$P_{33}\mathrm{}\left(1232\right)\mathrm{}$ resonance contribution to the amplitudes $M_{1+}^{3/2}{,E}_{1+}^{3/2}{,S}_{1+}^{3/2}$ from an analysis of the ${p(e,e}^{\prime }p){\pi }^0$ data at $Q^2=2.8,$ 3.2, and $4(\mathrm{GeV}{/c)}^2$ within the dispersion relation approach

Within the fixed-$t$ dispersion relation approach we have analyzed the TJNAF and DESY data on the exclusive ${p(e,e}^{\prime }p){\pi }^0$ reaction in order to find the $P_{33}\mathrm{}\left(1232\right)\mathrm{}$ resonance contribution to the multipole amplitudes $M_{1+}^{3/2}{,E}_{1+}^{3/2}{,S}_{1+}^{3/2}.$ As an input for the resonance and nonresonance contributions to these amplitudes the earlier obtained solutions of the integral equations which follow from dispersion relations are used. The obtained values of the ratio $E2/M1\mathrm{}$ for the ${\gamma }^{*}\overrightarrow{N}{P}_{33}\mathrm{}\left(1232\right)\mathrm{}$ transition are $0.039\pm 0.029,0.121\pm 0.032,$ $0.04\pm 0.031$ for $Q^2=2.8,$ 3.2, and $4(\mathrm{GeV}{/c)}^2,$ respectively. The comparison with the data at low $Q^2$ shows that there is no evidence for the presence of the visible perturbative QCD (PQCD) contribution into the transition $\gamma \overrightarrow{N}{P}_{33}\mathrm{}\left(1232\right)\mathrm{}$ at $Q^2=3\mathrm{}–4{\mathrm{GeV}}^2.$ The ratio $S_{1+}^{3/2}{/M}_{1+}^{3/2}$ for the resonance parts of multipoles is $-0.049\pm 0.029,$ $-0.099\pm 0.041,$ $-0.085\pm 0.021$ for $Q^2=2.8,$ 3.2, and $4(\mathrm{GeV}{/c)}^2,$ respectively. Our results for the transverse form factor $G_T{(Q}^2)$ of the ${\gamma }^{*}\overrightarrow{N}{P}_{33}\mathrm{}\left(1232\right)\mathrm{}$ transition are lower than the values obtained from the inclusive data. With increasing $Q^2,$ $Q^4{G}_T{(Q}^2)$ decreases, so there is no evidence for the presence of the PQCD contribution here too.