Antiparticle Contribution in the Cross Ladder Diagram for Bethe-Salpeter Equation in the Light-Front

We construct the homogeneous integral equation for the vertex of the bound state in the light front with the kernel approximated to order g ⁴. We will truncate the hierarchical equations from Green functions to construct dynamical equations for the two boson bound state exchanging interacting intermediate σ bosons and including pair creation process contributing to the crossed ladder diagram.