Mesonic decay constant and mass ratios and the conformal window

Two particular ratios related to mesons are proposed for the study of the conformal window in $SU(3)$ gauge theory and fundamental fermions. Lattice and other studies indicate that the lower end, $N_f^{\ast }$, is at around 7 - 13 flavors which is a wide range without a clear consensus. Here we propose the decay constant to mass ratios of mesons, $f_{PS,V}\mathrm{/}{m}_V$, as a proxy since below the conformal window lattice studies have shown that they are largely $N_f$-independent while at the upper end of the conformal window they are vanishing. The drop from the non-zero constant value to zero at $N_f=16.5$ might be indicative of $N_f^{\ast }$. We compute $f_V\mathrm{/}{m}_V$ to N${}^3$LO and $f_{PS}\mathrm{/}{m}_V$ to NNLO order in (p)NRQCD. The results are unambiguously reliable just below $N_f=16.5$, hence the results are expanded á la Banks-Zaks in $\epsilon =16.5-N_f$. The convergence properties of the series and matching with the non-perturbative infinite volume, continuum and chiral extrapolated lattice results at $N_f=10$ suggest that the perturbative results might be reliable down to $N_f=12$. A sudden drop is observed at $N_f=12$ and $N_f=13$ in $f_V\mathrm{/}{m}_V$ and $f_{PS}\mathrm{/}{m}_V$, respectively.