Baryon self-energy with qqq Bethe–Salpeter dynamics in the nonperturbative QCD regime: n-p mass difference

Abstract

A qqq BSE formalism based on DBχS of an input four-fermion Lagrangian of "current" u, d quarks interacting pairwise via gluon-exchange-propagator in its nonperturbative regime, is employed for the calculation of baryon self-energy via quark-loop integrals. To that end the baryon-qqq vertex function is derived under Covariant Instantaneity Ansatz (CIA), using Green function techniques. This is a three-body extension of an earlier qq̅ (two-body) result on the exact 3D–4D interconnection for the respective BS wave functions under 3D kernel support. The nonperturbative QCD feature of this approach (vis-a-vis chiral perturbation theory - see text for comparison) is preserved through the gluon exchange propagator in the infrared regime (characterized by two parameters C₀, ω₀), precalibrated to both qq̅ and qqq spectra plus other observables, together with a DBχ S mechanism to generate the dynamical mass m(p). The quark-loop integrals for the neutron (n) - proton (p) mass difference receive contributions from two sources: (i) the strong SU(2) effect arising from the d-u mass difference (4 MeV); (ii) the e.m. effect of the respective quark charges. The resultant n-p difference comes dominantly from the d-u effect (+1.71 MeV), which is mildly offset by e.m. effect (-0.44), subject to gauge corrections. For an estimate of the latter a general method for QED gauge corrections to an arbitrary momentum dependent vertex function is outlined (in App. C), and a calculation made for the (two-body) kaon as a test case, indicates an increase by 0.612 MeV in the earlier kaon e.m. value (1.032 MeV). This result is taken as a rough indication of the percentage gauge correction expected for the n-p case. A critical comparison is given with the results from QCD sum rules.