Rigorous phase-modulus correlations for forward scattering amplitude at high energies

Starting from the basic results of axiomatic field theory a series of new asymptotic theorems for the symmetric $F_S\mathrm{}\left(E\right)\mathrm{}$ and the antisymmetric $F_A\mathrm{}\left(E\right)\mathrm{}$ forward scattering amplitudes are derived, $E$ being the laboratory energy. The spins are neglected in the present paper. Assuming that the phase of $F_S$ or $F_A$ is asymptotically bounded within a certain range of values the corresponding bounds on $\left|F_S\right| or \left|F_A\right|$, respectively, are obtained. Further, we show that certain conditions on the ratio $\frac{\mathrm{Re}{F}_S\mathrm{}\left(E\right)\mathrm{}}{\mathrm{Im}{F}_S\mathrm{}\left(E\right)\mathrm{}}$ and on the signs of $\mathrm{Re}{F}_S\mathrm{}\left(E\right)\mathrm{}$ and $\mathrm{Im}{F}_S\mathrm{}\left(E\right)\mathrm{}$ for large enough energies can be formulated as criteria for the asymptotic behavior of the symmetric total cross section ${\sigma }_S\mathrm{}\left(E\right)\mathrm{}$. Analogous results are obtained for the antisymmetric forward scattering amplitude. Finally, useful necessary and sufficient conditions for the asymptotic rise or boundedness of ${\sigma }_S\mathrm{}\left(E\right)\mathrm{}$ and ${\sigma }_A\mathrm{}\left(E\right)\mathrm{}$ are derived. The results obtained are discussed in light of existing high-energy models and high-energy experiments.