$K^*$ mesons with hidden charm arising from $KX(3872)$ and $KZ_c(3900)$ dynamics

Inspired by the recent discovery of the pentaquark states $P_c(4450)$ and $P_c(4380)$, which can be viewed as excited nucleon states with hidden charm, we study the three-body interaction of a kaon and a pair of $D\bar{D}^*$ in isospin 0 and 1. We show that the two body interactions stringently constrained by the existence of the $D_{s0}^*(2317)$, $D^*_{s1}(2460)$, $X(3872)$, and $Z_c(3900)$, which are widely believed to contain large $DK$, $D^* K$, and $D\bar{D}^*$ components, inevitably lead to the existence of a heavy $K^*$ meson with hidden charm. Concrete coupled channel three-body calculations yield its mass and width as $(4307\pm 2)- i (9\pm 2)$ MeV with $I(J^P)=1/2(1^-)$. This state, if found experimentally, definitely cannot be accommodated in a $q\bar{q}$ picture, and therefore presents a clear case of an exotic hadron.

Understanding the nature of hadronic resonances/bound states is one of the most challenging issues in the frontiers of hadron physics. In recent years, experimental [1][2][3][4][5] and theoretical [6][7][8][9][10] efforts have been focusing on the nontraditional hadronic states, which cannot be (easily) explained either as qq or qqq states. One of the most recent claims on such kind of states is the P c (4380) and P c (4450) pentaquark states observed by the LHCb collaboration in the J/ψp invariant mass distribution of the Λ 0 b → J/ψK − p decay [11]. Curiously the existence of such states of molecularD(D * )Σ c /Λ c nature was predicted prior to the experimental claim [12].
The possible existence of such non-conventional mesons and baryons dates back to the original quark model of Gell-Mann and Zweig, in which the existence of multiquark states was already anticipated [13,14]. In spite of such a long lapse of time, the recent intensified theoretical and experimental efforts show clearly that the topic is still controversial.
Regardless of all these efforts, there is still a vast unexplored energy region and systems in which states of nonconventional quark content could be found, especially at energies of 4∼5 GeV. For instance, in the meson sector, heavy mesons of strangeness 0 with hidden charm, such as X(3872) or Z c (3900), have been found, and they are widely regarded, particularly the X(3872), as moleculelike states of DD * +c.c. in isospin 0 and 1, respectively (see, e.g., Refs. [15][16][17][18][19]). However, in the strange sector, there is surprisingly no experimental data available on heavy K or K * meson states around this energy region, leaving the heavy strange physics experimentally unexplored.
In this letter, we explore the possibility of the existence of K * moleculelike states (bound states/resonances) with hidden charm in a three-meson system formed from a kaon and a pair of DD * + c.c., when the latter is organized either as X(3872) or as Z c (3900). Different from other three-body studies, the interactions of the two-body subsystems in the present case, FIG. 1. Internal structure of the K * states found. The interaction DD * + c.c. forms the states X(3872) in isospin 0 and Zc(3900) in isospin 1. When a K is added to the system, the interaction between the KD (D) and KD * (D * ) systems is such that bound states around 4300 MeV are formed whose internal structure correspond to a K − X or K − Z moleculelike states. namely, the DK,DK, D * K,D * K, and DD * /DD * are stringently constrained by a large number of experimental as well as lattice QCD data. For instance, it is a known fact that the DK, D * K, and DD * /DD * interactions are attractive such that the D * s0 (2317), D * s1 (2460), X(3872), and Z c (3900) can be understood as molecular states of the respective pair of hadrons [17,[20][21][22][23][24][25][26]. In addition, studies in both lattice QCD as well as chiral perturbation theory show that theDK and D * K interactions in I = 0 are moderately attractive while in I = 1 are slightly repulsive [27,28]. Given such information, the existence of K(DD * + c.c.) bound states, depicted in Fig. 1, depend on the possibility of the attractive pair interactions dominating over the repulsive ones. It is the purpose of the present work to perform concrete coupled-channel three body studies to confirm such a scenario and to find their masses and widths.
Furthermore, it is interesting to note the similarity of our arXiv:1805.08330v1 [hep-ph] 22 May 2018 system with one of the most studied three-body system, thē KN N system (see, e.g., Refs. [29,30] and references cited therein).
In the present work, the KDD * systems are studied using the so-called fixed-center (FCA) approximation to solve the Faddeev equations, where one of the two-body subsystem is considered as a scattering center, whose properties do not get altered during the scattering. Such a formalism is especially relevant to systems where two of the three hadrons form a bound state, while the third hadron is a light hadron as compared to the mass of the bound system. Indeed, FCA has been successfully employed in describing theKd interaction at low energies [31][32][33]. Further, a comparison of the results obtained by solving the Faddeev equations, considering intermediate excitations of the bound system, and within FCA is done in Refs. [34], which shows that the two results are very similar, implying that the FCA is a good approximation in such cases. More recently, the FCA to the Faddeev equations has been used to study the formation of three-hadron resonances in several system, such as φKK, systems of one pseudoscalar and two vector mesons, ηKK and η KK, πKK * , ρKK, ρDD, ρD * D * , ρB * B * , the DKK and DKK systems, the BDD and BDD systems (see, for example, Refs. [35,36] and references therein).
We start by studying the KX(3872) configuration of the three-body system considering the description for X as a (DD * ) I=0 + c.c. system. In such a picture, we can treat DD * andDD * as clusters in the KX(3872) scattering, which does not get perturbed by the low energy scattering of K off the cluster and apply the FCA to get the three-body scattering matrix T . Within the FCA, T can be written as a sum of two of FIG. 2. Diagrams showing the scattering of the particle labeled "3" (K) on a cluster (X) made of particles 1 (D orD) and 2 (D * or D * ). the standard Faddeev partitions, where T 31 and T 32 represent the sum of the infinite series of diagrams with the particle "3" (kaon in the present case) first scattering off the particle "1" (D orD) and "2" (D * or D * ), respectively. The sum of the two series are illustrated as diagrams in Fig. 2, which can be expressed, mathematically, as a set of two coupled equations, I. Input two-body t 31(32) -matrices for total isospin 1/2 of the three-body system. The components of the vectors below have to be interpreted as a compact notation for writing KC b |t 31(32) |KCa as a linear combination of the two-body t-matrices in isospin I = 0 and I = 1 on the basis (t I=1 In the latter case, a global minus sign has to be considered for the non-diagonal transitions. In Eq. (2), t 31 , t 32 correspond to a weighted sum of the isospin 0 and 1 s-wave amplitudes of the KD (KD * ) and KD (KD * ) systems. The weights related to the total isospin I = 1/2 configuration of the three-body system are summarized in Table I. To obtain these values we write, for example, t 31 as where C i is the i-th cluster, and I i , I i z represent the isospin and its third component for the i-th cluster. The KD (KD), KD * (KD * ) amplitudes are obtained by solving the Bethe-Salpeter equation in a coupled channel approach, using a kernel obtained from a Lagrangian based on heavy-quark spin symmetry. As mentioned earlier, these coupled channel interactions generate the resonances D * s0 (2317) and D * s1 (2460). A normalization factor √ M a M b /m D(D * ) is included, with M a , M b being the masses related to the clusters in the initial and final states, respectively, in the definition of t 31 and t 32 . The origin of this factor, as explained in Refs. [37,38], lies in relating the S-matrix of the three-body system with the scattering of one particle on a cluster of the remaining two.
The loop function G 0 , in Eq. (2), represents the Green's function of the K meson propagating in the (DD * ) I + c.c. cluster, and can be expressed as where m K represents the mass of the kaon and q 0 is the onshell energy of the kaon in the center-of-mass frame of the kaon and the cluster: with √ s being the energy in the above mentioned frame. Note, that the form factor F a (q) introduced is related to the wave function of the cluster in terms of its internal DD * (DD * ) structure. We calculate this form factor following Refs. [37,38] as where N = F a (q = 0) is the normalization factor, and ω D (p) = » m 2 D + p 2 , ωD * (p) = » m 2 D * + p 2 . The upper integration limit Λ is chosen to be the same as the cutoff used to regularize the loop DD * /DD * to get the cluster (X(3872) or Z c (3900)). We take Λ ∼ 700 MeV from Refs. [17,39,40] and vary it up to 800 MeV to estimate the uncertainties involved in the results.
By using Eqs. (1) and (2), the total amplitude T can be written as T = T 31 + T 32 , with and is calculated as a function of the three-body invariant mass, √ s. For a given √ s, the two-body amplitudes are obtained at the invariant masses s 31 and s 32 of the relevant subsystem [41].
In Fig. 3 we show the results found for the T -matrix of the KX system for isospin 1/2 and spin-parity J P = 1 − . It can be seen from Fig. 3 that a narrow peak appears around 4303 MeV for the cutoff Λ = 750 MeV. It can also be seen that the results do not vary much with the cut-off. It should be noted that the peaks in Fig. 3 have a small but a nonzero width (∼ 0.7 MeV), even though the peak position lies below the KX and KDD * , KDD * thresholds. This small width comes from the intermediate open channels, like, πD sD * , which are implicitly considered in our formalism through the input KD amplitude in isospin 1. This amplitude is obtained by solving the Bethe-Salpeter equation considering KD and πD s as coupled channels. In fact, if the coupling to two-body open channels is switched off when getting the KD, KD * amplitudes in isospin 1, we indeed find a zero width state in the T -matrix.
In the last years, the existence of several exotic companions of the X(3872) has been claimed experimentally as well as theoretically (for a review see Ref. [10]). Particularly, Z c states, with isospin 1, have been reported in the same energy region of the X(3872), like the Z c (3900) found by the BE-SIII [42], or the Z c (3894) claimed by the Belle collaboration [43] or the Z c (3886) reported by the CLEO collaboration [44]. At the present moment it is unclear, given the experimental uncertainties in the masses and widths, if all these experimental findings do, or do not, correspond to the manifestation of the same state. Such a discussion is beyond the scope of the present work, but it would be interesting to study under the same formalism as for KX the existence of K * with hidden charm which could be interpreted as KZ c moleculelike states. Due to the present experimental uncertainty, we are using the name Z to denote the isospin 1 partner of X found in Ref. [40], which has a mass around 3872 MeV and width around 50 MeV. In case of the scattering of K with Z, to obtain reliable results, the width, Γ, of the Z can play a relevant role. In our formalism such information can be introduced by replacing the mass M of the cluster with M − iΓ/2 in the expression of the form factor. Since Γ Z ∼ 50 MeV (in agreement with the latest experimental data [42]) is not too large, and we are interested to study the formation of states below the KZ threshold, we can still rely on the FCA formalism to calculate the KZ → KZ amplitude.
In Fig. 4 we show the modulus squared amplitude for KZ scattering in isospin 1/2 (see Table I for the input two body t-matrices used in Eq. (8)). A clear signal for the formation of a state around 4268 MeV and a width of 25 MeV is seen. If we would have neglected the width of the Z state, a peak at 4287 MeV with a small width (∼ 0.6 MeV, which results from the presence of open channels, like πD sD * , in the system, as discussed in the case of the KX scattering) is observed. In both cases, the mass of the state is about 80-100 MeV below the KZ threshold (considering Z as a stable particle). This energy region is well within the range of the reliability of the results obtained within the FCA. Considering the uncertainty in the results due to the cut-off Λ, we find an isospin 1/2, J P = 1 − , state with M −iΓ/2 = (4268.3±3.4)−i(13.1±1.6) MeV in the KZ scattering.
The KZ system can also have total isospin 3/2. If a state appears in this case, it would be associated with an exotic strange meson with isospin 3/2 and spin-parity 1 − . We have studied this configuration of the KZ system but find no state formed in it.
Comparing the results of the KX and KZ systems in isospin 1/2, it can be concluded that both interactions result in formation of a state. However, the DD * (and its complex conjugate) system can reorganize itself in different isospin config- urations during the scattering with kaon, while conserving the total isospin of the three-body system producing transitions between the configurations KX and KZ. Such a possibility can be studied by treating KX and KZ as coupled channels.
In such a case the t 31 , t 32 and G 0 appearing in Eq. (8) are matrices in the coupled channel space (see Table I).
We have studied the effect of coupling the KX and KZ systems and, thus, allowing the transitions between them. The modulus squared amplitudes obtained for both systems, by solving the scattering equations within a coupled channel approach, are shown in Fig. 5.
In this calculation, the width of Z is taken into account. The mass and width of the KX state can now be written as M − iΓ/2 = (4337.0 ± 1.4) − i(3.3 ± 0.2) MeV, and of the KZ state is M − iΓ/2 = (4277.6 ± 2.8) − i(14.0 ± 2.5) MeV. It can be seen that the coupled channel calculation increases the mass of the KX state by ∼ 30 MeV and its width is better estimated to ∼ 6.5 MeV. The mass of the KZ state, on the other hand, increases by 10 MeV, while its width is similar to the one obtained in the single channel calculation.
In summary, we have studied the KDD * and KDD * systems where the DD * andDD * are treated as clusters forming X(3872) or Z c (3900). We find that both KX and KZ interactions lead to the generation of a new state, in each case, of molecular nature (see Fig. 1). Both states can be associated with K * -mesons with hidden charm. The masses of the two states are M −iΓ/2 = (4337.0±1.4)−i(3.3±0.2) MeV and (4277.6 ± 2.8) − i(14.0 ± 2.5) MeV. Interestingly, a recent study [45] solving the Schrödinger equation for the DD * K system, but with a very different dynamics than the one used here, found a state with a mass of 4317 MeV.
So far there is no experimental data available on K * states in the energy region investigated in the present work [46], so the results found here are predictions for K * mesons with hidden charm and of molecular three-body nature. Such states can be found at facilities, such as BEPC, in processes with final states, such asK 0 D + s D − . We hope that our work encourages such experimental investigations.