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A phenomenological model for the description of the single and double diffractive excitation in $pp$ collisions at high energies is presented. Considering the Good -- Walker approach, we propose a model for the eigenstates of the scattering operator and for the treatment of the interaction between them, with the high energy behavior of the cross section driven by perturbative QCD. The behavior of the total, elastic, single and double diffractive cross sections are analyzed and predictions for the energies of Run 3 of the LHC and those of the Cosmic Rays experiments are derived. We demonstrate that the model describes the current data for the energy dependence of the cross sections. A comparison with the recent data for the $\rho$ parameter and the differential elastic cross section are also presented and shortcomings of the current model are discussed., Comment: 13 pages, 5 figures, 1 table. Improved version published in Physical Review D

Inspired by the recent TOTEM data for the elastic proton -- proton ($pp$) scattering at $\sqrt{s} =$ 8 and 13 TeV, we update previous studies of the differential cross sections using the Phillips -- Barger (PB) model, which parametrizes the amplitude in terms of a small number of free parameters. We demonstrate that this model is able to describe the recent $pp$ data on a statistically acceptable way. Additionally, we perform separate fits of the $pp$ data for each center - of - mass energy and propose a parametrization for the energy dependence of the parameters present in the PB model. As a consequence, we are able to present the PB predictions for the elastic proton - proton cross section at $\sqrt{s} = 546$ GeV and $1.8$ TeV, which are compared with the existing antiproton -- proton ($\bar{p}p$) data. We show that the PB predictions, constrained by the $pp$ data, are not able to describe the $\bar{p}p$ data. In particular, the PB model predicts a dip in the differential cross section that is not present in the $\bar{p}p$ data. Such result suggests the contribution of the Odderon exchange at high energies., Comment: 6 pages, 4 tables, 2 figures, results updated, matches published version

In the present investigation we study the leading and subleading high-energy behavior of hadron-hadron total cross sections using a best-fit analysis of hadronic scattering data. The parametrization used for the hadron-hadron total cross sections at high energy is inspired by recent results obtained by Giordano and Meggiolaro (2014) using a nonperturbative approach in the framework of QCD and it reads $\sigma_\text{tot} \sim B\ln^2 s + C\ln s\ln\ln s$. Both $B$ and $C$ are obtained by means of best-fits to data for proton-proton and antiproton-proton scattering, including recent data obtained at the LHC, and also to data for other meson-baryon and baryon-baryon scattering processes. The results are compared to the theoretical predictions existing in the literature. In particular, following the above-mentioned nonperturbative QCD approach, we also consider fits where the parameters $B$ and $C$ are set to $B = \kappa B_\text{th}$ and $C = \kappa C_\text{th}$, where $B_\text{th}$ and $C_\text{th}$ are universal quantities related to the QCD stable spectrum, while $\kappa$ (treated as an extra free parameter) is related to the asymptotic value of the ratio $\sigma_\text{el}/\sigma_\text{tot}$. Different possible scenarios are then considered and compared., Comment: 33 pages, 7 tables, 7 figures

Forward amplitude analyses constitute an important approach in the investigation of the energy dependence of the total hadronic cross-section $\sigma_{tot}$ and the $\rho$ parameter. The standard picture indicates for $\sigma_{tot}$ a leading log-squared dependence at the highest c.m. energies, in accordance with the Froissart-Lukaszuk-Martin bound. Beyond this log-squared (L2) leading dependence, other amplitude analyses have considered a log-raised-to-gamma form (L$\gamma$), with $\gamma$ as a real free fit parameter. In this case, analytic connections with $\rho$ can be obtained either through dispersion relations (derivative forms), or asymptotic uniqueness (Phragm\'en-Lindel\"off theorems). In this work we present a detailed discussion on the similarities and mainly the differences between the Derivative Dispersion Relation (DDR) and Asymptotic Uniqueness (AU) approaches and results, with focus on the L$\gamma$ and L2 leading terms. We also develop new Regge-Gribov fits with updated dataset on $\sigma_{tot}$ and $\rho$ from $pp$ and $\bar{p}p$ scattering, in the region 5 GeV-8 TeV. The recent tension between the TOTEM and ATLAS results at 7 TeV and mainly 8 TeV is considered in the data reductions. Our main conclusions are: (1) all fit results present agreement with the experimental data analyzed and the goodness-of-fit is slightly better in case of the DDR approach; (2) by considering only the TOTEM data at the LHC region, the fits with L$\gamma$ indicate $\gamma\sim 2.0\pm 0.2$ (AU) and $\gamma\sim 2.3\pm 0.1$ (DDR); (3) by including the ATLAS data the fits provide $\gamma\sim 1.9\pm 0.1$ (AU) and $\gamma\sim 2.2\pm 0.2$ (DDR); (4) in the formal and practical contexts, the DDR approach is more adequate for the energy interval investigated than the AU approach. A review on the analytic results for $\sigma_{tot}$ and $\rho$ from the Regge-Gribov, DDR and AU approaches is presented., Comment: 41 pages, 3 tables, 8 figures, misprints corrected, matches published version

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., Comment: 14 pages, 3 figures, 6 tables, one appendix added (discussing some TOTEM data at 8 TeV), minor corrections, few comments added. Matches version accepted for publication in Nucl. Phys. A

Following earlier findings, we argue that the low-$|t|$ structure in the elastic diffractive cone, recently reported by the TOTEM Collaboration at $8$ TeV, is a consequence of the threshold singularity required by $t-$channel unitarity, such as revealed earlier at the ISR. By using simple Regge-pole models, we analyze the available data on the $pp$ elastic differential cross section in a wide range of c.m. energies, namely those from ISR to LHC8, obtaining good fits of all datasets. This study hints at the fact that the non-exponential behaviour observed at LHC8 is a recurrence of the low-$|t|$ "break" phenomenon, observed in the seventies at ISR, being induced by the presence of a two-pion loop singularity in the Pomeron trajectory., Comment: 11 pages, 6 figures, 7 tables. Invited contribution to the Gribov-85 Memorial Volume "Exploring Quantum Field Theory", to be published by World Scientific

The energy dependence of the total hadronic cross section at high energies is investigated with focus on the recent experimental result by the TOTEM Collaboration at 7 TeV and the Froissart-Martin bound. On the basis of a class of analytical parametrization with the exponent $\gamma$ in the leading logarithm contribution as a free parameter, different variants of fits to $pp$ and $\bar{p}p$ total cross section data above 5 GeV are developed. Two ensembles are considered, the first comprising data up to 1.8 TeV, the second also including the data collected at 7 TeV. We shown that in all fit variants applied to the first ensemble the exponent is statistically consistent with $\gamma$ = 2. Applied to the second ensemble, however, the same variants yield $\gamma$'s above 2, a result already obtained in two other analysis, by U. Amaldi \textit{et al}. and by the UA4/2 Collaboration. As recently discussed by Ya. I. Azimov, this faster-than-squared-logarithm rise does not necessarily violate unitarity. Our results suggest that the energy dependence of the hadronic total cross section at high energies still constitute an open problem., Comment: 20 pages, 10 figures, introduction extended and general references added to match editorial style, to appear in the Brazilian Journal of Physics

In the absence of a global description of the experimental data on elastic and soft diffractive scattering from the first principles of QCD, model-independent analyses may provide useful phenomenological insights for the development of the theory in the soft sector. With that in mind, we present an empirical study on the energy dependence of the ratio $X$ between the elastic and total cross sections; a quantity related to the evolution of the hadronic central opacity. The dataset comprises all the experimental information available on proton-proton and antiproton-proton scattering in the c.m. energy interval 5 GeV - 8 TeV. Generalizing previous works, we discuss four model-independent analytical parameterizations for $X$, consisting of sigmoid functions composed with elementary functions of the energy and three distinct asymptotic scenarios: either the standard black disk limit or scenarios above or below that limit. Our two main conclusions are the following: (1) although consistent with the experimental data, the black disk does not represent an unique solution; (2) the data reductions favor a semi-transparent scenario, with asymptotic average value for the ratio $\bar{X}$ = 0.30 $\pm$ 0.12. In this case, within the uncertainty, the asymptotic regime may already be reached around 1000 TeV. We present a comparative study of the two scenarios, including predictions for the inelastic channel (diffraction dissociation) and the ratio associated with the total cross-section and the elastic slope. Details on the selection of our empirical ansatz for $X$ and physical aspects related to a change of curvature in this quantity at 80 - 100 GeV, indicating the beginning of a saturation effect, are also presented and discussed., Comment: 41 pages, 10 figures, 13 tables; references [31], [90] and some misprints corrected

We present a model-independent analysis of the experimental data on the ratio $X$ between the elastic and total cross-sections from $pp$ and $\bar{p}p$ scattering in the c.m. energy interval 5 GeV - 8 TeV. Using a novel empirical parametrization for that ratio as a function of the energy and based on theoretical and empirical arguments, we investigate three distinct asymptotic scenarios: either the black-disk (BD) limit or scenarios above and below that limit. Our analysis favors a scenario below the BD, with asymptotic ratio $X = 0.36 \pm 0.08$., 5 pages, 1 figure added, extended text, matchs proceedings (Diffraction 2014)

We present an updated empirical analysis on the ratio of the elastic (integrated) to the total cross section in the c.m. energy interval from 5 GeV to 8 TeV. As in a previous work, we use a suitable analytical parametrization for that ratio (depending on only four free fit parameters) and investigate three asymptotic scenarios: either the black disk limit or scenarios above or below that limit. The dataset includes now the datum at 7 TeV, recently reported by the ATLAS Collaboration. Our analysis favors, once more, a scenario below the black disk, providing an asymptotic ratio consistent with the rational value 1/3, namely a gray disk limit. Upper bounds for the ratio of the diffractive (dissociative) to the inelastic cross section are also presented., Comment: 8 pages, 2 figures, 3 tables, references added, text slightly shortened, matches proceedings style (XIII Hadron Physics, IOP: Conf. Series). Presented at XIII Hadron Physics, Hotel do Bosque, Angra dos Reis, RJ, Brazil, March 22 - 27, 2015

We show that the proton-proton elastic differential cross section data at dip position and beyond can be quite well described by a parametrization based on the Tsallis distribution, with only five free fit parameters. Extrapolation of the results obtained at 7 TeV to large momentum transfer, suggests that hadrons may not behave as a black-disk at the asymptotic energy region., 3 pages, 1 figure, version matching proceedings style, XII Hadron Physics, 2012, AIP Proc. Conf

A forward amplitude analysis on $pp$ and $\bar{p}p$ elastic scattering above 5 GeV is presented. The dataset includes the recent high-precision TOTEM measurements of the $pp$ total and elastic (integrated) cross-sections at 7 TeV and 8 TeV. Following previous works, the leading high-energy contribution for the total cross-section ($\sigma_{tot}$) is parametrized as $\ln^{\gamma}(s/s_h)$, where $\gamma$ and $s_h$ are free \textit{real} fit parameters. Singly-subtracted derivative dispersion relations are used to connect $\sigma_{tot}$ and the rho parameter ($\rho$) in an analytical way. Different fit procedures are considered, including individual fits to $\sigma_{tot}$ data, global fits to $\sigma_{tot}$ and $\rho$ data, constrained and unconstrained data reductions. The results favor a rise of the $\sigma_{tot}$ faster than the log-squared bound by Froissart and Martin at the LHC energy region. The parametrization for $\sigma_{tot}$ is extended to fit the elastic cross-section ($\sigma_{el}$) data with satisfactory results. The analysis indicates an asymptotic ratio $\sigma_{el}/\sigma_{tot}$ consistent with 1/3 (as already obtained in a previous work). A critical discussion on the correlation, practical role and physical implications of the parameters $\gamma$ and $s_h$ is presented. The discussion confronts the 2002 prediction of $\sigma_{tot}$ by the COMPETE Collaboration and the recent result by the Particle Data Group (2012 edition of the Review of Particle Physics). Some conjectures on possible implications of a fast rise of the proton-proton total cross-section at the highest energies are also presented., Comment: 32 pages, 9 figures, discussion on the 8 TeV TOTEM data extended (2 tables and 1 figure added), typos/grammar corrected, conclusions unchanged, matches journal version

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., 40 pages, minor corrections, matches the published version

The rise of the total, elastic and inelastic hadronic cross sections at high energies is investigated by means of an analytical parametrization, with the exponent of the leading logarithm contribution as a free fit parameter. Using derivative dispersion relations with one subtraction, two different fits to proton-proton and antiproton-proton total cross section and rho parameter data are developed, reproducing well the experimental information in the energy region 5 GeV - 7 TeV. The parametrization for the total cross sections is then extended to fit the elastic (integrated) cross section data in the same energy region, with satisfactory results. From these empirical results we extract the energy dependence of several physical quantities: inelastic cross section, ratios elastic/total, inelastic/total cross sections, ratio total-cross-section/elastic-slope, elastic slope and optical point. All data, fitted and predicted, are quite well described. We find a statistically consistent solution indicating: (1) an increase of the hadronic cross sections with the energy faster than the log-squared bound by Froissart and Martin; (2) asymptotic limits 1/3 and 2/3 for the ratios elastic/total and inelastic/total cross sections, respectively, a result in agreement with unitarity. These indications corroborate recent theoretical arguments by Ya. I. Azimov on the rise of the total cross section., 35 pages, 12 figures, discussions improved with further clarifications, references added and updated, one note added, results and conclusions unchanged. Version to be published in J. Phys. G: Nucl. Part. Phys