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We study a chiral quark model of $\pi$, $\sigma$, $\rho$, and $a_1$ mesons inthe large-$N_c$ limit. We show that the quadratic $a_1-\pi$ mixing can be setto zero at zero momentum, thus protecting the $g_{\sigma\pi\pi}$ and$g_{\rho\pi\pi}$ couplings from being contaminated by the correspondingvertices with one or both pions replaced by the axial-vector meson. We furtherrequire the chiral-quark Lagrangian to feature an approximate classical scaleinvariance, the latter being only broken by dimension-two mass terms, and thatthe longitudinal vector meson scattering amplitudes grow at most like $s$,where $\sqrt{s}$ is the center-of-mass energy. This allows us to accuratelypredict the correct value of $g_{\sigma\pi\pi}$, $g_{\rho\pi\pi}$, and the$a_1$ decay constant $f_{a_1}$. We further show that strongly-coupled theorieswith near-conformal dynamics are expected to feature an approximate custodialchiral symmetry, with interesting phenomenological consequences.

The unobservable axial-pion mixing in QCD and near-conformal dynamics