Calculating the pion decay constant from \alpha_s(M_Z)
  We revisit the analysis of the improved ladder Schwinger-Dyson (SD) equation for the dynamical chiral symmetry breaking in QCD with emphasizing the importance of the scale ambiguity. Previous calculation done so far naively used one-loop MSbar coupling in the improved ladder SD equation without examining the scale ambiguity. As a result, the calculated pion decay constant f_\pi was less than a half of its experimental value f_\pi=92.4MeV once the QCD scale is fixed from the high energy coupling \alpha_s(M_Z). In order to settle the ambiguity in a proper manner, we adopt here in the present paper the next-to-leading-order effective coupling instead of a naive use of the MSbar coupling. The pion decay constant f_\pi is then calculated from high energy QCD coupling strength \alpha_s(M_Z)=0.1172 \pm 0.0020. Within the Higashijima-Miransky approximation, we obtain f_\pi=85--106MeV depending on the value of \alpha_s(M_Z) which agrees well with the experimentally observed value f_\pi=92.4MeV. The validity of the improved ladder SD equation is therefore ascertained more firmly than considered before. 
Michio Hashimoto, Masaharu Tanabashi