  A comparative study on some representative parametrizations for the total and
elastic cross-sections as a function of energy is presented. The dataset
comprises pp and \bar{p}p scattering in the c.m energy interval 5 GeV-8 TeV.
The parametrization for the total cross-section at low and intermediate
energies follows the usual reggeonic structure (non-degenerate trajectories).
For the leading high-energy pomeron contribution, we consider three distinct
analytic parametrizations: either a power (P) law, or a log-squared (L2) law or
a log-raised-to-gamma (Lgamma) law, where the exponent gamma is treated as a
real free fit parameter. The parametrizations are also extended to fit the
elastic (integrated) cross-section data in the same energy interval. Our main
conclusions are the following: the data reductions with the logarithmic laws
show strong dependence on the unknown energy scale involved, which is treated
here either as a free parameter or fixed at the energy threshold; the fit
results with the P law, the L2 law (free scale) and the Lgamma law (fixed scale
and exponent gamma above 2) are all consistent within their uncertainties and
with the experimental data up to 7 TeV, but they partially underestimate the
high-precision TOTEM measurement at 8 TeV; once compared with these results,
the L2 law with fixed scale is less consistent with the data and, in the case
of a free scale, this pomeron contribution decreases as the energy increases
below the scale factor (which lies above the energy cutoff); in all cases
investigated, the predictions for the asymptotic ratio between the elastic and
total cross-sections, within the uncertainties, do not exceed the value 0.430
(therefore, below the black-disc limit) and the results favor rational limits
between 1/3 and 2/5. We are led to conclude that the rise of the hadronic
cross-sections at the highest energies still constitutes an open problem.
