The $D\bar{D}^*$ interaction with isospin zero in an extended hidden gauge symmetry approach

The $D \bar{D}^*$ interaction via a $\rho$ or $\omega$ exchange is constructed within an extended hidden gauge symmetry approach, where the strange quark is replaced by the charm quark in the $SU(3)$ flavor space. With this $D \bar{D}^*$ interaction, a bound state slightly lower than the $D \bar{D}^*$ threshold is generated dynamically in the isospin zero sector by solving the Bethe-Salpeter equation in the coupled-channel approximation, which might correspond to the $X(3872)$ particle announced by many collaborations. This formulism is also used to study the $B \bar{B}^*$ interaction, and a $B \bar{B}^*$ bound state with isospin zero is generated dynamically, which has no counterpart listed in the review of the Particle Data Group. Furthermore, the one-pion exchange between the $D$ meson and the $\bar{D}^*$ is analyzed precisely, and we do not think the one-pion exchange potential need be considered when the Bethe-Salpeter equation is solved.


I. INTRODUCTION
The hidden gauge symmetry approach has been shown to be a successful method to include the vector meson in the Lagrangian [1][2][3][4]. Along these lines, the pseudoscalar meson and vector meson interaction [5], the vector meson and vector meson interaction [6,7], the vector meson and baryon octet interaction [8,9], and the vector meson and baryon decuplet interaction [10,11] in SU(3) flavor space have been studied in the coupled-channel unitary approximation. This method is extended to SU(4) space when the components related to c andc quarks are taken into account [12][13][14]. In the past few years, more and more XYZ states subspace of u, d and c(b) quark components. Many studies have been done on this topic [15][16][17], and it should especially be stressed that this replacement is used in the study of the generation of charm-beauty bound states of B(B * )D(D * ) and B(B * )D(D * ) interactions [18].
It is clear that the model becomes much simpler than those used in Refs. [12][13][14], where the SU(4) hidden gauge symmetry approach is discussed in detail.
The X(3872) state was first observed by the Belle Collaboration in 2003 [19], and then confirmed by many experimental collaborations. Finally, a mass of 3871.69 ± 0.17 MeV [20] and a decay width < 1.2 MeV [21] are given by fitting the experimental data, which is extremely close to the DD * threshold. A lot of theoretical research work has been done on the properties of X(3872). Some people suppose X(3872) to be a DD * /DD * bound state since its mass is very close to the DD * threshold [22][23][24][25]. X(3872) is also described as a virtual state of DD * /DD * [26,27], a tetraquark [28][29][30], a hybrid state [31] or a mixture of a charmonium χ c1 (2P ) with a DD * /DD * component [32,33]. Moreover, the X(3872) state is studied by using the pole counting rule method [34,35], and it is found that two nearby poles are necessary to describe the experimental data [36,37].
In the present work, we will replace the strange quark by the charm quark in the SU (3) hidden gauge symmetry approach, and then study the DD * interaction in the coupledchannel unitary approximation by solving the Bethe-Salpeter equation. Consequently, the X(3872) state is generated dynamically when the ρ and ω exchanges between D andD * mesons are taken into account.
One-pion exchange between D andD * mesons at the DD * threshold is addressed specially.
Since the mass of theD * meson is about one pion mass larger than the mass of the D meson, the intermediate pion might be regarded as a real particle at the DD * threshold, therefore the behavior of the DD * interaction through one-pion exchange is interesting. However, although it is divergent at the DD * threshold, the one-pion exchange potential of DD * becomes weaker when the total energy of the system departs from the DD * threshold. When the hidden gauge symmetry approach is considered, the ρ and ω meson exchange between D andD * mesons is dominant, and thus the one-pion exchange potential is neglected in the present work.
In addition, this model is extended to study the BB * interaction in the isospin zero sector by replacing the c quark with a b quark, and a new bound state is predicted, which is not listed in the review of the Particle Data Group (PDG) [20].
This article is organized as follows. The formulism is described in Section II, and then the implementation of unitarity is discussed in Section III, where the contribution from the longitudinal part of the vector meson propagator in the loop function of the Bethe-Salpeter equation is taken into account. The one-pion exchange potential of DD * is analyzed in Section IV. The calculation results on the DD * and BB * interactions are presented in Section V. Finally, a summary is given in Section VI.  In the hidden gauge symmetry approach, the DD * interaction would proceed through the exchange of a vector meson, as depicted in Fig. 1(a). Since the vector propagator contributes a factor of 1/M 2 V if the momentum transfer between the D meson and theD * meson can be neglected, the exchange of ρ and ω mesons is dominant, while the possible exchange of heavier vector mesons is suppressed.
The DDρ and DDω couplings can be obtained with the Lagrangian where with f π = 93MeV the pion decay constant and M V the mass of the ρ meson.
The matrices of vector mesons and pseudoscalar mesons take the form of and respectively, where only the relevant mesons are enumerated.
The Lagrangian density of vector mesons can be written as with According to Eq. (5), we can derive the D * D * ρ and D * D * ω couplings from the interaction Lagrangian Since the mass of the ω meson m ω = 782 MeV is similar to that of the ρ meson m ρ = 770 MeV, we suppose M V ≈ m ρ ≈ m ω , then the potential of the D meson andD * meson is simplified as with ε and ε * the polarization vectors of the initial and final vector mesons, and k 1 (p 1 ) and k 2 (p 2 ) the momenta of the initial and final D(D * ) mesons, respectively. The coefficients C ij in the different channels are shown in Table I.
The DD * pair with isospin I = 0 takes the form of where the C-parity of the DD * pair is assumed to be positive.
According to Eqs. (8) and (9), the potential of DD * with isospin I = 0 can be written According to Ref. [38], the kernelṼ t DD * →DD * can be obtained from the potential form in Eq. (10) when the Bethe-Salpeter equation is solved, i.e., where the ε · ε * structure has been eliminated.
Actually, the kernel in Eq. (11) can be written as where the Mandelstam variables s = (p 1 + k 1 ) 2 , t = (k 2 − k 1 ) 2 and u = (p 2 − k 1 ) 2 . In the derivation of Eq. (11), we have neglected the momentum transfer q = k 2 − k 1 compared to the mass of the vector meson M V , which would be a good approximation for the interaction relatively close to threshold where bound states or resonances are searched for, i.e., t = (k 2 − k 1 ) 2 = 0 is assumed in the approximation. Thus the kernel in Eq. (12) is only a function of the Mandelstam variables s, which is the square of the total energy in the center

III. IMPLEMENTATION OF UNITARITY
In the coupled-channel unitary approach, the unitarity can be implemented into the DD * interaction by solving the Bethe-Salpeter equation: whereṼ is the kernel of the DD * interaction provided by Eq. (11), andG is the DD * loop function. The loop function is logarithmically divergent and thus is calculated with a three-momentum cutoff [39,40], or by means of dimensional regularization [41]. Recently, a loop function of a pseudoscalar meson and a vector meson is derived in the dimensional regularization scheme, where the contribution of the longitudinal part of the vector meson propagator is taken into account in Ref. [38]. In the present work, this formula of the loop function will be applied to the DD * interaction in the hidden gauge symmetry approach.
The loop function can be written as where G D * D (s) is the original form of the loop function in Ref. [41], while the terms related to H 00 D * D (s) and H 11 D * D (s) stem from the longitudinal part of the vector meson propagator, and their analytical forms can be found in the appendix of Ref. [38].

IV. ONE-PION EXCHANGE
From the Lagrangian in Eq. (1), we can obtain the interaction Lagrangian for the D * Dπ coupling, which can be written as Therefore, the one-pion exchange potential of the D andD * mesons is obtained as as shown in Fig. 1(b). The coefficients D ij in Eq. (17) for different channels are listed in Table II. According to Eq. (9), the one-pion exchange potential of the D andD * mesons in the sector of isospin I = 0 can be written as where q = p 2 − k 1 = p 1 − k 2 , p 1 · ε = 0 and p 2 · ε * = 0 are used in the derivation.
Since theD * meson mass is about one pion mass larger than that of the D meson, M D * − M D ≈ m π , the intermediate pion can be regarded as a real particle approximately at the threshold of DD * , i.e., q 2 0 ≈ q 2 + m 2 π . The denominator in the one-pion exchange potential of the D andD * mesons in Eq. (18) can be written as approximately. However, the zero component of the polarization vector of theD * meson is in inverse proportion to theD * meson mass, and thus can be neglected in the calculation, so we have and q · ε ∼ | q|.
According to Eqs. (19), (20) and (21), although the one-pion exchange potential of the D andD * mesons is divergent at the DD * threshold, it can be neglected when the total energy of the system is far away from the DD * threshold.
In Ref. [22], the one-pion exchange potential of the D andD * mesons is assumed to be dominant in the generation of the X(3872) state. The formula of the one-pion exchange potential is given explicitly in the second term in Eq. (11) of Ref. [22], which is relevant to the external three-momentum in the center-of-mass frame. The potentials of the D and D * mesons as functions of the total energy of the system √ s are depicted in Fig. 2, and it can be found that the vector meson exchange potential is more important than the one-pion exchange potential of the D andD * mesons if the hidden gauge symmetry is taken into account. Therefore, the one-pion exchange potential of the D andD * mesons is neglected in the present work.

V. RESULTS
In Ref. [42], the DD * interaction is studied in the SU(4) flavor space, and an intermediate J/ψ exchange in the kernel is taken into account besides the ρ and ω exchanges. Actually, the J/ψ particle is heavier than the ρ and ω mesons, and the DD * interaction via a J/ψ exchange can be neglected in the calculation. Moreover, we suppose that the pion decay constant f π = 93 MeV in the DD * potential in Eq. (8). However, the f 2 π is replaced with f i f j in the potential in Eq. (4) of Ref. [42], related to the initial and final particles, respectively.
In the DD * → DD * process, both f i and f j take the value of the decay constant of the D meson, i.e., f i = f j = f D = 165 MeV.
Five channels of 1 are discussed in Ref. [42]. A potential is given by where ξ ij denotes the coefficient between these channels, and ǫ and ǫ * are 3-dimensional polarization vectors of the initial and final vector mesons, respectively. When the J/ψ exchange is neglected, the coefficients ξ ij for the 1 √ 2 (D * + D − − c.c.) and 1 √ 2 (D * 0D0 − c.c.) channels can be obtained from the values listed in Table I. It is apparent that ǫ· ǫ * is supposed to be −1 in Ref. [42], and thus the coefficients in these two channels take negative values in Eq. (5) of Ref. [42].
TheK * K threshold is far lower than the energy region where the X(3872) is detected.
Thus the 1 √ 2 (K * − K + − c.c.) and 1 √ 2 (K * 0 K 0 − c.c.) channels can be excluded when the generation of the X(3872) particle is discussed. Moreover, it should be emphasized that channel only contributes about 0.016 of the probability in the wave function of the X(3872) particle, as discussed in Ref. [42], so the 1 √ 2 (D * + D − − c.c.) and 1 √ 2 (D * 0D0 − c.c.) channels play an important role in the generation of the X(3872) particle. Therefore, it is reasonable that only the DD * interaction is taken into account in the present work.
The resonance state of DD * corresponds to the condition det(I −ṼG) = 0.
In a single channel, Eq. (24) leads to poles in theT amplitude whenṼ −1 =G. Figure 3 shows D meson and theD * meson is analyzed precisely. Since the mass of theD * meson is just one pion mass larger than that of the D meson, the intermediate pion can be treated as a real particle at the DD * threshold. Thus the diagram of the one-pion exchange between the D meson and theD * meson is divergent and supplies a singularity at the DD * threshold.
However, this one-pion exchange potential becomes trivial when the total energy of the DD * system is far away from the threshold, so it is neglected in this work.
A kernel of the DD * interaction by exchanging a ρ or ω meson is derived, and then this kernel is used to solving the Bethe-Salpeter equation in the coupled-channel unitary approximation. In the isospin I = 0 sector, a DD * bound state with a mass about 3872 MeV is produced, which is slightly lower than the DD * threshold and can be regarded as a counterpart of the X(3872) particle. This method is also extended to study the BB * interaction by replacing the corresponding c quarks with b quarks, respectively, and a bound state is produced in the isospin I = 0 sector, which has no counterpart in the PDG data. It should be emphasized that the regularization scale takes different values from the DD * case when Although a DD * bound state can be generated dynamically in the isospin I = 0 sector, which stems from the ρ and ω meson exchange due to the hidden gauge symmetry approach, the DD * interaction in the isospin I = 1 sector is unfortunately zero, and thus no bound state can be generated dynamically. This implies that other mechanisms need to be considered besides the hidden gauge symmetry approach.