Ward-Takahashi identity, soft photon theorem and the magnetic moment of the $\Delta$ resonance

Abstract

Starting from the Ward-Takahashi identity for the radiative $\pi N$ scattering amplitude a generalization of the soft photon theorem approach is obtained for an arbitrary energy of an emitted photon. The external particle radiation part of the $\pi N\to {\gamma }^{\prime }{\pi }^{\prime }{N}^{\prime }$ amplitude is analytically reduced to the double $\Delta$ exchange amplitude with the $\Delta \to {\gamma }^{\prime }{\Delta }^{\prime }$ vertex function. We have shown, that the double $\Delta$ exchange amplitudes with internal $\Delta$ radiation is connected by current conservation with the corresponding part of the external particle radiation terms. Moreover according to the current conservation the internal and external particle radiation terms with the $\Delta -{\gamma }^{\prime }{\Delta }^{\prime }$ vertex have a opposite sign i.e. they must cancel each other. Therefore we have a screening of the internal double $\Delta$ exchange diagram with the $\Delta -{\gamma }^{\prime }{\Delta }^{\prime }$ vertex by the external particle radiation. This enables us to obtain a model independent estimation of the dipole magnetic moment of ${\Delta }^{+}$ and ${\Delta }^{++}$ resonances ${\mu }_{\Delta }$ through the anomalous magnetic moment of the proton ${\mu }_p$ as ${\mu }_{{\Delta }^{+}}=\frac{M_{\Delta }}{m_p}{\mu }_p$ and ${\mu }_{{\Delta }^{++}}=\frac{3}{2}{\mu }_{{\Delta }^{+}}$ in agreement with the values obtained from the fit of the experimental cross section of the ${\pi }^{+}p\to {\gamma }^{\prime }{\pi }^{+}p$ reaction.

Comments: 28 pages, 2 figures, 1 table