Three-Quark Bethe-Salpeter vertex function under pairwise gluon-exchange-like interaction: application to n-p mass difference

A qqq BSE formalism based on an input 4-fermion Lagrangian of 'current' u, d quarks interacting pairwise via a gluon-exchange-like propagator in its nonperturbative regime, is employed for the construction of a relativistic qqq-wave function under the Covariant Instantaneity Ansatz (CIA). The chiral invariance of the input Lagrangian is automatically ensured by the vector character of the gluonic propagator, while the 'constituent' masses are the low momentum limits of the dynamical mass function m(p) generated by the standard mechanism of DBχS in the solution of the Schwinger Dyson Equation (SDE). The CIA gives an exact reduction of the BSE to a 3D form which is appropriate for baryon spectroscopy, while the reconstructed 4D form identifies the hadron quark vertex function as the key ingredient for evaluating transition amplitudes via quark-loop integrals. In this paper the second stage of this 'two-tier' BSE formalism is extended from the 4D qq̅-meson to the 4D qqq-baryon vertex reconstruction through a reversal of steps offered by the CIA structure. As a first application of this 4D qqq wave function, we evaluate the quark loop integrals for the neutron (n) - proton (p) mass difference which receives contributions from two sources : i) the strong SU(2) effect arising from the u-d mass difference (4 MeV); ii) the e.m. effect of the respective quark charges. The resultant n-p difference works out at 1.28 MeV (vs. 1.29 expt), with two free parameters C0, ω0 characterizing the infrared structure of the gluonic, which have been precalibrated from a common fit to qq̅ and qqq spectra as well as several other observable quark loop integrals. A formal derivation, based on Green's function techniques for 3 spinless quarks, of the CIA structure of the 4D qqq-baryon vertex function as employed in the text, is given for completeness in Appendix B