[Paper Review] A toy model of elastic scattering of high energy protons

I. Dremin|arXiv (Cornell University)|Feb 21, 2017

High-Energy Particle Collisions Research1 references3 citations

TL;DR

This paper proposes a simplified analytical toy model for high-energy proton-proton elastic scattering, relying on a single key parameter—the ratio of total to elastic cross sections—to describe the spatial inelastic profile. By neglecting the real part of the scattering amplitude and focusing on the imaginary part's behavior, the model links the critical value of the inelastic profile parameter ζ to experimental observables, showing that ζ ≈ 1 at LHC energies, indicating near-complete inelastic opacity at central impact parameters.

ABSTRACT

The ratio of elastic to total proton cross sections is related to the darkness of the spatial profile of inelastic interactions by a single parameter in the framework of a simple analytical model. Their critical values at LHC energies are discussed. Two possible variants of their asymptotical behavior are described.

Motivation & Objective

• To develop a minimal analytical model for elastic proton-proton scattering with fewer adjustable parameters than existing phenomenological models.
• To clarify the role of the real part of the scattering amplitude, showing it contributes negligibly to integral observables like r and ζ.
• To explore the energy dependence of the inelastic profile parameter ζ and its implications for the asymptotic behavior of elastic scattering.
• To provide a direct link between measurable experimental quantities (σ_tot, B, t_dip) and the degree of inelastic opacity at central collisions.

Proposed method

• The model assumes the real part of the elastic amplitude is negligible at high energies, justified by kfk-model results and dispersion relations.
• The imaginary part of the amplitude is modeled as f_I(s,t) = σ_tot(s)/(4√π) × (1 - (t/t₀(s))²) × exp(B(s)t/2), capturing the diffraction cone and a single zero.
• The parameter ζ = σ_el / (4πB) is derived from the integral of f_I², representing the normalized inelastic profile at b=0.
• The dip position t_dip ≈ t₀ is used as a proxy for the zero of the imaginary amplitude, linking to experimental data.
• The model uses the optical theorem and unitarity condition to relate σ_tot, B, and t₀ to ζ and r = σ_el / σ_tot.
• Analytical estimates are derived for the contribution of the tail beyond the diffraction cone, showing it is negligible (Δζ ≈ -2×10⁻³ at LHC).

Experimental results

Research questions

• RQ1 How does the neglect of the real part of the elastic amplitude affect the accuracy of integral observables like r and ζ?
• RQ2 What is the role of the imaginary amplitude's zero at t₀ in shaping the differential cross section and determining ζ?
• RQ3 Can a single-parameter model accurately describe the energy evolution of elastic scattering and inelastic profile at LHC energies?
• RQ4 What are the implications for the asymptotic behavior of ζ if r > 1 at higher energies?
• RQ5 How do experimental uncertainties in σ_tot, B, and t_dip translate into precision requirements for estimating ζ?

Key findings

• The real part of the amplitude contributes less than 1.5% to the total elastic cross section, justifying its neglect in the model.
• The contribution of the tail region (|t| > |t₀|) to ζ is negligible and slightly negative, Δζ ≈ -2×10⁻³ at LHC energies.
• At LHC energies, the inelastic profile parameter ζ is estimated to be ≈1.02 ± 0.04, indicating near-complete inelastic opacity at central impact parameters.
• The ratio r = σ_el / σ_tot ranges from 1.01 ± 0.06 to 1.06 ± 0.06 at LHC, placing it close to the critical value of 1.
• The model shows that precise measurements of σ_tot, B, and t_dip are essential to determine whether ζ will exceed 1 asymptotically.
• The critical behavior at LHC suggests that future data with improved precision could reveal whether ζ approaches 1 or exceeds it, signaling incomplete inelastic attenuation at central collisions.