Residues of Regge Poles and the Diffraction Peak

It is shown, on the basis of potential theory, that for a Regge pole with position $\alpha \mathrm{}\left(t\right)>\frac{1}{2}$ the reduced residue $\frac{\beta \mathrm{}\left(t\right)\mathrm{}}{{\nu }^{\alpha \mathrm{}\left(t\right)\mathrm{}}}$ behaves essentially as $\left[\frac{{\alpha }^{\prime }\mathrm{}\left(t\right)\mathrm{}}{R}\right]R^{2\alpha \mathrm{}\left(t\right)\mathrm{}}$ near threshold ($\nu \lesssim 0$), where $R$ is the effective radius interaction. The quantity $\gamma \mathrm{}\left(t\right)={\left(\frac{{m_0}^2}{\nu }\right)}^{\alpha }\beta \mathrm{}\left(t\right)\mathrm{}$ which occurs in the asymptotic term $\gamma \mathrm{}\left(t\right)\mathrm{}{\left(\frac{s}{2{m_0}^2}\right)}^{\alpha }$ can therefore be considered as a slowly varying function of $t$ only if one takes $m_0=R^{-1\mathrm{}}$. Assuming that our threshold expression gives a qualitative description in the relativistic case, we note that if $\mathrm{Mr}>\mathrm{}1\mathrm{}$, where $M$ is the nucleon mass, then the normalization $m_0=M$ used in high-energy phenomenology will give rise to an exponential falloff of $\gamma \mathrm{}\left(t\right)\mathrm{}$ with ${\left(\mathrm{width}\right)}^{-1\mathrm{}}\approx 2{\alpha }^{\prime }\mathrm{}\left(0\right)\mathrm{} \ln \mathrm{}\left(\mathrm{MR}\right)\mathrm{}$. The values of $R$ in the $t$ channel of $\pi \pi$, $\pi N$, and $\mathrm{NN}$, estimated from the knowledge of the nearest left-hand branch points or factorization, occur in increasing order, and for each of them $\mathrm{MR}>\mathrm{}1\mathrm{}$. Our results for the Pomeranchuk trajectory roughly reproduce the exponential diffraction shape and indicate that a larger total cross section implies a sharper falloff of the amplitude in the diffraction region. A connection between $f^0$ and the diffraction width is discussed, as well as the question of zeros in $\frac{\beta }{{\nu }^{\alpha }}$.