Chiral Lagrangian for excited pions

We construct a chiral Lagrangian containing, in addition to the usual pion field $\left(\pi \right)$, also its first radial excitation $\left({\pi }^{\prime }\right)$. The Lagrangian is derived in the large-$N_c$ limit from a Nambu–Jona-Lasinio (NJL) quark model with separable nonlocal interactions, with form factors corresponding to three-dimensional ground- and excited-state wave functions. Chiral symmetry breaking is governed by the NJL gap equation. The effective Lagrangian for $\pi$ and ${\pi }^{\prime }$ mesons shows the decoupling of the Goldstone pion and the vanishing of the ${\pi }^{\prime }$ leptonic decay constant $f_{{\pi }^{\prime }}$ in the chiral limit, as required by axial-vector current conservation. We derive the excited-state contribution to the axial-vector current of the model using Noether’s theorem. For a finite pion mass and ${\pi }^{\prime }$ masses in the range of 750–1300 MeV, $f_{{\pi }^{\prime }}{/f}_{\pi }$ is found to be of the order of 1%.