fϱ/mϱ and fπ/m

The ${f}_{\ensuremath{\varrho}}/{m}_{\ensuremath{\varrho}}$ ratio is calculated at ${\mathrm{N}}^{3}\mathrm{LO}$ 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 $ϵ=16.5\ensuremath{-}{N}_{f}$ 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 ${\mathrm{N}}^{3}\mathrm{LO}$ 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 $2\ensuremath{\le}{N}_{f}\ensuremath{\le}10$. Assuming a monotonous dependence on ${N}_{f}$ leads to an estimate for the lower end of the conformal window, ${N}_{f}^{*}\ensuremath{\simeq}12$, by matching the nonperturbative and our perturbative results. In any case an abrupt change is observed in ${f}_{\ensuremath{\varrho}}/{m}_{\ensuremath{\varrho}}$ at twelve flavors. As a cross-check we also consider the ${f}_{\ensuremath{\pi}}/{m}_{\ensuremath{\varrho}}$ 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 ${\mathrm{N}}^{3}\mathrm{LO}$ correction of ${f}_{\ensuremath{\varrho}}/{m}_{\ensuremath{\varrho}}$ for estimating the same for ${f}_{\ensuremath{\pi}}/{m}_{\ensuremath{\varrho}}$ leads to the estimate ${N}_{f}^{*}\ensuremath{\simeq}13$.