Ultra-high Energy Predictions of Proton-Air Cross Sections from Accelerator Data: an Update

At $\sqrt{s}=57\pm 7$ TeV, the Pierre Auger Observatory (PAO) collaboration has recently measured the proton-air inelastic production cross section ${\sigma }_{\mathrm{p}-\mathrm{a}\mathrm{i}\mathrm{r}}$. Assuming a helium contamination of 25%, they subtracted 30 mb from their measured value, resulting in a p-air inelastic production cross section, $\sigma_{\rm p-air}=475 \pm 22\ ({\rm stat.})\pm^{20}{15}
({\rm syst.})$ mb, exclusive of helium contamination. Using this result in a Glauber calculation to obtain the $pp$ inelastic cross section, they found the inelastic $pp$ cross section $\sigma{\rm inel}= 90\pm 7\ ({\rm stat.}) \pm^9_{11} ({\rm syst.}) \pm 1.5 {\rm \ (Glaub.})$mb\mathrm{.}Parameterizationofthe$\bar pp$and$pp$ cross sections incorporating analyticity constraints and unitarity has allowed us to make accurate extrapolations to ultra-high energies, and using Glauber calculations, accurately predict cosmic ray results for $\text{spai}$. In this update for 57 TeV, we predict i) a $pp$ total cross section, ${\sigma }_{\mathrm{t}\mathrm{o}\mathrm{t}}=133.4\pm 1.6$ mb, using high energy predictions from a saturated Froissart bound parameterization of accelerator data on forward $\bar{p}p$ and $pp$ scattering amplitudes and ii) a p-air inelastic production cross section, ${\sigma }_{\mathrm{p}-\mathrm{a}\mathrm{i}\mathrm{r}}=483\pm 3$ mb, by using ${\sigma }_{\mathrm{t}\mathrm{o}\mathrm{t}}$ together with Glauber theory, allowing us to determine independently that the helium contamination was 19%, in reasonable agreement with their estimate of 25%. Our predictions agree with all available cosmic ray extensive air shower measurements, both in magnitude and in energy dependence. By using our value for the $pp$ total cross section at 57 TeV, Block and Halzen \cite{blackdisk} have predicted that the $pp$ inelastic cross section is $\sigma_{\rm inel}= 92.9\pm 1.6$ mb, in agreement with the measured POA value.