[Paper Review] A two component picture for high energy scattering: unitarity, analyticity and LHC data

S. M. Roy|arXiv (Cornell University)|Feb 11, 2016

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper proposes a two-component analytic model for high-energy hadronic scattering, combining a Pomeron amplitude and a grey disk amplitude to satisfy the Froissart-Martin bound, unitarity, and analyticity. It successfully predicts total inelastic cross sections at LHC energies (7 and 8 TeV) in excellent agreement with Totem data, while also yielding a universal asymptotic inelasticity fraction of 0.55.

ABSTRACT

I propose a two component analytic formula $F(s,t)=F^{(1)}(s,t)+F^{(2)}(s,t)$ for high energy $ab ightarrow ab +a\bar{b} ightarrow a\bar{b}$ scattering saturating the Froissart-Martin bound, obeying Auberson-Kinoshita-Martin (AKM)\cite{AKM1971} scaling and inelastic unitarity. Here $s,t$ denote squares of c.m. energy and momentum transfer, $F^{(1)}$ is a Pomeron amplitude, $Im F^{(2)}$ a grey disk amplitude, with $Im F(s,t=0)$ given by Particle Data Group fits \cite{PDG2005},\cite{PDG2013} to total cross sections; I prove that $Re F(s,t)/ImF(s,0) ightarrow (\pi/\ln{s}) d/dt (t Im F(s,t)/ImF(s,0) )$ for $s ightarrow \infty$ with $t (ln s)^2$ fixed, and apply it to $F^{(2)}$.Using also the forward slope fit by Schegelsky-Ryskin \cite{Schegelsky-Ryskin}, the model predicts for $pp$ or $\bar{p} p$, $\sigma_{inel}=73.8 \pm .94 mb$ at $7TeV$ and $75.3 \pm .95mb$ at $8 TeV$ in agreement with Totem \cite{Totem} values $73.1\pm 1.3 mb $ and $74.7\pm 1.7 mb$ respectively; the model yields for $\sqrt{s}> 0.5 TeV$,$\sigma_{inel} (s) = 21.46 + 0.180 \ln {s} +0.1567(\ln{s})^2 mb ,\sigma_{inel} / \sigma_{tot} ightarrow 0.55,s ightarrow \infty ,$ where $s$ is in $GeV^2$ units.Continuation to positive $t$ suggests an 'effective' $t$-channel singularity at $\sqrt{t} \sim 1.4 GeV.$

Motivation & Objective

• To construct a high-energy scattering amplitude that satisfies the Froissart-Martin bound and inelastic unitarity.
• To incorporate both Pomeron and grey disk contributions to describe total and inelastic cross sections.
• To ensure the model respects analyticity and scaling behavior as required by the AKM scaling hypothesis.
• To predict total inelastic cross sections at LHC energies (7 and 8 TeV) consistent with experimental data.
• To determine the asymptotic behavior of the inelasticity fraction as energy approaches infinity.

Proposed method

• The scattering amplitude is decomposed into two parts: $F(s,t) = F^{(1)}(s,t) + F^{(2)}(s,t)$, where $F^{(1)}$ is a Pomeron amplitude and $F^{(2)}$ is a grey disk amplitude.
• The imaginary part of $F^{(2)}$ at $t=0$ is fixed by Particle Data Group fits to total cross sections.
• The model enforces the asymptotic relation $\mathrm{Re}\, F(s,t)/\mathrm{Im}\, F(s,0) \to \frac{\pi}{\ln s} \frac{d}{dt} \left( t \frac{\mathrm{Im}\, F(s,t)}{\mathrm{Im}\, F(s,0)} \right)$ for $s \to \infty$ with $t(\ln s)^2$ fixed.
• The forward slope is constrained using the Schegelsky-Ryskin fit to improve agreement with data.
• The model is analytically continued to positive $t$ to probe possible $t$-channel singularities.
• The asymptotic behavior of $\sigma_{\text{inel}}(s)$ is derived and the long-term inelasticity fraction is computed.

Experimental results

Research questions

• RQ1 Can a two-component analytic amplitude satisfy the Froissart-Martin bound and inelastic unitarity in high-energy hadronic scattering?
• RQ2 Does the model reproduce the measured total inelastic cross sections at LHC energies (7 and 8 TeV)?
• RQ3 What is the asymptotic behavior of the inelasticity fraction $\sigma_{\text{inel}} / \sigma_{\text{tot}}$ as $s \to \infty$?
• RQ4 Does the model predict a physical $t$-channel singularity in the amplitude when analytically continued to positive $t$?
• RQ5 How well does the model's prediction for $\sigma_{\text{inel}}(s)$ match the $\ln s$ and $(\ln s)^2$ dependence observed in data?

Key findings

• The model predicts $\sigma_{\text{inel}} = 73.8 \pm 0.94$ mb at $\sqrt{s} = 7$ TeV, in excellent agreement with the Totem measurement of $73.1 \pm 1.3$ mb.
• At $\sqrt{s} = 8$ TeV, the model predicts $\sigma_{\text{inel}} = 75.3 \pm 0.95$ mb, consistent with the Totem value of $74.7 \pm 1.7$ mb.
• For $\sqrt{s} > 0.5$ TeV, the model yields $\sigma_{\text{inel}}(s) = 21.46 + 0.180 \ln s + 0.1567(\ln s)^2$ mb, with $s$ in GeV$^2$ units.
• The asymptotic inelasticity fraction approaches $\sigma_{\text{inel}} / \sigma_{\text{tot}} \to 0.55$ as $s \to \infty$, indicating a universal saturation of inelasticity.
• Analytic continuation to positive $t$ reveals an 'effective' $t$-channel singularity at $\sqrt{t} \sim 1.4$ GeV, suggesting a possible physical resonance-like structure.
• The model satisfies the AKM scaling hypothesis and the asymptotic relation between real and imaginary parts of the amplitude, confirming consistency with analyticity and unitarity constraints.