Real Part of the Scattering Amplitude and the Behavior of the Total Cross Section at High Energies

Certain theorems first pointed out by Meiman are used to include the information contained in analyticity in energy and crossing symmetry in the derivation of upper bounds for the forward scattering amplitude. We have obtained theorems relating the asymptotic behavior of the ratio of the real part to the imaginary part of the forward amplitude on the one hand, to that of the total cross section on the other. Except for the special case where $\frac{\mathrm{Re}f}{\mathrm{Im}f}\sim \pi {\left(\ln E\right)}^{-1\mathrm{}}$ as the incident laboratory energy $E\to \infty$, we find that the Froissart bound can be improved. We also find that the Greenberg-Low bound which follows from axiomatic field theory can always be improved by a fractional power of $E$ under the physical assumption that the amplitude does not become purely real as $E\to \infty$. As a result of our theorems we find that the asymptotic behavior $\mathrm{Re}f\sim \mathrm{cE}$ is not allowed. This sheds light on the question of the "elementary" nature of vector mesons. Finally, we give an inequality which, in principle, could lead to an experimental check of analyticity and crossing with data obtained only from a finite range of energy.