Part Of Elastic Scattering Amplitude Research Articles

The real part of the elastic-scattering amplitude at the $$S\bar ppS$$ and predictions at LHC and SSCand predictions at LHC and SSC

A precise measurement of\(\bar pp\) elastic scattering in the Coulomb strong-interaction interference region was performed at the CERN\(S\bar ppS\) Collider at a centre-of-mass energy of 541 GeV. The ratio of the real-to-imaginary part of the forward elastic-scattering amplitude was found to be ρ=0.135±0.015. The slope of the exponential fall-off of the strong-interaction part was also measured to beb=15.5±0.1 GeV−2. Using this new result, an overall fit to the data on the total cross-section and on the real part for\(\bar pp\) and pp was performed using dispersion relations. Numerical predictions are presented for total cross-sections at LHC and SSC energies.

Bounds on the Moments of Absorptive Parts of Elastic Scattering Amplitudes

A rigorous set of bounds on the moments of the absorptive part of elastic scattering amplitudes and its derivatives with respect to the cosine of the scattering angle is presented for pion-pion and pion-nucleon scattering. These bounds are similar to unitarity bounds found by Singh and Roy, and involve integrals over absorptive parts and their derivatives in the physical scattering region only.

Another Bound on the Absorptive Part of Elastic Scattering Amplitudes

Via Lagrange multipliers for inequality constraints, an upper bound on the absorptive part of the elastic scattering amplitudes is derived assuming unitarity, a fixed total and elastic cross section, and the condition that the partial waves decrease monotonically with increasing angular momentum. Numerical results are given.

Bounds on the Absorptive Part of Elastic Scattering Amplitudes with a Monotonicity Constraint

Bounds on the Absorptive Part of Elastic Scattering Amplitudes with a Monotonicity Constraint

Improved Upper Bound on the Imaginary Part of Elastic Scattering Amplitudes

The method of Singh and Roy, which obtains a bound on the imaginary part of the elastic scattering amplitude, given ${\ensuremath{\sigma}}_{\mathrm{tot}}$ and ${\ensuremath{\sigma}}_{\mathrm{el}}$, is generalized to allow inclusion of other given data such as $\frac{d\ensuremath{\sigma}}{\mathrm{dt}}$ at fixed points. A calculation is described in which one additional experimental point is given. The upper bound is compared with the experimental data for ${\ensuremath{\pi}}^{\ifmmode\pm\else\textpm\fi{}}p$, ${K}^{\ifmmode\pm\else\textpm\fi{}}p$, $\mathrm{pp}$, and $p\overline{p}$ elastic scattering at various energies in the nearforward direction. It is found that this single extra piece of given data extends by a factor of about 3 the range for which the original bound remained close to the data and at the largest $t$ value considered ($t\ensuremath{\approx}\ensuremath{-}0.5$ Be${\mathrm{V}}^{2}$) improves this bound by a factor of about 4.

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

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.