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

TL;DR

This paper develops a new formulation of current conservation in pion-nucleon bremsstrahlung, revealing a cancellation mechanism that leads to model-independent relations for the magnetic moments of the $\Delta$ resonances, consistent with experimental data.

Contribution

It introduces a novel current conservation approach that explains the screening effect and derives model-independent relations for $\Delta$ magnetic moments without quark degrees of freedom.

Findings

Double $\Delta$ exchange cancels with longitudinal external radiation.

Derived relations connect $\Delta$ magnetic moments to proton's magnetic moment.

Results agree with experimental cross section data.

Abstract

The pion-nucleon bremsstrahlung $\pi+N\Longrightarrow\gamma'+\pi'+N'$ 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 $\Delta-\gamma' \Delta'$ 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 $\mu_{\Delta^+}={{M_{\Delta}}\over {m_p}} \mu_p$ and $\mu_{\Delta^{++}}={3\over 2}\mu_{\Delta^+}$ in agreement with the values extracted from the fit for the experimental cross section of the $\pi^+ p\to\gamma'\pi^+ p$ reaction.