Validity of Scaling Relations between Nucleon Form Factors

We prove, within the framework of analytic $S$-matrix theory and the physical mass spectrum, that the following scaling relations cannot be rigorously true for all $t$: (i) ${G_M}^V\mathrm{}\left(t\right)\mathrm{}\propto {G_E}^V\mathrm{}\left(t\right)\mathrm{}$, (ii) ${G_M}^V\mathrm{}\left(t\right)=R_1\mathrm{}\left(t\right)\mathrm{}{G_E}^V\mathrm{}\left(t\right)+R_2\mathrm{}\left(t\right)\mathrm{}$, where $R_{1,2}\mathrm{}\left(t\right)\mathrm{}$ are rational functions of $t$, (iii) $F_{\pi }\mathrm{}\left(t\right)\mathrm{}\propto {G_E}^V\mathrm{}\left(t\right)\mathrm{}$, and (iv) $F_{\pi }\mathrm{}\left(t\right)\mathrm{}\propto {G_M}^V\mathrm{}\left(t\right)\mathrm{}$.