  The system of semi-relativistic particles coupled to a scalar Bose field is
considered. A scaled total Hamiltonian for the system is a self-adjoint
operator on a tensor product of a square-integrable function space and a boson
Fock space. We consider the strong resolvent limit of a renormalized
Hamiltonian, which is defined by subtracting a divergent term from the scaled
total Hamiltonian. Applying an abstract scaling limit theory and a unitary
transformation, we derive the Yukawa potential and Coulomb potential as
effective potentials.
