f/m and f/m ratios and the conformal window

The f(rho)/m(rho) ratio is calculated at (NLO)-L-3 order within perturbative (p)non-relativistic quantum chromodynamics (NRQCD) with N-f flavors of mass degenerate fermions. The massless limit of the ratio is expanded a la Banks-Zaks in epsilon = 16.5 - N-f leading to reliable predictions close to the upper end of the conformal window. The comparison of the next-to-next-to leading order (NNLO) and (NLO)-L-3 results indicate that the Banks-Zaks expansion may be reliable down to twelve flavors. Previous lattice calculations combined with the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin (KSRF) relations provide us with the same ratio for the range 2 <= N-f <= 10. Assuming a monotonous dependence on N-f leads to an estimate for the lower end of the conformal window, N-f* similar or equal to 12, by matching the nonperturbative and our perturbative results. In any case an abrupt change is observed in f(rho)/m(rho) at twelve flavors. As a cross-check we also consider the f(pi)/m(rho) ratio for which lattice results are also available. The perturbative calculation at present is only at the NNLO level which is insufficient for a reliable and robust matching between the low N-f and high N-f regions. Nonetheless, using the relative size of the (NLO)-L-3 correction of f(rho)/m(rho) for estimating the same for f(pi)/m(rho) leads to the estimate N-f* similar or equal to 13.