In-Medium Chiral Perturbation Theory and Pion Weak Decay in the Presence of Background Matter

Two-point functions related to the pion weak decay constant f_\pi are calculated from the generating functional of chiral perturbation theory in the mean-field approximation and the heavy-baryon limit. The aim is to demonstrate that Lorentz invariance is violated in the presence of background matter. This fact manifests itself in the splitting of both f_\pi and the pion mass into uncorrelated time- and spacelike parts. We emphasize the different in-medium renormalizations of the correlation functions, show the inequivalence between the in-medium values of f_\pi deduced from Walecka-type models, on the one hand, and QCD sum rules, on the other hand, and elaborate on the importance for some nuclear physics observables.