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

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 $\sigma_{\rm tot}$ and $\sigma_{\rm inel}$ 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 $\sigma_{\rm tot}$ and $\rho$, 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 $\sigma_{\rm tot}$ and $\sigma_{\rm inel}$ measured by the TOTEM experiment at the LHC cms (center of mass) energy of $\sqrt s=7$ TeV, as well as those of $\sigma_{\rm inel}$ 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 $\sigma_{\rm tot}$ and $\sigma_{\rm inel}$ behave as $\ln^2s$, saturating the Froissart bound, (ii) the forward scattering amplitude becomes pure imaginary (iii) the ratio $\sigma_{\rm inel}/\sigma_{\rm tot}=0.509 \pm 0.021$, compatible with the black disk value of 1/2, and (iv) proton interactions become flavor blind.Comment: 4 pages, 3 figure

Forward hadronic scattering at 8 TeV: predictions for the LHC

The Large Hadron Collider (LHC) recently started operating at 8 TeV. In this note, we update our earlier LHC forward hadronic scattering predictions \cite{physicsreports,update7, blackdisk}, giving new predictions, including errors, for the $pp$ total and inelastic cross sections, the $\rho$-value, the nuclear slope parameter $B$, $d\sigma_{\rm el}/dt$, and the large gap survival probability at 8 TeV.Comment: 4 pages, 1 figure, 1 table. arXiv admin note: substantial text overlap with arXiv:1102.316

Decoupling the coupled DGLAP evolution equations: an analytic solution to pQCD

Using Laplace transform techniques, along with newly-developed accurate numerical inverse Laplace transform algorithms, we decouple the solutions for the singlet structure function $F_s(x,Q^2)$ and $G(x,Q^2)$ of the two leading-order coupled singlet DGLAP equations, allowing us to write fully decoupled solutions: F_s(x,Q^2)={\cal F}_s(F_{s0}(x), G_0(x)), G(x,Q^2)={\cal G}(F_{s0}(x), G_0(x)). Here ${\cal F}_s$ and $\cal G$ are known functions---found using the DGLAP splitting functions---of the functions $F_{s0}(x) \equiv F_s(x,Q_0^2)$ and $G_{0}(x) \equiv G(x,Q_0^2)$, the chosen starting functions at the virtuality $Q_0^2$. As a proof of method, we compare our numerical results from the above equations with the published MSTW LO gluon and singlet $F_s$ distributions, starting from their initial values at $Q_0^2=1 GeV^2$. Our method completely decouples the two LO distributions, at the same time guaranteeing that both distributions satisfy the singlet coupled DGLAP equations. It furnishes us with a new tool for readily obtaining the effects of the starting functions (independently) on the gluon and singlet structure functions, as functions of both $Q^2$ and $Q_0^2$. In addition, it can also be used for non-singlet distributions, thus allowing one to solve analytically for individual quark and gluon distributions values at a given $x$ and $Q^2$, with typical numerical accuracies of about 1 part in $10^5$, rather than having to evolve numerically coupled integral-differential equations on a two-dimensional grid in $x, Q^2$, as is currently done.Comment: 6 pages, 2 figure