Renormalizability of effective pionic Lagrangians in the nuclear medium

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Abstract
We show the renormalizability of a Lagrangian of nucleons and pions interacting via a pseudoscalar coupling in the presence of a medium. We extend the proof to the effective Lagrangian obtained from the previous one through the functional integration over the fermionic fields. Notably in the latter case the renormalization takes place within each order of the corresponding loop expansion. This result is of significance in view of the connection between the loop expansion and the conventional many-body approximation methods.