HAL QCD potentials with non-zero total momentum and an application to the $I=2$ $\pi\pi$ scattering

We consider the HAL QCD method in the system with non-zero total momentum (laboratory frame). We derive a relation between the NBS wave function in the laboratory frame and the energy-independent non-local potential (HAL QCD potential), and propose the time-dependent method to extract the potential from correlation functions in the laboratory frame. We then apply this formulation to the $I=2$ $\pi\pi$ system to calculate the corresponding potential in the laboratory frame, employing the 2+1 flavor gauge configuration on a $32^3\times 64$ lattice at the lattice spacing $a\simeq 0.091$ fm and $m_\pi \simeq 700$ MeV. While statistical errors are larger, the effective leading order (LO) potentials and corresponding phase shift agree with those from the HAL QCD potential in the center of mass (CM) frame. We also demonstrate the consistency in scattering phase shifts between the HAL QCD method in several frames and the finite volume method. The HAL QCD method in the laboratory frame enlarges applicabilities of the method to investigate hadron interaction including mesonic resonances such as $\rho$ and $\sigma$.