The flavor dependence of $m_\varrho / f_\pi$

We calculate the $m_\varrho / f_\pi$ ratio in the chiral and continuum limit for $SU(3)$ gauge theory coupled to $N_f = 2,3,4,5,6$ fermions in the fundamental representation. Keeping all systematic effects under full control we find no statistically significant $N_f$-dependence; $m_\varrho / f_\pi = 7.95(15)$. Assuming the KSRF-relations we conclude that 3 other low energy quantities related to the vector meson are also $N_f$-independent within errors including the $\varrho\pi\pi$ coupling $g_{\varrho\pi\pi}$. If the model is thought of as a strong dynamics inspired composite Higgs model our results indicate that the experimentally most easily accessible new composite particle, the vector meson, and its properties may be robust and independent of the fermion content of the model as long as the gauge group is $SU(3)$, provided $N_f$-independence extends all the way to the conformal window.