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

We study a chiral quark model of $\pi$, $\sigma$, $\rho$, and $a_1$ mesons in the large-$N_c$ limit. We show that the quadratic $a_1-\pi$ mixing can be set to zero at zero momentum, thus protecting the $g_{\sigma\pi\pi}$ and $g_{\rho\pi\pi}$ couplings from being contaminated by the corresponding vertices with one or both pions replaced by the axial-vector meson. We further require the chiral-quark Lagrangian to feature an approximate classical scale invariance, the latter being only broken by dimension-two mass terms, and that the longitudinal vector meson scattering amplitudes grow at most like $s$, where $\sqrt{s}$ is the center-of-mass energy. This allows us to accurately predict 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 theories with near-conformal dynamics are expected to feature an approximate custodial chiral symmetry, with interesting phenomenological consequences.