Asymptotic bounds on the absorptive parts of the elastic scattering amplitudes

We establish exact bounds on the absorptive parts A(s,t) of an elastic scattering amplitude (spinless case) and evaluate them for positive t values lying within the Lehmann-Martin ellipse [the major axis = $2(1+\frac{t_0}{2k^2})$]. These bounds are used to derive a number of asymptotic results; e.g., (i) the "diffraction-peak width" W is larger than $W_{min}~\frac{4t_0}{{(1+\lambda )}^2}\times (1-\frac{1}{2}\sigma ){\left(\ln s\right)}^2$ (for s→∞); (ii) the leading Regge trajectory for t₀>t>0 lies below $[1+{\left(\frac{t}{t_0}\right)}^{\frac{1}{2}}]-\lambda [1-{\left(\frac{t}{t_0}\right)}^{\frac{1}{2}}]$; (iii) there are no complex zeros of A(s,t) for $\left|t\right|<\frac{4t_0}{{(1+\lambda )}^2}{e}^2{\left(\ln s\right)}^2$ (for s→∞) and no real zeros for t₀>t>-W_min, where $\lambda ={lim}_{s\to \infty }\frac{{\ln \sigma }_{tot}\left(s\right)}{\ln s}$ and $\sigma ={lim}_{s\to \infty }{t}_0\frac{{\sigma }_{tot}\left(s\right)}{4\pi {\left(\ln s\right)}^2}$.