Calculation of the pion decay constant in the framework of the Bethe-Salpeter equation

We elaborate on the calculation of the pion decay constant $f_{\pi }$ in the context of QCD and in the limit of purely spontaneous chiral-symmetry breaking. We use the expressions given by relativistic boundstate theory which require the simultaneous solution of the Schwinger-Dyson equation and of the Bethe-Salpeter equation developed to first order in the pion momentum. The equations are solved by numerical iteration in the Landau gauge and in the ladder approximation, but without any further approximations. We use a model for the gluon propagator with ultraviolet behavior determined by renormalization-group considerations and discuss its influence on the determination of $f_{\pi }$. We analyze approximations that have been used in the literature for the calculation of $f_{\pi }$ and discuss their limitations.