Asymptotic of grazing collisions for the spatially homogeneous Boltzmann equation for soft and Coulomb potentials

We give an explicit bound for the Wasserstein distance with quadratic cost between the solutions of the Boltzmann and Landau equations in the case of soft and Coulomb potentials. This gives an explicit rate of convergence for the grazing collisions limit. Our result is local in time for very soft and Coulomb potentials and global in time for moderately soft potentials.