πJ/ψ − DD̄∗ potential described by the quark exchange diagram

Exotic hadrons reported in the heavy flavor sector have attracted much interest in hadron and nuclear physics. Especially near the thresholds, the hadron-hadron interaction is important to understand these exotic structures. In this talk, the short range πJ/ψ − DD̄∗ potential whose importance is indicated by the Lattice QCD is studied by the D(∗) meson exchange model, and the Bornorder quark exchange model. We find the large difference between the results of the D(∗) meson exchange and the quark exchange calculations.


Introduction
Exotic hadrons reported in the heavy flavor sector have been one of the interesting topics in hadron and nuclear physics. In the experiments, such as BaBar, Belle, BESIII, and LHCb, unexpected mesons called XYZ states have been reported, whose mass spectra have not been explained by the standard constituent quark picture being qqq as a baryon and qq as a meson [1,2]. Especially for the exotic meson near the thresholds, those states could be realized as a hadronic molecule which is a loosely bound or resonant state of multi-mesons. For instance, X(3872) reported by Belle in 2003 has been discussed as the DD * molecule [3,4]. On the other hand, some of the exotics near the thresholds might be a cusp by the kinematical effect [2], which is not a physical bound state.
Z c (3900) is also an exotic state reported in the Y(4260) → ππJ/ψ decay by BESIII [5], Belle [6] and other facilities [7,8]. One of the interesting properties of Z c (3900) is that it has nonzero electric charge which cannot be possessed by the standard cc state. Therefore, Z c (3900) is a genuine exotic state which must have a multiquark component such as ccud.
Recently HALQCD collaboration performed Lattice QCD simulation for the coupledchannel system including πJ/ψ − ρη c − DD * corresponding to Z c (3900) at the non physical pion mass m π = 410 − 700 MeV [15]. This simulation indicates that Z c (3900) is a virtual state, and it is induced by the strong πJ/ψ − DD * potential. It is an interesting result because such charm quark (charmed meson) exchange process is expected to be suppressed due to the large mass of the exchange particle. * e-mail: yasuhiro.yamaguchi@riken.jp In this talk, we study the short range πJ/ψ − DD * interaction for the quantum number J PC = 1 +− , with the total angular momentum J, parity P and C−parity C. The πJ/ψ − DD * interaction is described by (i) the meson exchange model, and (ii) quark exchange model, and we compare the results obtained by these models. In the meson exchange model, the D ( * ) meson exchange is introduced in the πJ/ψ − DD * off-diagonal term. In the quark exchange model, the πJ/ψ − DD * interaction is given by the Born-order quark exchange diagram [16,17].

Quark exchange model
To describe the short range interaction in πJ/ψ−DD * , the Born-order quark exchange diagram is employed [16,17]. The scattering amplitude is given by the sum of four diagrams called capture 1, capture 2, transfer 1, and transfer 2, as shown in Fig. 1. The capture (transfer) diagram is induced by the quark-quark and antiquark-antiquark (quark-antiquark) interactions. For instance, the scattering amplitude of the capture 1 is given by where ( A, B, C, D) stand for the momenta of mesons A, B, C, D, and ( a, b, c, d) for the momenta of quarks inside mesons, function V q is the quark-quark interaction by the constituent quark model and φ i (i = A, B, C, D) is the meson wave function. The meson wave function is obtained by the constituent quark model Hamiltonian [16,17], The parameters are fixed to reproduce the meson masses, summarized in Tab. 1. Above K q is the kinetic term, and F i = λ i /2 with the Gell-Mann matrix λ i . In this study, the single Gaussian function is employed for the meson wave function, The Gaussian parameter b G is determined to minimize E(b G ) = φ|H q |φ . The obtained meson masses and Gaussian parameter are summarized in Tab. 2.

Numerical results
In this section, the cross section of the πJ/ψ − DD * transition is shown, and the results of the meson exchange and quark exchange models are compared. In Fig. 2 [16,17].
The charm quark mass m c is also determined to fit the charmed meson masses   Fig. 2 (a), the large contribution from the spin-spin term is obtained. This contribution is given by the light quark dynamics, because the spin-spin term in the charm quark interaction is suppressed due to the large mass of the charm quark. Fig. 2 (b) shows the cross section by the meson exchange model. In comparison with the result of the quark exchange, the cross section in the meson exchange model is very small. This observation would be useful to understand the short range πJ/ψ − DD * interaction. The D ( * ) exchange potential plays a minor role, while the very short range interaction such as the quark exchange has the important role in the πJ/ψ − DD * scattering.

Summary
We have investigated the short range πJ/ψ−DD * potential by two models: the quark exchange and the meson exchange models. Understanding the short range interaction is important to investigate the exotic structure which is realized as a bound state or cusp near the hadron thresholds. In this talk, the cross sections in the πJ/ψ − DD * transition by the two models have been compared. The cross section in the quark exchange is dominated by the spin-spin term which is contributed by the light quark dynamics rather than the charm quark one. In comparison with the quark exchange, the cross section obtained by the meson exchange is very small. The large difference between two models is obtained, and it would be useful to understand the short range interaction in the πJ/ψ − DD * channel.