We develop and apply methods for solving non-perturbative quantum field theories in the Hamiltonian formalism. The current work is a first step towards an ab initio approach to QCD bound-state problems.

In particular, we investigate heavy quarkonium within the basis light-front quantization approach. We implement a phenomenological confinement from the Light-Front Holographic QCD and a theoretically derived one-gluon exchange effective interaction. We adopt the holographic light-front wavefunctions as our basis and solve the bound-state problem by diagonalizing the Hamiltonian matrix. We obtain the mass spectrum for charmonium and bottomonium. We also compute the decay constants and the elastic form factors for selected mesons. The results compare favorably with experimental measurements and with other established methods.

We also address systematic non-perturbative renormalization in a simpler model, the scalar Yukawa model, using the a Fock sector dependent renormalization scheme. We apply the Fock sector truncation up to four constituent particles. The eigenvalue equation is properly renormalized and a set of coupled integral equations are derived. We solve these equations by a parallel numerical iterative procedure. We find that the lowest (one- and two-body) Fock sectors dominate the physical state up to a non-perturbative coupling α ≈ 1.7. By comparing with lower sector truncations, we show that the form factor converges with respect to the Fock sector expansion in the perturbative and non-perturbative regime. This calculation demonstrates the use of the systematic Fock sector expansion with a proper non-perturbative renormalization as an ab initio approach to solve light-front quantum field theory.

These results initiate a pathway for solving the strong interaction bound-state problems from first principles.