Rummukainen-Gottlieb formula on a two-particle system with different masses

A proposal by Luscher enables us to extract the elastic scattering phases from two-particle energy spectrum in a cubic box using lattice simulations. Rummukainen and Gottlieb further extend it to the moving frame, which is devoted to the system of two identical particles. In this work, we generalize Rummukainen-Gottlieb's formula to the generic two-particle states where two particles are explicitly distinguishable, namely, the masses of the two particles are different. Their relations with the elastic scattering phases of two-particle energy spectrum in the continuum are obtained for both C-4 upsilon and C-2 upsilon symmetries. Our analytical results will be very helpful for the study of some resonances, such as kappa, vector kaon, and so on.