$D^*\bar{D}_1(2420)$ and $D\bar{D}'^*(2600)$ interactions and the charged charmonium-like state $Z(4430)$

The $D^*\bar{D}_1(2420)$ and $D\bar{D}'^*(2600)$ interactions are studied in a one-boson-exchange model. Isovector bound state solutions with spin parity $J^P=1^{+}$ are found from the $D^*\bar{D}_1(2420)$ interaction, which may be related to the observed charged charmonium-like state $Z(4430)$. There is no bound state solution found from the $D\bar{D}'^*(2600)$ interaction.


I. INTRODUCTION
A resonant structure near 4.43 GeV in the π ± ψ ′ invariant mass distribution was first observed by the Belle Collaboration [1], which is the first evidence of the existence of charged charmonium-like states. The mass M = 4433±4(stat)±2(syst) MeV and width Γ = 45 +18 −13 (stat) +30 −13 (syst) MeV were extracted by using a Breit-Wigner resonance shape. A higher mass M = 4485 +22+28 −22−11 MeV and a larger width Γ = 200 +41+26 −46−35 MeV were reported by Belle Collaboration through a full amplitude analysis of B 0 → ψ ′ K + π − decay and it spin-parity J P = 1 + was favored over other hypotheses [2]. Recently, the LHCb Collaboration released their new result about B 0 → ψ ′ π − K + decay, which confirmed the existence of the 1 + resonant structure Z(4475) with a mass 4475 ± 7 +15 −25 MeV and a width 172 ± 13 +37 −34 MeV with high significance [3]. The Z(4475) was observed in the ψ ′ π invariant mass spectrum, which suggests that it should be a exotic state beyond conventional cc charmonium state which has a neutral charge. Many theoretical efforts have been paid to understand the internal structure of the Z(4475) and a number of explanations have been offered. Since the Z(4475) carries the charge the hybrid interpretation is excluded [4]. It is natural to explain the charge carrier Z(4430) as a multiquark system in which except for cc there exist other light quarks. The first type of multiquark explanation is the excited tetraquark [5][6][7][8][9] where four quarks are in one colour singlet. Another type of multiquark explanation is loosely bound molecular state composed of two charmed mesons [10,11], or charmed baryons [12]. There also exist several nonresonant explanations, such as, the threshold cusp effect [13], cusp in the D 1 D * channel [14].
The mass of the Z(4475) measured by Belle Collaboration [1], 4433 ± 4(stat) ± 2(syst) MeV, is close to the D * D 1 (2420) threshold. It is popular to attempt to explain Z(4475) as a S wave D * D 1 molecular state with J P = 0 − after its first observation. The analyses based on the long range one pion exchange mechanism conclude that the isovec-tor D * D 1 (2420) system can not form a J P = 0 − or 1 − bound state [15,16] and a probable isovector 1 + [15]. Further inclusion of medium range σ meson exchange leads to S wave binding for D * D 1 (2420) with J P = 0 − , 1 − and 2 − [17], and to binding for the D * D 1 (2420) configuration but only for J P = 0 − . A calculation in the context of QCD sum rule [18] also favors the molecular D * D 1 (2420) bound state explanation with spin-parity 0 − . The new Belle and LHCb results suggest the spin-parity of Z(4475) is 1 + . So, the previous theoretical conclusions are inconsistent to the new experimental results. A new calculation by Barnes et al. suggests that the Z(4475) is either a D * D 1 state dominated by long-range π exchange, or a DD * (1S , 2S ) state with short-range components [19]. It is also suggested that the Z(4475) may be from the interaction DD ′ * 1 (2600) in S wave because the mass of Z(4475) is very close to the DD ′ * 1 (2600) threshold [20]. The Bethe-Salpeter equation is a powerful tool to study the bound state problem, such as deuteron [21]. A Bethe-Salpeter formalism has been developed and applied to study the Y(4274) and its decay pattern [22,23] and Σ c (3250) as D * 0 (2400)N system [24]. In Refs. [25,26], the BB * /DD * system was also studied by solving Bethe-Salpeter equation with boson exchange mechanism to explore the possible relationship between the recent observed Z b (10610) and Z c (3900) and the BB * /DD * interaction. It was found that for a system composed of two constituents with different masses and spins, the cross diagram can not be transformed to the schrödinger equation with potential in coordinate space V(r) [25]. As shown in Ref. [16], that the cross diagram is more important in the π exchange mechanism in the DD 1 (2420) system. Besides, in the DD ′ * 1 (2600) interaction where only cross diagram exist in π exchange mechanism due to forbiddance of DDπ vertex. Hence, it is of interesting to study the D * D 1 (2420) and the DD ′ * (2600) systems in the Bethe-Salpeter equation approach.
In this paper, we will study the Z(4475) by solving the Bethe-Salpeter equation. The mass of the Z(4475) is close to the threshold of configurations, D * D′ 1 (2430), D * D 1 (2420), DD ′ * (2600), and D * D′ * (2550). The large width of D ′ 1 (2430) , Γ = 384 +130 −110 MeV [27], which means very short lifetime, make it difficult to bind it and D * together to form a bound state with width about 170 MeV. The configuration D * D′ * (2550) is also related to Z(4475) in the literature. However, its threshold is about 100 MeV higher than the mass of Z(4475). In our theoretical frame only loosely bound states are considered. Hence, in this paper, only two configurations, D * D 1 (2420) and DD ′ * 1 (2600), are included in calculation. The paper is organized as follows. In next section we develop a theoretical frame to study the system with both D * D 1 and DD ′ * configurations (we omit the number for mass 2420 and 2600 here and hereafter) by solving the coupled channel Bethe-Salpeter equation. In Section II, the potential in π and σ exchange mechanisms is derived with the help of the effective Lagrangian from the heavy quark effective theory. The numerical results are given in Section IV. In the last section, a summary is given.

II. COUPLED CHANNEL BETHE-SALPETER EQUATION
We start from the Bethe-Salpeter equation for the vertex |Γ , where the V and G are the potential kernel and the propagator for two constituents of the system. The vertex function of system with two configurations can be written as where Γ D * D 1 and Γ DD ′ * are the vertex functions after separating out the flavor parts |D * D 1 and |DD ′ * . In this paper the S U(2) symmetry is considered, so the same vertex function is used for one configuration.
The explicit flavor structure for |D * D 1 system is [16] where c = ± corresponds to C-parity C = ∓ respectively. The flavor structure for DD ′ * system is analogous to the D * D   1 system. The vertex function can be rewritten as with i = 1, 2 for configuration D * D 1 or DD ′ * , a for different components in a configuration. The δ i,a is the factor for |i, a in one configuration in Eq. (3). After multiplying i, a|, the Bethe-Salpeter equation becomes As in Ref. [25], we adopt the spectator covariant theory to make quasipotential approximation of the 4-dimensional Bethe-Salpeter equation to 3-dimensional equation. With the help of on-shellness of the heavyer constituent 2, D 1 /D ′ * , the numerator of the propagator P µν 2 = λ 2 ǫ µ 2λ 2 ǫ ν † 2λ 2 with ǫ µ 2λ being the polarization vector with helicity λ 2 . Different from Ref. [25] where DD * system is considered, the constituent 1, D * , for the system D * D 1 is also a vector meson. In this paper, we make an approximation P µν 1 = λ 1 ǫ µ 1λ 1 ǫ ν † 1λ 1 with polarization ǫ µ 1λ 1 on shell. Such approximation will introduce an uncertainty about several percent in the propagator, which will be further smeared by introduction of the form factors. Now, the equation for the vertex is in the form with the rest of propagator G j 0 for particle 1 and 2 in j configuration with mass m j 1 and m j 2 written down in the center of mass frame where P = (W, 0) is where The integral equation can be written explicitly as where the reduced potential kernel with a factor as The normalized wave function can be related to the vertex as |φ i For the wave function, we have the relation φ i 1,2 and J T,1,2 are the parity and spin for the total system, constituent1 or constituent 2 in the configuration i. We only consider the independent wave functions for a system with certain spin-parity, which are listed below, The DD ′ * can not form a system with 0 + system, so for 0 + only D * D 1 configuration is involved.

III. LAGRANGIAN AND POTENTIAL
The effective Lagrangians describing the interaction between the light pseudoscalar meson P and heavy flavor mesons are constructed with the help of the chiral symmetry and heavy quark symmetry, The octet pseudoscalar meson matrices reads as which corresponds to D = (D 0 , D + , D + s ) andD = (D 0 , D − , D − s ). The coupling constant g can be extracted from the experimental width of D * with value g = 0.59 [28]. Falk and Luke obtained an approximate relation k = g in quark model [29]. With the available experimental information, Casalbuoni and collaborators extracted h ′ = (h 1 + h 2 )/Λ χ = 0.55 GeV −1 [30]. The coupling constant for D ′ * decaying into Dπ and D 1 π can be extracted from the decay widths obtained in quark model as Γ D ′ * 1 →Dπ = 10.84 MeV and Γ D ′ * 1 →D 1 π = 0.28 MeV [31]. The values are g ′ = 0.086 and h ′′ = (h ′ 1 +h ′ 2 )/Λ χ = 0.42 GeV −1 .
For a loosely bound system the long-range interaction through π exchange should be more important than the shortrange interaction. Moreover, in this paper, the constituent is unstable, so the short range interaction should be further suppressed due to the short interaction time. Hence, the heavy meson exchange, such as ρ and ω mesons are not considered in this paper. The σ exchange which mediate the medium range interaction is included as in Ref. [17]. The Lagrangians for heavy meson and scalar σ meson read, The coupling constant g σ = g ′ σ = − 1 2 √ 6 g π with g π = 3.73 [32].
With the above Lagrangians, we can obtain the potential for direct and cross diagrams, where p ( ′ ) 1,2 is the initial (final) momentum for constituent 1 or 2. The flavor factor I i j c,d is listed in Table. I. The form factor is introduced to compensate the offshell effect of heavy meson h(k 2 ) = [ nΛ 4 nΛ 4 +(m 2 −q 2 ) 2 ] n with n = 2. In the propagator of the meson exchange we make a replacement q 2 → −|q 2 | to remove the singularities as Ref. [33]. So, the form factor for the light meson is chosen as f (q 2 ) = Λ 2 −m 2 Λ 2 +|q 2 | .
Corresponding to the independent wave function, the po-tential can be written in the matrix form as The potentials for J P = 1 ± and 2 ± can be obtained analogously.

IV. NUMERICAL RESULTS
To search the bound state from the D * D 1 and DD ′ * coupled channel system, we need to solve the coupled channel 3-dimensional integral equation. We follow the procedure in Ref. [25]. First, we make a partial wave expansion to reduce the 3-dimensional integral equation to a one-dimensional equation, where k/l is the number of the wave function with certain spin-parity. After discretion of |k| and |k ′ | by the Gauss quadrature, the recursion method in Refs. [23,25,34] is adopted to solve the nonlinear spectral problem. The numerical results are listed in Table II. First, we consider the case with D * D 1 configuration only. As shown in Table II, There exists bound solution with quantum number J P = 0 − with cutoffs about 1 GeV. Such S wave D * D 1 molecular state has been related to the Z(4475) with assumption that it carries spin-parity J P = 0 − . However, the new experimental results favor 1 + , which correspond to a P wave D * D 1 bound state. In this work, all quantum number J ≤ 2 will be considered in the range of cutoff 0.8 < Λ < 2 GeV.
Different from Ref. [16], the π exchange is dominant in the D * D 1 interaction in our model, and the effect of σ exchange is negligible compared with π exchange. In the π exchange, the contributions from D * D 1 → D 1D * diagram, that is, the cross diagram, is more important than the contribution from the direct diagram D * D 1 → D * D 1 . So, the contribution from cross diagram of π exchange is dominant in the D * D 1 interaction. Considered the flavor factors listed in Table I, the interaction through π exchange in isoscalar sector is three time larger than these in isovector sector, which is also found in Ref [19]. The numerical results in Table II support such observation. More bound states are produced in isoscalar sector, which is also consistent with Ref [19]. In isovector sector, only two bound states are produced from D * D 1 interaction. One of them has quantum number with I G (J P ) = 1 − (1 + ) compared with the experimental observed quantum number of the Z(4475), J P = 1 + .
The S wave DD ′ * system has a spin-parity 1 + which is consistent with the new experimental results and its mass is very II: The binding energies E for D * D 1 and DD ′ * and coupled channel systems with different cutoffs Λ. "-" means that no bound state is found. The cutoff Λ and binding energy are in the units of GeV and MeV, respectively. . close to the mass measured in the new experiment [3]. Hence, in Ref. [35] it was suggested to be the best candidate of the Z(4475). However, In our calculation, no bound state solution is found from DD ′ * interaction with coupling constant h ′′ = 0.42 GeV −1 . In this work, the coupling constant is determined from the decay width predicted in the quark model [31]. It is of interesting to test the effect of variation of coupling constant h ′′ . We change the h ′′2 to larger values and find that even with 10h ′′2 , there is no bound state produced.
In Table II, we also present the results with both configurations, D * D 1 and DD ′ * . The results are almost the same as these with the configuration D * D 1 only, which suggests that the interaction between DD ′ * through π and σ exchange mechanism is much weaker than the interaction between D * D 1 . Hence, for the system with two configurations, D * D 1 and DD ′ * , the cross diagram contribution from π exchange for channel D * D 1 → D 1D * is dominant. So, the results is sensitive to square of the coupling constant, h ′2 , for D 1 → D * π. The value h ′ = 0.55 GeV −1 is extract from the old experimental data, which corresponds to decay width Γ tot (D 1 (2420)) ≈ 6 MeV [30]. Compared with the new suggested value in PDG, 25 ± 6 MeV [27], the largest possible value of h ′ is about 1 GeV −1 . It is of interest to check the variation of results especially the bound state with Z(4475) quantum numbers with the variation of coupling constant h ′ . As shown in Table III, if we adopt larger h ′ , both 1 ++ and 1 +− state, which is related to Z(4475) can be generated with smaller cutoffs. With a coupling constant h ′ = 0.95, the isovector bound sates with J P = 1 ++ and J P = 1 +− are generated with cutoffs about 1.3 GeV and 1.5 GeV, respectively. With such coupling constant, the cutoffs needed to generate the isoscalar bound state are smaller. Hence, if the bound states in isovector with J P = 1 + sector is the Z(4475) observed in experiment, there should be a rich spectrum in isoscalar vector.

V. SUMMARY AND DISCUSSION
The new experimental results released by LHCb Colaboration exclude the S wave D * D 1 molecular state explanation with quantum number J P = 0 − for the Z(4475). In this paper we discuss the possibility to interpret the Z(4475) as D * D 1 or DD ′ * molecular state with quantum number J P = 1 + in a coupled channel Bethe-Salpeter equation approach. The long range π exchange and medium range σ exchange mechanisms are adopted to describe the interactions of D * D 1 (2420) and DD ′ * (2600).
It was suggested that the Z(4474) is a good candidate of S wave DD ′ * molecular state. However, in our calculation it is found that the interaction between DD ′ * from the π and σ exchange mechanism is very weak and not strong enough to generate the bound state. In Ref. [19], a 40% larger quarkpion coupling is sufficient to generate a isovector bound state which can be related to Z(4475). However, in our model a 10 times lager h ′′2 is still not enough to generate any bound states. It indicts that the π exchange in DD ′ * in our model is much weaker than these in Ref. [19], which may be partly from the different treatment of the cross diagram contribution. As discussed in Ref. [25], the potential for cross diagram can not be converted to a potential V(r) in coordinate space even if the potential is only dependent on the exchange momentum q in momentum space.
A calculation with both D * D 1 and DD ′ * channels is performed, and it is found that the results is almost the same as these obtained from D * D 1 channel only. For D * D 1 system, the cross diagram of π exchange is dominant and other contributions from direct diagram of π and σ exchanges are small. With reasonable coupling constants and cutoffs, the isovector bound state solutions are found in the D * D 1 interaction with J P = 1 + , which can be related to the experimental observed Z(4475). Due to the dominance of the cross diagram contribution with π exchange, the flavor factors lead to a three time stronger interaction between D * D 1 for the isoscalar sector than these for the isovector sector, which is consistent with the discussion in Ref. [19]. Hence, if the Z(4475) is assigned as the isovector bound state with J + from D * D 1 interaction, a rich spectrum for isoscalar bound state is predicted.