2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/187598We give a general $SU(2)_L\times SU(2)_R$ $\times U(1)_{EM}$ sigma model with external sources, dynamical breaking and spontaneous vacuum symmetry breaking, and present the general formulation of the model. It is found that $σ$ and $π^0$ without electric charges have electromagnetic interaction effects coming from their internal structure. A general Lorentz transformation relative to external sources $J_{gauge}$ $=(J_{A_μ},J_{A_μ^κ})$ is derived, using the general Lorentz transformation and the four-dimensional current of nuclear matter of the ground state with $J_{gauge}$ = 0, we give the four-dimensional general relations between the different currents of nuclear matter systems with $J_{gauge}\neq 0$ and those with $J_{gauge}=0$. The relation of the density's coupling with external magnetic field is derived, which conforms well to dense nuclear matter in a strong magnetic field. We show different condensed effects in strong interaction about fermions and antifermions, and give the concrete scalar and pseudoscalar condensed expressions of $σ_0$ and $π_0$ bosons. About different dynamical breaking and spontaneous vacuum symmetry breaking, the concrete expressions of different mass spectra are obtained in field theory. This paper acquires the running spontaneous vacuum breaking value $σ_0^{\prime},$ and obtains the spontaneous vacuum breaking in terms of the running $σ_0^{\prime}$, which make nucleon, $σ$ and $π$ particles gain effective masses. We achieve both the effect of external sources and nonvanishing value of the condensed scalar and pseudoscalar paticles. It is deduced that the masses of nucleons, $σ$ and $π$ generally depend on different external sources.12 pagesHigh Energy Physics - Theorytext