariant masses squared of the external K meson with D andD inside the cluster of X(3720), respectively. In the rest frame of the X(3720), s DK and sD K are given by Before proceeding further, a normalization factor is needed for the two body scattering amplitudes t 1 and t 2 , With all the above ingredients, the total scattering amplitude of DDK three body system can be easily obtained, From the analysis of the K(DD) X(3720) scattering amplitude T one can identify dynamically generated resonances with peaks in |T | 2 .

III. NUMERICAL RESULTS AND CONCLUSIONS
In this section we show the numerical results obtained for the DDK three body scattering amplitude squared with total isospin I = 1/2 and spin-parity J P = 0 − . We evaluate the scattering amplitude T and search for peaks in |T | 2 , which are identified with three body states generated from the K(DD) X(3720) system. In fig. 5 we show the modulus squared |T | 2 for the KX(3720) → KX(3720) scattering. One can see that there is a clear and narrow peak around 4191 MeV, which could be associated with the state, K cc (4180), obtained in Ref. [23]. However, it was found that the root-mean-square radius of the DK subsystem in the DDK bound state is about 1.26 fm, while those of theDK and DD subsystems are much larger, yielding 2.27 and 2.10 fm, respectively. Here, we take the DD subsystem as a bound state of X(3720). Furthermore, taking √ s = 4191 MeV, we get √ s DK = √ sD K = 2342 MeV. At this energy, the interactions of DK andDK are strong. It is interesting to mention that a study of Ref. [60], within QCD sum rules, do not find aDD * s0 (2317) bound state, which may indicate that the K cc (4180) is a three body DDK molecule, where the DD, DK, andDK interactions are important. Though, it is clear that the existence of a DDK bound state seems to be a robust prediction.

IV. SUMMARY
In summary, we have investigated the DDK three body system assuming that there is a primary clustering of particles KX(3720) with X(3720) as a bound state of DD subsystem. By using the fixed center approximation to the Faddeev equations, we have obtained the KX(3720) → KX(3720) scattering amplitude. It is found that there is a clear and narrow peak in |T | 2 around 4180 MeV indicating the formation of a bound DDK state around this energy. These results are in agreement with those obtained in Ref. [23] using the Gaussian expansion method. Thus, the main value of the present work is not only to provide extra support for the existence of K cc (4180) state but also to test the reliability of the FCA to deal with the DDK system.
Finally, we would like to stress that, thanks to the strong s-wave interactions of DD and DK, the DDK system can bind. This support the existence of a excited hidden charm K meson, K cc (4180), with quantum numbers I(J P ) = 1 2 (0 − ) and mass about 4180 MeV. So far there is no experimental data available on this state [53]. It is expected that these theoretical results motivate its search in the future experimental measurements (see more discussions in Refs. [23,61,62]).