Angular Dependence of Scattering at High Energies

Experimental evidence indicates that for nonforward (and nonbackward) scattering angles $\theta$, differential scattering cross sections fall off exponentially with increasing barycentric energy $\surd s$; namely, $\frac{d\sigma }{d\Omega }\sim {}_{s\to \infty }{}^{}{}_{}{}^{}e_{}^{-\phi (s,cos\theta )}{}_{}{}^{}$, where $\phi (s, cos\theta )$ is a positive quantity increasing as some power of $s$. From assumptions of boundedness and analyticity in $cos\theta$, we obtain certain constraints on the $cos\theta$ dependence of the exponent function $\phi (s, cos\theta )$.