Pion and quark electromagnetic self-masses in the Nambu-Jona-Lasinio model.

Abstract

The mass difference between the charged and the neutral pions that arises from chiral symmetry-breaking effects on these particles considered as Goldstone modes of the two-flavor Nambu--Jona-Lasinio model is studied in the Hartree plus random phase approximation. Most of the pion mass difference originates from the pions' electromagnetic self-energy that is due to the electromagnetic interactions of the constituent quarks that make up the pion, with only a very small additional ``mechanical'' contribution, proportional to the square of their mass difference, coming from the current quark masses. We discuss the gauge invariance of this self-energy and the residual chiral symmetry of the theory in the presence of electromagnetic fields in the limit of vanishing current quark masses, and prove the Goldberger-Treiman relation between O(${\mathrm{\ensuremath{\alpha}}}_{\mathrm{EM}}$)-corrected quantities in this limit for the neutral pion. We elucidate the special role that the electromagnetic contributions to the constituent quark mass play in keeping the neutral pion massless in the chiral limit in the presence of its own internal electromagnetic fields in agreement with Dashen's theorem. The lowest order nonchiral electromagnetic corrections to the mass shifts are repulsive and contain ``chiral logarithms'' of the meson mass ${\mathit{m}}_{\mathrm{\ensuremath{\pi}}}$ of the form \ensuremath{\alpha}${\mathit{m}}_{\mathrm{\ensuremath{\pi}}}^{2}$ln${\mathit{m}}_{\mathrm{\ensuremath{\pi}}}^{2}$ in the charged channel in accord with the Langacker-Pagels general analysis. By contrast, the neutral pion's mass is only shifted by terms of O(\ensuremath{\alpha}${\mathit{m}}_{\mathrm{\ensuremath{\pi}}}^{2}$), resulting in a very weak nonchiral breaking of Dashen's theorem. We evaluate the corrections numerically and show that the relative sizes of the electromagnetic chiral, the nonchiral logarithm, and the remaining O(\ensuremath{\alpha}${\mathit{m}}_{\mathrm{\ensuremath{\pi}}}^{2}$) plus mechanical contributions make up approximately 90%, 9%, and 1% of the observed pion mass splitting, respectively.