Baryonic form factors of the pion and kaon in a chiral quark model

The baryonic form factor of charged pions is studied in detail in the Nambu--Jona-Lasinio model with constituent quarks, where the spontaneously broken chiral symmetry is the key dynamical ingredient guaranteeing the would-be Goldstone boson nature of the pseudoscalar mesons octet. In general, this form factor arises when the isospin symmetry is broken, which is the case if the $u$ and $d$ quark masses are split, as in the real world, or if the electromagnetic effects were taken into account. We obtain estimates for this basic property of the pion resulting from the quark mass splitting for a range of model parameters, and importantly, for different pion masses, going up to the values used in lattice QCD. We find very stable model results, with the mean square radius of $\pi^+$ in the range $(0.05-0.07~{\rm fm})^2$. From charge conjugation, the baryonic form factor of $\pi^+$ and $\pi^-$ are equal and opposite. We also obtain the transverse-coordinate, relativistically invariant baryonic density of the charged pion. In $\pi^+$, the inner region carries a negative, and the outside -- a positive baryon number density, both cancelling to zero, as obviously the pion carries no net baryon charge. We also carry out an analogous analysis for the kaon, where the effect is much larger due to the sizable $s$ and $u/d$ quark mass splitting. We discuss the prospects of lattice QCD measurements of the baryonic form factors of charged pions and kaons.