Structure of Forward pp and p=p Elastic Amplitudes at Low Energies

Exact analytical forms of solutions for Dispersion Relations for Amplitudes and Dispersion Relations for Slopes are applied in the analysis of pp and $rm {p bar p}$ scattering data in the forward range at energies below $sqrt(s)approx 30 GeV$. As inputs for the energy dependence of the imaginary part, use is made of analytic form for the total cross sections and for parameters of the $t$ dependence of the imaginary parts, with exponential and linear factors. A structure for the $t$ dependence of the real amplitude is written, with slopes $B_R$ and a linear factor $rho-mu_R t$ that allows compatibility of the data with the predictions from dispersion relations for the derivatives of the real amplitude at the origin. A very precise description is made of all $dsigma/dt$ data, with regular energy dependence of all quantities. It is shown that a revision of previous calculations of total cross sections, slopes and $rho$ parameters in the literatures is necessary, and stressed that only determinations based on $dsigma/dt$ data covering sufficient $t$ range using appropriate forms of amplitudes can be considered as valid.