Pomeron pole plus grey disk model: real parts, inelastic cross sections and LHC data

Abstract

I propose a two component analytic formula $F(s,t)=F^{(1)}(s,t)+F^{(2)}(s,t)$ for $(ab\rightarrow ab) +(a\bar{b}\rightarrow a\bar{b})$ scattering at energies $\ge 100 GeV$ ,where $s,t$ denote squares of c.m. energy and momentum transfer.It saturates the Froissart-Martin bound and obeys Auberson-Kinoshita-Martin (AKM) \cite{AKM1971} scaling. I choose $Im F^{(1)}(s,0)+Im F^{(2)}(s,0)$ as given by Particle Data Group (PDG) fits to total cross sections. The PDG formula is extended to non-zero momentum transfers using partial waves of $Im F^{(1)}$ and $Im F^{(2)}$ motivated by Pomeron pole and 'grey disk' amplitudes . $Re F(s,t)$ is deduced from real analyticity: I prove that $Re F(s,t)/ImF(s,0) \rightarrow (π/\ln{s}) d/dτ(τIm F(s,t)/ImF(s,0) )$ for $s\rightarrow \infty$ with $τ=t (ln s)^2$ fixed, and apply it to $F^{(2)}$.Using also the forward slope fit by Schegelsky-Ryskin , the model gives real parts,differential cross sections for $(-t)<.3 GeV^2$, and inelastic cross sections in good agreement with data at $546 GeV, 1.8 TeV,7 TeV$ and $ 8 TeV $. It predicts for inelastic cross sections for $pp$ or $\bar{p} p$, $σ_{inel}=72.7\pm 1.0 mb$ at $7TeV$ and $74.2 \pm 1.0mb$ at $8 TeV$ in agreement with $pp$ Totem experimental values $73.1\pm 1.3 mb $ and $74.7\pm 1.7 mb$ respectively, and with Atlas values $71.3\pm 0.9 mb$ and $71.7\pm 0.7mb$ respectively. The predictions at $546 GeV$ and $1800 GeV$ also agree with $\bar{p} p$ experimental results of Abe et al \cite{Abe} at $546 GeV$ and $1800 GeV$. The model yields for $\sqrt{s}> 0.5 TeV$, with PDG2013 total cross sections , and Schegelsky-Ryskin slopes as input, $σ_{inel} (s) =22.6 + .034 ln s + .158 (ln s)^2 mb , and σ_{inel} / σ_{tot} \rightarrow 0.56, s\rightarrow \infty ,$ where $s$ is in $GeV^2$