Recent data from LHC13 by the TOTEM Collaboration indicate an unexpected decrease in the value of the $\rho$ parameter and a ${\sigma }_{\mathrm{tot}}$ value in agreement with the trend of previous measurements at 7 and 8 TeV. These data at 13 TeV are not simultaneously described by the predictions from Pomeron models selected by the COMPETE Collaboration but show agreement with the maximal Odderon dominance, as recently demonstrated by Martynov and Nicolescu. Here, we present a detailed analysis on the applicability of Pomeron dominance by means of a general class of forward scattering amplitude, consisting of even-under-crossing leading contributions associated with single, double, and triple poles in the complex angular momentum plane and subleading even and odd Regge contributions. The analytic connection between ${\sigma }_{\mathrm{tot}}$ and $\rho$ is obtained by means of singly subtracted dispersion relations, and we carry out fits to $pp$ and $\overline{p}p$ data in the interval 5 GeV–13 TeV. The data set comprises all the accelerator data below 7 TeV, and we consider two independent ensembles by adding either only the TOTEM data or the TOTEM and ATLAS data at the LHC energy region. In the data reductions to each ensemble, the uncertainty regions are evaluated with both one and two standard deviations ($\sim 68\%$ and $\sim 95\%\mathrm{CL}$, respectively). Besides the general analytic model, we investigate four particular cases of interest, three of them typical of outstanding models in the literature. We conclude that, within the experimental and theoretical uncertainties and both ensembles, the general model and three particular cases are not able to describe the ${\sigma }_{\mathrm{tot}}$ and $\rho$ data at 13 TeV simultaneously. However, if the discrepancies between the TOTEM and ATLAS data are not resolved, one Pomeron model, associated with double and triple poles and with only 7 free parameters, seems not to be excluded by the complete set of experimental information presently available.

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 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-γ (Lγ) law, where the exponent γ 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: (1) 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; (2) the fit results with the P law, the L2 law (free scale) and the Lγ law (fixed scale and exponent γ 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; (3) 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); (4) 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, demanding further and detailed investigation.

We discuss some formal and practical aspects related to the replacement of Integral Dispersion Relations (IDR) by derivative forms, without high-energy approximations. We first demonstrate that, for a class of functions with physical interest as forward scattering amplitudes, this replacement can be analytically performed, leading to novel Extended Derivative Dispersion Relations (EDDR), which, in principle, are valid for any energy above the physical threshold. We then verify the equivalence between the IDR and EDDR by means of a popular parametrization for total cross sections from proton-proton and antiproton-proton scattering and compare the results with those obtained through other representations for the derivative relations. Critical aspects on the limitations of the whole analysis, from both formal and practical points of view, are also discussed in some detail.