We investigate the relativistic bound states in the non-perturbative regime with the light-front Hamiltonian formalism. Our studies focus on two systems, the relativistic Yukawa model and the light unflavored mesons.

We apply the light-front Hamiltonian approach to the two-fermion bound state in the (3+1)-dimensional Yukawa model. Within the single-particle coordinate representation, we construct the effective Hamiltonian in the lowest Fock sector with Basis Light-Front Quantization (BLFQ) and solve the eigenvalue equation of the relativistic bound state with the scalar boson exchange as well as the self-energy correction. We propose a working scheme to regularize the self-energy correction. From the Hamiltonian, we obtain the full mass spectroscopy and the light-front wave functions. We calculate the spin components and the distribution functions for the selected triplet and singlet bound states. We compare our numerical results with those of other models whenever available.

We study the light unflavored mesons also using the BLFQ approach. We improve the Hamiltonian used by others in the heavy quarkonium system in the valence quark sector and extend the framework to study the light unflavored meson. Within the relative-particle coordinate, we diagonalize the effective Hamiltonian in our basis function representation to obtain the mass spectrum and the light-front wave functions. Based on these light-front wave functions, we then study the structure of selected mesons by computing the electromagnetic form factors, the decay constants, the parton distribution amplitudes, and the parton distribution functions. In particular, we perform the respective evolutions of these distributions from a low energy scale to a high energy scale. Our results are compared with available experiments and other theoretical models.

Those studies serve as crucial steps for future applications of relativistic bound states in the non-perturbative quantum field theory.