Effects of diffractive channels on hadronic cross sections

We consider the effects of diffractive channels on total, elastic, and diffractive cross sections in a model in which an elementary particle scatters on a composite system of $A$ constituents. We find that the effect of diffractive channels is to lower $\frac{{\ensuremath{\sigma}}_{\mathrm{el}}}{{\ensuremath{\sigma}}_{T}}$ below the value it would have were no diffractive channels present. We find both $\frac{{\ensuremath{\sigma}}_{\mathrm{el}}}{{\ensuremath{\sigma}}_{T}}$ and $\frac{({\ensuremath{\sigma}}_{\mathrm{el}}+{\ensuremath{\sigma}}_{\mathrm{dif}})}{{\ensuremath{\sigma}}_{T}\ensuremath{\le}\frac{1}{2}}$. In the limit $A\ensuremath{\rightarrow}\ensuremath{\infty}$, we find $\frac{{\ensuremath{\sigma}}_{\mathrm{el}}}{{\ensuremath{\sigma}}_{T}}\ensuremath{\rightarrow}\frac{1}{2}$ and $\frac{{\ensuremath{\sigma}}_{\mathrm{dif}}}{{\ensuremath{\sigma}}_{T}}\ensuremath{\rightarrow}0$, for both the optical and the overlap cases discussed in the text.