Generalized Lüscher formula in multichannel baryon-meson scattering

L\"uscher's formula relates the elastic scattering phase shifts to the two-particle energy levels in a finite cubic box. The original formula was obtained for elastic scattering of two massive spinless particles in the center-of-mass frame. In this paper, we consider the case for the scattering of a spin $1/2$ particle with a spinless particle in a multichannel scattering. A generalized relation between the energy of the two-particle system and the scattering matrix elements is established. We first obtain this relation using quantum mechanics in both the center-of-mass frame and in a general moving frame. The result is then generalized to quantum field theory, using methods outlined by Hansen and Sharpe [Phys. Rev. D 86, 016007 (2012)]. We verify that the results obtained using both methods are equivalent up to terms that are exponentially suppressed in the box size.