Analytic models and forward scattering from accelerator to cosmic-ray energies

Analytic models for hadron-hadron scattering are characterized by analytical parameterizations for the forward amplitudes and the use of dispersion relation techniques to study the total cross section, $\sigma_{tot}$, and the $\rho$ parameter. In this paper we investigate four aspects related with the application of the model to $pp$ and $\bar{p}p$ scattering, from accelerator to cosmic-ray energies: 1) the effect of different estimations for $\sigma_{tot}$ from cosmic-ray experiments; 2) the differences between individual and global (simultaneous) fits to $\sigma_{tot}$ and $\rho$; 3) the role of the subtraction constant in the dispersion relations; 4) the effect of distinct asymptotic inputs from different analytic models. This is done by using as framework the Single Pomeron and the Maximal Odderon parameterizations for the total cross section. Our main conclusions are the following: 1) Despite the small influence from different cosmic-ray estimations, the results allow to extract an upper bound for the soft pomeron intercept: $1 + \epsilon = 1.094$; 2) Although global fits present good statistical results, in general, this procedure constraint the rise of $\sigma_{tot}$; 3) the subtraction constant as free parameter affects the fit results at both low and high energies; 4) Independently of the cosmic-ray information used and the subtraction constant, global fits with the odderon parameterization predict that, above $\sqrt s \approx 70$ GeV, $\rho_{pp}(s)$ becomes greater than $\rho_{\bar{p}p}(s)$ and this result is in complete agreement with all the data presently available. In particular, we infer $\rho_{pp} = 0.134 \pm 0.005$ at $\sqrt s = 200$ GeV and $0.151 \pm 0.007$ at 500 GeV (RHIC energies)