Hadron-hadron potentials coupled to quark degrees of freedom for exotic hadrons

We study the properties of the hadron-hadron potentials and quark-antiquark potentials from the viewpoint of the channel coupling. We demonstrate that, for finite quark masses, the coupling to the two-hadron continuum induces the imaginary part of the quark-antiquark potential, in contrast to the string-breaking phenomena in the static limit. It is also shown that the elimination of the different degrees of freedom induces the nonlocality and energy dependence of the effective potentials. For the obtained nonlocal potentials, we apply two methods of the local approximation proposed previously, the formal derivative expansion and the derivative expansion in the HAL QCD method, by carefully examining the energy dependence of the potential. As an application, we construct a coupled-channel model of $c\bar{c}$ and $D^{0}\bar{D}^{*0}$ to describe $X(3872)$, and discuss the property of the effective $D^{0}\bar{D}^{*0}$ potentials. We confirm that the local approximation by the HAL QCD method works better than the formal derivative expansion also for the energy-dependent potential. At the same time, we show that, in the HAL QCD method, the resulting phase shift is sensitive to the choice of the wavefunction to construct the local potential when the system has a shallow bound state such as $X(3872)$.