Unitarity Upper and Lower Bounds on the Absorptive Parts of Elastic Scattering Amplitudes

Abstract

We derive upper and lower bounds on the imaginary part of the elastic scattering amplitude of two spinless particles in the physical region, in terms of the elastic cross section ${\ensuremath{\sigma}}_{\mathrm{el}}$ and the total cross section ${\ensuremath{\sigma}}_{\mathrm{tot}}$, using unitarity alone. The bounds derived are the best possible ones, given only the stated unitarity constraints. The upper bound for high energies and small values of the momentum transfer squared $t$ has a particularly simple and universal form, $\frac{\mathrm{Im}F(s, t)}{\mathrm{Im}F(s, 0)}l~1\ensuremath{-}\frac{1}{9}\ensuremath{\rho}+\frac{3}{8}{(\frac{\ensuremath{\rho}}{9})}^{2}\ensuremath{-}(\frac{21}{320}){(\frac{\ensuremath{\rho}}{9})}^{3}+\ensuremath{\cdots}$ if $2.5g~\ensuremath{\rho}\ensuremath{\equiv}(\ensuremath{-}\frac{t}{4\ensuremath{\pi}})\frac{{{\ensuremath{\sigma}}_{\mathrm{tot}}}^{2}(s)}{{\ensuremath{\sigma}}_{\mathrm{el}}(s)}$, which depends on the particular scattering process and on the energy and momentum transfer only through the dimensionless parameter $\ensuremath{\rho}$. We give explicit formulas and numerical values for the upper bound up to $\ensuremath{\rho}=8.42$. We compare the experimental curve of $\frac{(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}{{(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}_{t=0}}$ versus $4(\ensuremath{-}t)\frac{{(\frac{d\ensuremath{\sigma}}{\mathrm{dt}})}_{t=0}}{{\ensuremath{\sigma}}_{\mathrm{el}}}$ with the theoretical upper bound on ${[\frac{\mathrm{Im}F(s, t)}{\mathrm{Im}F(s, t=0)]}]}^{2}$ versus $\ensuremath{\rho}$. The quantities plotted in the experimental and theoretical curves are the same if the unpolarized cross sections are spin-independent and purely absorptive in the diffraction-peak region. We find that the experimental points for $\mathrm{pp}$, $\overline{p}p$, ${\ensuremath{\pi}}^{+}p$, and ${\ensuremath{\pi}}^{\ensuremath{-}}p$ scattering in the lab momentum range 6-13 GeV/c fall on a curve lying only slightly below the theoretical upper-bound curve, the difference being less than 10% for $\ensuremath{\rho}$ in the range (0,3) and less than 25% for $\ensuremath{\rho}$ in the range (3,5). We further notice that this experimental curve is universal. We also derive unitarity lower bounds on the $n\mathrm{th}$ derivatives of the absorptive part at $t=0$, and on the absorptive part for positive values of $t$ within the Lehmann-Martin ellipse, in terms of ${\ensuremath{\sigma}}_{\mathrm{el}}$ and ${\ensuremath{\sigma}}_{\mathrm{tot}}$. The corresponding bounds if ${\ensuremath{\sigma}}_{\mathrm{tot}}$ alone is known are also derived.