Mesonic cloud contribution to the nucleon and $\Delta$ masses

Pion-nucleon elastic scattering in the dominant $P_{33}$ channel is examined in the model in which the interaction is of the form $\pi +N\leftrightarrow N,\hspace{0.5em}\Delta \mathrm{}\left(1232\right).$ New expressions are found for the elastic pion-nucleon scattering amplitude which differ from existing formula both in the kinematics and in the treatment of the renormalization of the nucleon mass and coupling constant. Fitting the model to the phase shifts in the $P_{33}$ channel does not uniquely fix the parameters of the model. The cutoff for the pion-nucleon form factor is found to lie in the range $\beta =750\mathrm{}\pm 350\hspace{0.5em}\mathrm{MeV}/c.\mathrm{}$ The masses of the nucleon and the $\Delta$ which would arise if there were no coupling to mesons are found to be $m_{{}_N}^{\mathrm{}\left(0\right)\mathrm{}}=1200\mathrm{}\pm 200\hspace{0.5em}\mathrm{MeV}$ and $m_{\Delta }^{\mathrm{}\left(0\right)\mathrm{}}=1500\mathrm{}\pm 200\hspace{0.5em}\mathrm{MeV}.$ The difference in these bare masses, a quantity which would be accounted for by a residual gluon interaction, is found to be $\delta m^{\mathrm{}\left(0\right)\mathrm{}}=350\mathrm{}\pm 100\hspace{0.5em}\mathrm{MeV}.$