Chiral Lagrangian for excited pions

Abstract

We construct a chiral Lagrangian containing, in addition to the usual pion field $(\ensuremath{\pi})$, also its first radial excitation $({\ensuremath{\pi}}^{\ensuremath{'}})$. 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 $\ensuremath{\pi}$ and ${\ensuremath{\pi}}^{\ensuremath{'}}$ mesons shows the decoupling of the Goldstone pion and the vanishing of the ${\ensuremath{\pi}}^{\ensuremath{'}}$ leptonic decay constant ${f}_{{\ensuremath{\pi}}^{\ensuremath{'}}}$ 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 ${\ensuremath{\pi}}^{\ensuremath{'}}$ masses in the range of 750--1300 MeV, ${f}_{{\ensuremath{\pi}}^{\ensuremath{'}}}{/f}_{\ensuremath{\pi}}$ is found to be of the order of 1%.