One loop corrections to quantum hadrodynamics with vector mesons

The renormalized elastic $\pi\pi$ scattering amplitude to one loop is calculated in the chiral limit in the $\sigma$ model and in a Quantum Hadrodynamic model (QHD-III) with vector mesons. It is argued that QHD-III reduces to the linear $\sigma$ model in the limit that the vector meson masses become large. The pion decay constant is also calculated to 1-loop in the $\sigma$ model, and at tree level in QHD-III; it is shown that the coefficient of the tree level term in the scattering amplitude equals $F_\pi^{-2}$. The 1-loop correction of $F_\pi$ in QHD-III violates strong isospin current conservation. Thus,it is concluded that QHD-III can, at best, only describe the strongly interacting nuclear sector.