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

Abstract

The pion-nucleon bremsstrahlung $\pi +N⟹{\gamma }^{\prime }+{\pi }^{\prime }+N^{\prime }$ is studied in a new form of current conservation. According to this condition, the internal and external particle radiation parts of the $\pi N$ 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 $\pi N$ bremsstrahlung. In particular, it is shown that the double $\Delta$ exchange diagram with the $\Delta -{\gamma }^{\prime }{\Delta }^{\prime }$ 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 ${\Delta }^{+}$ and ${\Delta }^{++}$ resonances and the anomalous magnetic moment of the proton ${\mu }_p$ is obtained, where ${\mu }_{\Delta }$ is expressed by ${\mu }_p$ as ${\mu }_{{\Delta }^{+}}=\frac{M_{\Delta }}{m_p}{\mu }_p$ and ${\mu }_{{\Delta }^{++}}=\frac{3}{2}{\mu }_{{\Delta }^{+}}$ in agreement with the values extracted from the fit for the experimental cross section of the ${\pi }^{+}p\to {\gamma }^{\prime }{\pi }^{+}p$ reaction.

Comments: 37 pages, 2 figures and 1 table