  We propose a strategy to access the $q\bar{q}$ component of the $\rho$
resonance in lattice QCD. Through a mixed action formalism (overlap valence on
domain wall sea), the energy of the $q\bar{q}$ component is derived at
different valence quark masses, and shows a linear dependence on $m_\pi^2$. The
slope is determined to be $c_1=0.505(3)\,{\rm GeV}^{-1}$, from which the
valence $\pi \rho$ sigma term is extracted to be $\sigma_{\pi \rho}^{(\rm
val)}=9.82(6)$ MeV using the Feynman-Hellman theorem. At the physical pion
mass, the mass of the $q\bar{q}$ component is interpolated to be
$m_\rho=775.9\pm 6.0\pm 1.8$ MeV, which is close to the $\rho$ resonance mass.
We also obtain the leptonic decay constant of the $q\bar{q}$ component to be
$f_{\rho^-}=208.5\pm 5.5\pm 0.9$ MeV, which can be compared with the
experimental value $f_{\rho}^{\rm exp}\approx 221$ MeV through the relation
$f_{\rho}^{\rm exp}=\sqrt{Z_\rho}f_{\rho^\pm} $ with $Z_\rho\approx 1.13$ being
the on-shell wavefunction renormalization of $\rho$ owing to the $\rho-\pi$
interaction. We emphasize that $m_\rho$ and $f_\rho$ of the $q\bar{q}$
component, which are obtained for the first time from QCD, can be taken as the
input parameters of $\rho$ in effective field theory studies where $\rho$ acts
as a fundamental degree of freedom.
