  Recent measurements of the proton-proton total cross section $\sigma_{tot}$
at 7 and 8 TeV by the TOTEM and ATLAS Collaborations are characterized by some
discrepant values: the TOTEM data suggest a rise of the cross section with the
energy faster than the ATLAS data. Attempting to quantify these different
behaviors, we develop new analytical fits to $\sigma_{tot}$ and $\rho$ data
from $pp$ and $\bar{p}p$ scattering in the energy region 5 GeV - 8 TeV. The
dataset comprises all the accelerator data below 7 TeV and we consider three
ensembles by adding: either only the TOTEM data (T), or only the ATLAS data
(A), or both sets (T+A). For the purposes, we use our previous RRPL$\gamma$
parametrization for $\sigma_{tot}(s)$, consisting of two Reggeons (RR), one
critical Pomeron (P) and a leading log-raised-to-gamma (L$\gamma$) contribution
(with $\gamma$ as a free fit parameter), analytically connected to $\rho(s)$
through singly-subtracted derivative dispersion relations and energy scale
fixed at the physical threshold. The data reductions with ensembles T and A
present good agreement with the experimental data analyzed and cannot be
distinguished on statistical grounds. The quality of the fit is not as good
with ensemble T+A. The fit results provide $\gamma \sim 2.3 \pm 0.1$ (T), $2.0
\pm 0.2$ (A), $2.1 \pm 0.2$ (T+A), with $\chi^2/\mathrm{DOF} \sim 1.07$ (T),
$1.09$ (A), $1.14$ (T+A), suggesting extrema bounds for $\gamma$ given by 1.8
and 2.4. Fits with $\gamma = 2$ (fixed) are also developed and discussed.
