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Current conservation, screening and the magnetic moment of the $Δ$ resonance. -- 1. Formulation without quark degrees of freedom

The pion-nucleon bremsstrahlung $π+N\Longrightarrowγ'+π'+N'$ is studied in a new form of current conservation. According to this condition, the internal and external particle radiation parts of the $πN$ radiation amplitude have opposite signs, i.e., they contain terms which must cancel each other. Therefore, one has a screening of the internal and external particle radiation in the $πN$ bremsstrahlung. In particular, it is shown that the double $Δ$ exchange diagram with the $Δ-γ' Δ'$ vertex cancel against the appropriate longitudinal part of the external particle radiation diagrams. Consequently, a model independent relation between the magnetic dipole moments of the $Δ^+$ and $Δ^{++}$ resonances and the anomalous magnetic moment of the proton $μ_p$ is obtained, where $μ_Δ$ is expressed by $μ_p$ as $μ_{Δ^+}={{M_Δ}\over {m_p}} μ_p$ and $μ_{Δ^{++}}={3\over 2}μ_{Δ^+}$ in agreement with the values extracted from the fit for the experimental cross section of the $π^+ p\toγ'π^+ p$ reaction.