Interaction potentials for two-particle states with nonzero total momenta in lattice QCD

Abstract

In this study, we extend the HAL QCD method to a case where a total momentum of a two-particle system is nonzero and apply it to the 𝐼=2 S-wave 𝜋⁢𝜋 scattering in order to confirm its validity. We derive a fundamental relation of an energy-independent nonlocal potential defined in the center of mass frame with Nambu-Bethe-Salpeter wave functions in a laboratory frame. Based on the relation, we propose the time-dependent method to extract potentials, often used in practice for the HALQCD method in the center of mass frame. For numerical simulations in the 𝐼 =2 𝜋⁢𝜋 system, we employ (2+1)-flavor gauge configurations on a 32³×64 lattice at the lattice spacing 𝑎≈0.0907 fm and 𝑚[𝜋]≈700 MeV. Both effective leading order (LO) potentials and corresponding phase shifts obtained in laboratory frames agree with those obtained in the center-of-mass frame by the conventional HAL QCD method within somewhat larger statistical errors. In addition, we observe a consistency in scattering phase shifts between ours and results by the finite-volume method as well. The HAL QCD method with nonzero total momenta, established in this study, brings more flexibility to the HAL QCD method, which enables us to handle systems having the same quantum numbers with a vacuum or to access energy regions prohibited in the center of mass frame.