X(3872) as a hybrid state of the charmonium and the hadronic molecule

In order to understand the structure of the X(3872), the $c \bar c$ charmonium core state coupling to the $D^0 \bar D^{\ast 0}$ and $D^+ D^{\ast -}$ molecular states are studied. The strengths of the couplings between the charmonium state and the hadronic molecular states are determined so as to reproduce the observed mass of the the X(3872). The isospin symmetry breaking is introduced by the mass differences of the neutral and charged $D$ mesons. The obtained structure of the X(3872) is that about 7% is $\ccbar$ charmonium, 75% is the isoscalar $D \bar D^\ast$ molecule and 18% is the isovector $D \bar D^\ast$ molecule, which explains observed properties of the X(3872) well. The structure of the X(3872) we have obtained is not just a $D^0 \bar D^{ast 0}$ hadronic molecule nor the compact tetraquark state but the charmonium-hadronic molecule hybrid state.

As for the spin-parity quantum numbers of the X(3872), the angular distributions and correlations of the π + π − J/ψ final state have been studied by CDF [9] and they concluded that the pion pairs originate from ρ 0 mesons and that the favored quantum numbers of the X(3872) are J P C = 1 ++ and 2 −+ . Recent analyses support J P C = 1 ++ interpretation [10]. BABAR has found the evidence of the radiative decays of X(3872) → γJ/ψ with 3.4-3.6 σ significance [11,12], which implies that the C-parity of X(3872) is positive. Though we assume that X(3872) is J P C = 1 ++ , whether the quantum number is 1 ++ or 2 −+ is still an issue of the discussion and more experimental data are certainly necessary.
Since the first observation of the X(3872), it has received much attention because its features are difficult to explain if a simple cc bound state of the quark potential model is assumed [13]. X(3872) is one of the promising candidates of the exotic states reviewed in Ref. [14][15][16]. Many kinds of structures have been suggested for the X(3872) from the theoretical side, such as a tetraquark structure [17][18][19][20], a D 0D * 0 molecule [21][22][23][24][25][26][27][28] and a charmonium-molecule hybrid [29][30][31]. We also employ this hybrid picture and argue that that is most appropriate.
Usual size of the isospin symmetry breaking is at most a few %. It is interesting to know what is the origin of this strong isospin symmetry breaking. In [35], this problem has been studied by using the chiral unitary model and the effect of the ρ-ω mixing has been discussed in [36]. It was reported that both of the approaches can explain the observed ratio given in Eq. (1), but at present, consensus on the mechanism of the large isospin symmetry breaking has yet to be reached. We will show in this work that the mass difference of the D 0D * 0 and the D + D * − thresholds gives enough amount of the isospin violation to explain the experiments. The production processes have been studied in [37][38][39][40]. It seems that a pure molecule picture cannot explain the production rate of the X(3872) in the pp collision well [39]. There, the production rate of the X(3872) is about 1/20 of the rate of ψ(2S), which suggests that X(3872) has to have, by a very rough estimate, the order of 5% of the cc component.
The hadronic decays of the X(3872) are investigated in [41][42][43][44][45][46][47][48][49]. As for the radiative decays, as seen in [50][51][52][53][54][55][56][57][58], the existence of the core seems to be required, but the results depend on details of the wave function. Here we assume that the cc core is created by the weak interaction in the B decay as B → (cc) + K, and investigate the transfer strength from the cc to DD * . We only investigate the hadronic mode and will discuss the radiative decay elsewhere.
The X(3872) exists above the open charm threshold, DD. Below this threshold, the cc mass spectrum is well predicted by a simple quark model. The model, however, failed to predict the masses above the open charm threshold for the DD or BB S-wave sector. In this work, we also show that the cc peak above the threshold can actually disappear by introducing the cc-DD * coupling.
It is also an important issue that whether the charged partner of the X(3872) exists as a measurable peak or not. BABAR has searched such a state in the X → π − π 0 J/ψ channel and found no signal [59]. The hybrid picture, where the coupling to the cc core is essential to bound the neutral X, is consistent with the absence of the charged X.
Recently, Z b (10610) ±,0 and Z b (10650) ±,0 (J P =1 + ) resonances have been found in the Υ(5S) decay to Υ(nS)π + π − (n=1,2,3) and h b (mP )π + π − (m=1,2) reactions [60]. The masses of these resonances are just above the BB * and the B * B * thresholds, respectively; the main component is considered to be the B ( * )B * two-meson state. This means that there exists an almost-zero-energy bound state (or resonance) in each of the DD * and the B ( * )B * systems. In order to make such states, the attraction in the DD * system is considered to be about 2.7 times as strong as that of the B ( * )B * system because the reduced mass of the DD * system, 967 MeV, is about 2.7 times as light as that of the B ( * )B * system, 2651 MeV. On the other hand, the interaction between the D andD * mesons is probably about the same size as that between the B ( * ) andB * mesons. We argue that the extra attraction required for the X(3872) comes, at least mainly, from its coupling to the cc core, which is absent in these isovector Z b systems.
In this article, we present a hybrid picture where X(3872) is J P C = 1 ++ and consists of D 0D * 0 , D + D * − , and the 2 3 P 1 cc core, which stands for the χ c1 (2P ) if observed. A separable DD * interaction is introduced, whose strength is determined so as to give a zero-energy bound state when it applied to the B ( * )B * systems. The rest of the required attraction to form the X(3872) are assumed to come from the cc-DD * coupling. The coupling strength is determined so as to give the observed X(3872) mass. The cc core mass is taken from the quark model result, and the cutoff is chosen by considering the cc core size. As we will discuss later, the behaviors of the X(3872) do not depend strongly on the detail of the interactions. Main parameters of the present model are the overall strength of the two-meson interaction and that of the coupling, which are essentially determined from the masses of X(3872) and Z b 's. This simple picture, however, is found to be consistent with many of the experiments, such as the isospin symmetry breaking in the X(3872) decay, the production rate of X(3872) in the pp collision, and the absence of χ c1 (2P ) peak or the charged X, in addition to the mass of X(3872) and Z b 's, which are the inputs.
It should be noted here that the quark number is not the conserved quantity in QCD and our treatment of taking the cc and DD * as the orthogonal states is an approximate one. In the low-energy QCD, the spontaneous chiral symmetry breaking occurs and the light quarks get the dynamical masses. In such a situation, the treatment of taking the cc and ccuū (ccdd) as the orthogonal states seems to be acceptable, since these two states are energetically different. In order to study the structure of the exotic hadrons, how to count the quark number is an issue of the discussions. Three methods have been proposed to observe the number of the valence quarks in the hadron. The first one is to measure the elliptic flow in relativistic heavy ion collisions [61] while the second one is to measure the nuclear modification ratios in heavy ion collisions [62]. The last one is to use the fragmentation functions [63]. We hope some of these methods will be applied to the X(3872) and the quark component of the X(3872) will be determined experimentally.
Let us briefly mention features of our work in comparison to those that also employ the charmoniummolecule hybrid model [29][30][31]. In Ref. [29], the hybrid structure of the X(3872) has been studied in the QCD sum rule approach by considering a mixed charmonium-molecular current. They found a very deeply bound X(3872), m X = (3.77 ± 0.18) GeV, 97% of whose component is a charmonium. As we shall show in Sec.2, the structure of the X(3872) certainly depends on the binding energy strongly. In Ref. [30,31], the effective hadronic models with the charmonium-DD * molecule transition interaction have been used as well as in the present work. Danilkin and Simonov have studied the DD * production spectrum [30]. They obtained the strength of the cc-DD * coupling from the heavy quarkonium decay calculated by a quark model with a small adjustment. They certainly found a steep rise near the threshold. We have studied the B → X(3872) K or DD * K weak decay spectrum at almost same time independently in the very similar approach in [64,65]. Here, we also introduce the interaction between D andD * , and look into the features of the shallowly bound X(3872). The work in Ref. [31] they examined the effects of the Okubo-Zweig-Iizuka forbidden ρ 0 J/ψ and ωJ/ψ channels. Since the main decay modes of the X(3872) are the X(3872) → π + π − J/ψ and X(3872) → π + π − π 0 J/ψ, their inclusion is certainly important. As shown in [31], however, this effect on the pole position seems rather small. To avoid the complication we discuss it elsewhere. In the present study, we have introduced the attractive interaction between D andD * mesons with the coupling strength being consistent with the observed Z b (10610) and Z b (10650) masses. This point is new to the previous two studies and we consider that we can successfully draw the consistent picture of the observed exotic hadrons X(3872), Z b (10610) and Z b (10650).
One of the authors (S. T.) has studied the X(3872) using a quark potential model by introducing an extra (qq) pair to a cc system [66] and found a shallow bound state of qqcc with J P C = 1 ++ . Recently, an elaborate study has been done in the quark potential model [67]. They have performed the coupled channel calculations including two and four-quark configurations using the 3 P 0 model and found a good agreement with the experimental data. The purpose of the present work is to make the situation of the X(3872) clearer by studying the role of the cc core state, which couples to the D 0D * 0 and D + D * − molecular states, with a simple hadronic model. This approach will complement the picture given by the quark model approach.
This paper is organized as follows. In Sec.2, the calculation of the X(3872) state is given. In Sec.3, we discuss the transition strength of the weak decay of B meson: B → X(3872)K or DD * K using the Green's function approach. In Sec.4, we study the effects of the interaction between the D andD * mesons. We discuss the possibility of the other exotic hadrons by the present mechanism in Sec.5. Finally, Sec.6 is devoted to summary of this paper.

X(3872)
We argue that the X(3872) state is a superposition of the cc core state, the D 0D * 0 hadronic molecular state, and the D + D * − hadronic molecular states. So, the wave function of the X(3872) in the center of the mass frame is represented by The D 0D * 0 and D + D * − molecular states are given by where q q q represents the relative momentum of the D andD * mesons. The normalization of the states are Here ϕ 0 (q q q) and ϕ + (q q q) are the momentum representation of the wave functions of the D 0D * 0 and D + D * − hadronic molecular states, respectively. The charge conjugation is assumed to be positive throughout this paper. We assume these three states (|cc , |D 0D * 0 and |D + D * − ) are the orthonormal states. If |D 0D * 0 and |D + D * − are the spatially wide objects, this assumption seems to be reasonable. As we shall show in Fig. 1, indeed |D 0D * 0 and |D + D * − are the wide objects.
We introduce a coupling between the cc core state and the DD * states in the isospin symmetric manner. Since we are looking into the low energy region, the results do not depend much on the shape of the interaction. Thus, we take a monopole-type coupling as: The interaction we have introduced above causes effectively an attraction for the X(3872) because its energy is lower than the mass of the cc core, m cc . In this section, we ignore the direct interactions between the D and D * mesons; as we will discuss later in Sec.4, the coupling to the cc core seems more important to make the X(3872) than the direct DD * attraction.
Here we consider only the relative S-wave states of these two mesons in the non-relativistic scheme because the X(3872) is close to the threshold. The Schrödinger equation to solve is Since the interaction we employ is separable, we can solve this Schrödinger equation analytically. The bound state energy is obtained by solving the following equation. with and For later convenience, we define α 0 and α + by and where m X represents the observed mass of the X(3872).
In order to obtain the numerical results, we use the D meson masses given in the 2012 Review of Particle Physics [8] (Table 1). Since we have introduced the isospin symmetric interaction V in Eq. (7), the only origin of the isospin violation in the present model is the mass difference between the charged and neutral D and D * mesons.
As for the cc core state, we consider that it corresponds to the J P C = 1 ++ charmonium state with the mass m cc = 3.950 GeV, the closest cc core to X. This value is taken from the Godfrey and Isgur's results of the quark potential model calculation for the 2 3 P 1 cc state [68].
In the following, we will show the results as well as their dependence on the various assumptions.
There are two free parameters in the present model: the cutoff Λ and the dimensionless coupling constant g. We take typical hadron sizes for Λ: e.g., Λ = 0.3 GeV, 0.5 GeV and 1.0 GeV. Then, for a given Λ, the coupling constant g is determined so that the model reproduces the observed mass of the X(3872), namely, 3.87168 GeV. The results are given in Table 2. The wave function we have obtained: The values of c's are shown in Table 3 for each of the cutoff values. It seems that the overall feature of the admixture of each component does not depend much on the value of Λ, which is not surprising because a very shallow state does not depend much on detail of the potential. The main component of the X(3872) state is Table 1: Meson masses and the thresholds. All the entries are in GeV.   Table 3: |D 0D * 0 , reflecting the fact that the mass of the X(3872) is only 0.16 MeV below the D 0D * 0 threshold. The amplitude of the |D + D * − component is much smaller because the D + D * − threshold is 8.22 MeV above the mass of the X(3872). The size of the isospin symmetry breaking we have obtained for the averaged mass, 3.87168 GeV, seems to be roughly consistent with the one estimated from the experiments given by Eq. (2).
Let us emphasize that we have obtained a measurable amount of the |cc component. In the present scheme, the origin of the attraction is the coupling between the DD * component and cc core state. So, it is natural to have a certain amount of |cc component in the X(3872) state.
The mass of the cc core is taken from the quark model calculation. It may differ because of the model assumption. The slight change of its mass, however, does not change our results much. For example, for m cc = 3.93 or 3.97 GeV, which we take because ± 20 MeV is typical ambiguities of the quark models, the cc component becomes 11% and 7%, respectively. There is no drastic change in the results.
We take only this state as the cc core state in this paper because it has the closest mass to that of X(3872). The mass of the 1 3 P 1 cc state is, for example, around 3.5 GeV and therefore its coupling to the X(3872) will be suppressed [67]. In our calculation which includes the lower cc state with the same coupling size, the probability of the 1 3 P 1 cc state is found to be about 1/20 of that of the 2 3 P 1 cc state. The existence of another core component may change the nature of the γ-decay of X(3872), where a large cancellation occurs and results are very sensitive to the wave function [69,70]. We, however, look into such observables elsewhere, and concentrate on the bulk feature of X(3872) in this work.
The S-wave state of the D * D * channel is able to couple with the J P C = 0 ++ charmonium state and the threshold of the D * D * channel is about 140 MeV above the X(3872) mass. We, therefore, should examine whether the D * D * channel can contribute to the structure of the X(3872). We have performed the calculation of the X(3872) structure with the D * D * channel and the result has been that the D * D * component of the X(3872) is about 2%, reasonably small.
Experimental uncertainty of the X(3872) mass still exists. So, we solve the system also for m X = 3.8687 GeV. This mass is the one determined from the neutral B meson decay data, and the lightest mass among the ones given by the experiments. Now the binding energy becomes 3.14 MeV instead of the one corresponding to the average mass, 0.16 MeV. The value of g to form the lighter mass becomes 0.05625, which is 1.1 times as large as that of the average mass, 0.05110. In order to form the more deeply bound X, the dimensionless coupling constant g is required to be larger. The coefficients of the wave function are listed in Table 3. The size of the cc-core component also becomes larger: it changes from (0.293) 2 ≃ 0.086 to (0.522) 2 ≃ 0.272 as m X changes from 3.87168 to 3.8687 GeV for the case of Λ = 0.5 GeV. The size of the cc-core component in the X(3872) is found to be sensitive to the binding energy of the state. The amount of the isospin symmetry breaking depends also on the binding energy of X; The symmetry breaking occurs because of the difference of the binding energies of the X(3872) from the two thresholds, i.e., D 0D * 0 and D + D * − . For m X = 3.87168 and 3.8687 GeV, the ratios of the size of the isovector to the isoscalar DD * components are 0.315 and 0.057, respectively. When the mass of the X(3872) becomes smaller, namely, the binding energy becomes larger, the effects of the threshold difference becomes smaller, and the isospin violation becomes smaller.
Let us show the shape of the obtained wave functions. The explicit expressions of the wave functions in the coordinate space are and rϕ(r) with Each of the neutral and the charged DD * components of the wave function of the bound state with Λ = 0.5 GeV and m X = 3.87168 GeV is shown in Fig. 1. It is also found that the radius of the D + D * − component is much smaller than that of D 0D * 0 . In Fig. 2, we show the wave function of the X(3872) also for m X = 3.8687 GeV. One finds that the size of the bound state, especially the size of D 0D * 0 component, becomes much smaller than that in Fig. 1 though it is still much larger than the usual charmonium, whose rms 1 fm [68].

Spectrum
In this section, we investigate the transition strength S(E) of the B meson weak decay: B → ccK and cc to X(3872) or DD * . The X(3872) appears as a bound state in the spectrum. This spectrum does not correspond directly to the observed pion distribution in the X(3872) → J/ψπ n experiments. By looking into the DD * spectrum, however, one can see that the strength of DD * gathers around the threshold, and that the peak corresponding to the cc core actually disappears.
In this article, we assume that the observed X(3872) corresponds to a very shallow bound state. To have such a bound state, the interaction must be attractive but maybe a rather weak one. As mentioned in Sect. 2, we have fixed the strength of the DD * -cc coupling, g, so as to reproduce the observed X(3872) mass. In such a situation, the cc core state becomes a resonance appearing in the DD * continuum. Since no sharp resonance is observed experimentally around 3.95 GeV, the width of this resonance should be large. One of the issues in this section is whether such a 'weak' attraction can give a resonance with a large decay width.
The S(E) is normalized so that the production of the 2 3 P 1 cc state by the weak decay is equal to one. The vertex of the weak decay process, B → cc + K, and the probability that the cc is in the 2 3 P 1 configuration are factorized out. We assume that among the cc states produced by the weak decay, the 2 3 P 1 cc state plays a major role to form the X(3872) and the DD * spectrum up to around E ∼ 4 GeV because the predicted mass of the 2 3 P 1 cc state is 3.95 GeV. Again we use the non-relativistic scheme with the relative S-wave, because the reduced mass of the system is about 1 GeV and we only consider here up to about 0.1 GeV above the threshold.
Then, the S(E) is expressed as follows.
with the Green's function; Here, E represents the energy transfer andĤ is the full Hamiltonian of the cc-core and DD * system. The state |cc represents the center of mass system of the cc state with the normalization cc|cc = 1, This normalization leads the energy sum rule Using the free Green's functions and the interaction given in Eq. (7), the Green's function is represented as follows.
The calculated transition strength for the cutoff Λ = 0.3 GeV with the mass of the X(3872) m X = 3.87168 GeV is shown in Fig. 3. The spectrum has a sharp cusp above the D 0D * 0 threshold. The resonance which corresponds to the χ c1 (2P ) becomes very broad. The bound X(3872) is not plotted in the figure because it does not have a width in the scheme. If we consider the experimental inaccuracy of the energy and the X(3872) → J/ψππ decay width, the bound X(3872) peak and the threshold cusp will be merged into one single peak, which corresponds to the observed X(3872) in the J/ψππ spectrum. By integrating S(E) to the DD * continuum state, one can obtain the transfer strength from the cc-core to the bound state. In this case, the former is 0.949 while the latter is 0.051. The cc core state of the bare mass of 3.950 GeV becomes a resonance state of E = (3.974 − i 2 0.067) GeV; its peak position is by 24 MeV shifted upward. In Fig. 4, we show the transition strength for the cutoff Λ = 0.5 GeV. The spectrum is almost flat at around E = 3.95 GeV. In the case of this harder cutoff, the cc core state couples to the DD * continuum of more wider energy range. As a result, the bump around 3.95 GeV found for the Λ = 0. moves to E = (3.971 − i 2 0.147) GeV. The strength from the cc-core to the bound state becomes slightly larger, ı.e., 0.087.
Let us now show the effect of the difference in the binding energy. In Fig. 5, we plot the transition strength S(E) in the case of the cutoff Λ = 0.5 GeV and the mass of the X(3872) is m X = 3.8687 GeV, ı.e., the more deeply binding case. The S-wave threshold cusp becomes much smaller as the bound state position moves away from the threshold. The transfer strength to the bound state in this case is 0.269, much larger than the previous cases.
Since no peak is found around E = 3.95 GeV experimentally, the Λ = 0.5 GeV or more is favorable in that sense. This corresponds to the hadron size ∼0.4 fm, which is a reasonable value. By setting cutoff of this  Fig. 3.
size, the shallow bound state and the large decay width for the cc peak can be realized simultaneously. In the following calculation, we use Λ = 0.5 GeV.

Effect of the interaction between D andD *
In this section, we introduce the interaction between the D andD * mesons. We use the Yamaguchi separable potential [71] for the interaction, namely, where Λ is the cutoff, and λ is the strength of the interaction. The Yamaguchi separable potential has been first introduced to study the deuteron, the shallow bound state of one proton and one neutron. So, we consider this interaction is suitable for the present case. The cutoff Λ determines the interaction range and therefore, the range is chosen to the typical hadron size here. For simplicity, we take the same value for the cutoff Λ in Eq. (26) as that of Eq. (7) in the following calculation. In order to give a zero-energy bound state only by the two-meson interaction, the strength should be with the reduced mass of the system, µ MM ′ . For Λ=0.5 GeV and µ BB * =2.651 GeV, this strength becomes 0.1886, which we denote λ B below. As for the DD * , the required strength to have a zero-energy bound state becomes 0.5712. First let us make a rough estimate of the size of the DD * attraction using the information from the B ( * )B * system. Each of the BB * and B * B * systems has a J P = 1 + resonance by about 2∼3 MeV above the thresholds: MeV. This strongly suggests that the two-meson attraction is barely strong enough to make a zero-energy bound state (or somewhat weaker), and that there are almost no mixing between the BB * and B * B * 1 + states. The physical origin of the two-meson interaction is probably the light-meson exchange and/or the gluonic interaction. In either case, the strength of the two-meson interaction for the DD * system has a similar size to that of the BB * system, because the bosons exchanging between the light quarks is considered to give the largest contribution. So, also for the two-meson interaction between the D andD * , we employ the one with the strength which gives a zero-energy bound state for the BB * systems, λ B .
Thus the DD * interaction we employ is: Though, we look into the effects of the DD * attraction by changing the value of λ from λ B .
To use λ B also for the interaction between D andD * mesons means that we assume the attraction is independent of the isospin as well as of the heavy quark masses. Let us make a brief comment why we do not employ the pion-exchange (OPE) interaction, and accordingly a spin-isospin dependent interaction. The spin-isospin factor of the OPE interaction between the light quark and the anti-quark is −(τ ·τ )(σ · σ) [72]. The factor −(τ ·τ )(σ · σ) becomes +1 for the BB * or the B * B * diagonal states; i.e. the Yukawa term is repulsive here. (See Table 4, where we also show those for the DD * systems). Both of the values corresponds to those obtained from the heavy meson effective lagrangian [72,73]. It has been reported that the OPE interaction (with the tensor term and higher order partial wave states) makes a bound state [73]. There, however, they found that one bound state below the BB * threshold and one resonance above the B * B * threshold rather than two similar resonances. This occurs because the factor σ · σ will also cause the mixing between the BB * and B * B * states. Thus, the spin dependence of OPE interaction seems inconsistent with the B ( * )B * experiments, where the energy difference of the two peaks is almost the same as that of the two thresholds. As was pointed out in Ref. [72], the Yukawa term and the δ-function term in the OPE interaction tend to cancel each other. We consider the OPE interaction is small and the effects of the spin-isospin independent attraction are dominant in the present systems.
Let us go back to the Schrödinger equation, which now includes the two-meson interaction, U : Now the model has one more parameter, λ, which stands for the coupling strength of the interaction between the D andD * , in addition to the cutoff Λ and the coupling strength between the cc core and the two-meson state, g. The λ = 0 limit corresponds to the results in Sec. 2, where we determined the coupling strength g so as to reproduce the observed X(3872) mass without introducing the direct DD * attraction. We call that value g 0 in the following and use it as a reference.
In order to study the effects of the interaction between the D andD * , we change the value of the coupling constant λ. For a positive λ, g should be smaller than g 0 in order to reproduce the observed mass of the X(3872). Or, equivalently, when (g/g 0 ) 2 < 1, one has to take λ > 0 to compensate the weakened coupling. At the g = 0 limit, the X(3872) becomes a pure D 0D * 0 hadronic molecular state. There is no charmonium component nor the D + D * − component in the X(3872). There will be a similar bound state in the D + D * − system also, and the cc core becomes a sharp resonance at around 3.95 GeV. We consider the actual situation is in-between of the two λ = 0 and g = 0 limits. Table 4: The spin-isospin matrix elements of the OPEP by the two-meson states: DD * J P C = 1 ++ and B ( * )B * I(J P ) = 1(1 + ). In Fig. 6, we show the size of each of the cc, the D 0D * 0 and the D + D * − components in the X(3872) wave function in our calculation. For each values of (g/g 0 ) 2 , we re-adjust the value of λ to fit the mass of the X(3872) to be 3.87168 GeV. In Fig. 7, we also plot the sizes of each of the isovector and the isoscalar DD * components. As the interaction between the D andD * becomes larger (ı.e., (g/g 0 ) 2 becomes smaller), the isovector DD * component in the X(3872) wave function becomes larger while the isoscalar DD * component reduces to 0.5.
As was mentioned before, experimentally the isovector component seems to be about one forth of the isoscalar component (see eq. (2)). Also, the production process of X(3872) suggests that there should be a measurable cc component. From Fig. 7, one can find that these requirements are fulfilled when (g/g 0 ) 2 is close to 1, namely the λ = 0 limit.
When the DD * interaction is switched on, and its strength becomes λ = λ B , the coupling to the cc core becomes g = 0.0427315, which corresponds to (g/g 0 ) 2 = 0.699. This point also gives the appropriate size of the isospin symmetry breaking as well as the measurable cc component. There each of the components of the X(3872) wave function is: This result means that about 6% of the X(3872) is the charmonium, about 69% is the isoscalar DD * molecule and 26% is the isovector DD * molecule. Provided that the rhs of Eq. (2) corresponds faithfully to the ratio of the isovector to the isoscalar DD * molecular components in the X(3872) wave function as it is, the state expressed by Eq. (29) is consistent with the experiment. This situation seems to depend on the (g/g 0 ) 2 value only mildly. We have also solved the system where the mass m X = 3.8687 GeV, namely, by about 3 MeV more bound case. The components in such a case are shown in Fig. 8. Here, the cc component is much larger than that of m X = 3.87168 GeV. The size of the cc core component is sensitive to the value of the binding energy. To make the mass m X = 3.8687 GeV, the strength becomes g = 0.04873, which corresponds to (g/g 0 ) 2 = 0.750. The core component becomes large in this situation, though the isovector component becomes somewhat smaller.
In Fig. 9, we plot the transition strength S(E) for the Λ = 0.5 GeV and m X = 3.87168 GeV with (g/g 0 ) 2 = 0.699 case. Also when the DD * interaction is introduced, it is found that the strength gathers close to the ! " thresholds. The strength to the X(3872) is 0.056. The peak around the cc core disappears due to the coupling between the two-meson states and the cc core. It becomes a resonance of E = (3.966 − i 2 0.091) GeV. Thus, we conclude that in case of the X(3872), rather small amount of the interaction is coming from the direct interaction between the D and D * mesons and that the rest of the attraction is coming from the coupling to the cc core state. Then we have the right size of the isospin symmetry breaking as well as a measurable cc component, both of which are key features to explain the experiments. Also, this picture is consistent with the existence of Z b resonances and absence of the charged X.

Application to the other systems
In this section, we discuss the possibility to apply the present method to investigate the existence of other exotic hadrons as well as the absence of the QQ states above the threshold. If there is a charmonium or a bottomonium state (QQ) above the Qq and the qQ meson threshold and if the quantum numbers of the system allows QQ to couple to those two mesons, then this coupling causes the effective attraction between the two mesons by the same mechanism as the present approach. Moreover, if the coupling occurs in S-wave, the effective attraction can be larger and the QQ state gains a large decay width.
For the overview, we show the observed mass spectra of χ cJ (nP ) and χ bJ (nP ) with the quark model results for the QQ systems in Table 5 with the lowest S-wave threshold of the Qq and the qQ mesons. The potential in the quark model consists of the color-Coulomb, linear confinement, and the color-spin interactions. The values of the parameters in the interactions are taken from [68]. Since we have neglected the spin-orbit interaction and the tensor terms, all of the obtained masses of the 3 P J are the same. One can see from the table that the Table 5: The observed mass spectra of cc and bb systems with J P C = J ++ (J = 0, 1, 2) [8] and of the quark model with the color-Coulomb, linear confinement, and the color-spin interactions (QQ) with their lowest S-wave DD or BB threshold. Parameters in the interactions are taken from [68]. All entries are in MeV.
observed states below the S-wave threshold roughly correspond to those calculated by the quark model. Above the threshold, however, simple QQ states are not observed any more. We argue that they disappear because they have a large width due to the coupling to the two-meson scattering states.
From the X(3872) case, we have learned that the QQ state by about 80 MeV above the threshold can contribute to form such an exotic state assuming that the size of the coupling is similar to the X(3872) case. Let us check whether such a state exists in the other systems.
We next investigate the J P C = 0 ++ charmonium states. The ground state is χ c0 (1P ), and its mass is (3414.75 ± 0.31) MeV. The χ c0 (2P ) state has not been observed and the theoretical estimation of its mass is 3920 MeV [68], whose mass is by 44 MeV lighter than our calculation due to the noncentral force. The main S-wave decay channel of the χ c0 (2P ) state is the DD, whose thresholds is 3730 MeV. The cc state is by about 200 MeV above the threshold; its effects may be attractive, but the size is probably small.
As for the J P C = 2 ++ charmonia, the situation is different from the 0 ++ or 1 ++ charmonia. In this channel, the first radially excited state, χ c2 (2P ), has been observed, while only the ground states have been observed in the 0 ++ and 1 ++ channels. The reason of this difference is simple in the present picture. The 2 ++ channel can couple only to the D * D * systems in S-wave; their threshold, 4014 MeV, is rather high and 87 MeV above the χ c2 (2P ) mass. The calculated mass of the χ c2 (3P ) is 4325 MeV, which is about 300 MeV above the D * D * threshold. So, its effects may be repulsive in this channel.
In summary, the X(3872) is found to be surprisingly special. Although there may be exotic hadrons with the higher partial wave, or one has to consider the rearrangement meson channels such as QQ-qq systems, the cc 1 ++ channel seems the only promising candidate to form an S-wave exotic hadron by the present mechanism: a hybrid state of the charmonium and the hadronic molecule.

Conclusion
In this work, we have studied the structure of the X(3872) as well as the transfer strength from the cc core to the DD * scattering state. The system consists of D 0D * 0 , D + D * − , and the 2 3 P 1 cc core, which stands for the χ c1 (2P ) if observed. We have introduced the direct interaction between the two mesons, which is just as attractive as the one which makes a zero-energy bound state if applied to the B ( * )B * system. Namely, we assume that this two-meson interaction gives the Z b (10610) and Z b (10650) resonances. This interaction, however, is not strong enough to make a bound state in the DD * systems alone. In this model, the coupling between the cc core and the DD * two-meson state is also introduced, which effectively produces the attraction between the D andD * . We assume that this coupling provides the rest of the attraction required to make a bound state in the DD * system, X(3872). Both of the interaction and the coupling are assumed to be isospin independent. The isospin symmetry breaking in this model solely comes from the mass difference between the neutral and charged D and the D * mesons.
In the obtained wave function of the X(3872), there is about 6% of the cc core component. This size is consistent with a rough estimate from the X(3872) production rate in the pp collision. As for the DD * components of the X(3872) wave function, 69% is isoscalar and 26% is isovector; the ratio of the isovector to the isoscalar DD * components is also consistent with the experiments of the final π 2 to π 3 decay ratio. The present work shows that the structure of the X(3872) is not a simple cc nor a simple D 0D * 0 bound state. It is charmonium-hadronic molecule hybrid, which is certainly an exotic state.
Since the cc core cannot couple to the charged DD * states, such as D +D * 0 , the present picture can explain why there exists no charged partners of the X(3872). Also, it can explain why the 2 3 P 1 cc core, or χ c1 (2P ), is not found experimentally though it has been predicted by the quark model which gives correct mass spectrum below the open charm threshold; this core couples strongly to the DD * two-meson state and becomes a resonance with a very broad width.
In order to confirm the present picture of the X(3872), we consider that the inclusion of the ρJ/ψ and ωJ/ψ channels is important because the X(3872) is mainly observed in the X(3872) → ρJ/ψ → ππJ/ψ and X(3872) → ωJ/ψ → πππJ/ψ channels. We are now performing such calculations and the results will be reported soon.
Recently, Belle Collaboration reported the results of the radiative decays of the X(3872) [74]. They searched the X(3872) → ψ ′ γ in B decays, but no significant signal has been found. On the other hand, BABAR Collaboration has reported that B(X(3872) → ψ ′ γ) is almost 3 times that of B(X(3872) → J/ψγ) [12]. To make the situation clear, it is useful to calculate the radiative decays of the X(3872) in the present model including the charmonium structure. It is left as the future study.