New experimental evidence that the proton develops asymptotically into a black disk

Abstract

Recently, the Auger group has extracted the proton-air cross section from observations of air showers produced by cosmic ray protons (and nuclei) interacting in the atmosphere and converted it into measurements of the total and inelastic $pp$ cross sections $σ_{\rm tot}$ and $σ_{\rm inel}$ at the super-LHC energy of 57 TeV. Their results reinforce our earlier conclusions that the proton becomes a black disk at asymptotic energies, a prediction reached on the basis of sub-LHC $\pbar p$ and $pp$ measurements of $σ_{\rm tot}$ and $ρ$, the ratio of the real to the imaginary part of the forward scattering amplitude [M. M. Block and F. Halzen, Phys. Rev. Lett. {\bf 107}, 212002 (2011)]. The same black disk description of the proton anticipated the values of $σ_{\rm tot}$ and $σ_{\rm inel}$ measured by the TOTEM experiment at the LHC cms (center of mass) energy of $\sqrt s=7$ TeV, as well as those of $σ_{\rm inel}$ measured by ALICE, ATLAS and CMS, as well as the ALICE measurement at 2.76 TeV. All data are consistent with a proton that is asymptotically a black disk of gluons: (i) both $σ_{\rm tot}$ and $σ_{\rm inel}$ behave as $\ln^2s$, saturating the Froissart bound, (ii) the forward scattering amplitude becomes pure imaginary (iii) the ratio $σ_{\rm inel}/σ_{\rm tot}=0.509 \pm 0.021$, compatible with the black disk value of 1/2, and (iv) proton interactions become flavor blind.

Figures (3)

• Figure 1: Froissart-bounded analytic amplitude fits to $\rho$, the ratio of the real to the imaginary portion of the forward scattering amplitude, vs. $\sqrt s$, the cms energy in GeV, taken from BH blockhalzen2. The $\bar{p}p$ data used in the fit are the (red) circles and the $pp$ data are the (blue) squares.
• Figure 2: Froissart-bounded analytic amplitude fits to the total cross section, $\sigma_{\rm tot}$, for $\bar{p}p$ (dashed curve) and $pp$ (dot-dashed curve) from Eq. (\ref{['sigmapmpp']}), in mb vs. $\sqrt s$, the cms energy in GeV, taken from BH blockhalzen2. The $\bar{p}p$ data used in the fit are the (red) circles and the $pp$ data are the (blue) squares. The fitted data were anchored by values of $\sigma_{\rm tot}^{\bar{p}p}$ and $\sigma_{\rm tot}^{pp}$, together with the energy derivatives ${d\sigma_{\rm tot}^{\bar{p}p}/ d\nu}$ and ${d\sigma_{\rm tot}^{pp}/ d\nu}$ at 6 GeV using FESR, as described in Ref. blockhalzen2. It should be noted that our ultra-high energy total cross section predictions that are made from our analytic amplitude fit use only total cross section data that are in the lower energy range $6\le \sqrt s \le 1800$ GeV.
• Figure 3: Predictions for $\sigma_{\rm tot}$ and $\sigma_{\rm inel}$ vs. $\sqrt s$ for $\bar{p}p$ and $pp$. For $\sigma_{\rm tot}$, we have compared our predictions with recent $pp$ TOTEM data at 7 TeV and Auger data at 57 TeV, while for $\sigma_{\rm inel}$, we have compared our results with 2.76 TeV $pp$ data from ALICE, 7 TeV $pp$ data from ALICE, ATLAS, CMS and Totem, as well as with the 57 TeV $pp$ inelastic cross section. The upper solid (black) curve is the central-value prediction for $\sigma_{\rm tot}$ and the lower solid (red) curve is the central-value prediction for $\sigma_{\rm inel}$. The dotted curves are the errors ($\pm 1 \sigma$) in our predictions, due to the correlated errors of the fitting parameters. We emphasize that none of the datum points in this plot have been used in our predictions.