PION AND QUARK ELECTROMAGNETIC SELF-MASSES IN THE NAMBU-JONA-LASINIO MODEL

The mass difference between the charged and the neutral pions that ari ses from chiral symmetry breaking effects on these particles considere d as Goldstone modes of the two-flavor Nambu-Jona-Lasinio model is stu died 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 qu arks that make up the pion, with only a very small additional ''mechan ical'' contribution, proportional to the square of their mass differen ce, coming from the current quark masses. We discuss the gauge invaria nce of this self-energy and the residual chiral symmetry of the theory in the presence of electromagnetic fields in the limit of vanishing c urrent quark masses, and prove the Goldberger-Treiman relation between O(alpha(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 massles s in the chiral Limit in the presence of its own internal electromagne tic fields in agreement with Dashen's theorem. The lowest order nonchi ral electromagnetic corrections to the mass shifts are repulsive and c ontain ''chiral logarithms'' of the meson mass m(pi) of the form alpha m(pi)(2) In m(pi)(2) in the charged channel in accord with the Langac ker-Pagels general analysis. By contrast, the neutral pion's mass is o nly shifted by terms of O(alpha m(pi)(2)), resulting in a very weak no nchiral breaking of Dashen's theorem. We evaluate the corrections nume rically and show that the relative sizes of the electromagnetic chiral , the nonchiral logarithm, and the remaining O((alpha m(pi)(2)) plus m echanical contributions make up approximately 90%, 9%, and 1% of the o bserved pion mass splitting, respectively.