[Paper Review] QCD Mini-jet contribution to the total cross section

A. Achilli, R. Godbole+3 more|ArXiv.org|Jul 6, 2009

Particle physics theoretical and experimental studies6 references3 citations

TL;DR

This paper proposes a QCD mini-jet model for proton-proton total cross-section at the LHC, combining perturbative mini-jet production with resummed soft gluon radiation via an eikonal formalism. It demonstrates that the model respects the Froissart bound by showing the total cross-section grows logarithmically with energy, stabilized by the infrared behavior of the strong coupling constant αₛ.

ABSTRACT

We present the predictions of a model for proton-proton total cross-section at LHC. It takes into account both hard partonic processes and soft gluon emission effects to describe the proper high energy behavior and to respect the Froissart bound.

Motivation & Objective

• To predict the proton-proton total cross-section at LHC energies with a model grounded in QCD mini-jet production.
• To incorporate soft gluon radiation effects to prevent unphysical high-energy growth in the mini-jet cross-section.
• To ensure the model respects the Froissart bound, which limits total cross-section growth to log²s behavior.
• To link the asymptotic high-energy behavior of the total cross-section to the infrared structure of the strong coupling constant αₛ.

Proposed method

• Uses the perturbative QCD expression for mini-jet cross-section σ_jet(s, p_tmin) with parton distribution functions (PDFs) such as GRV, MRST, and CTEQ.
• Applies a resummation of soft gluon radiation via the eikonal formalism, modeling multiple partonic collisions through a Poisson-distributed number of interactions.
• Introduces an energy-dependent overlap function A(b,s) to account for spatial parton distribution and non-perturbative effects.
• Implements a phenomenological infrared-regularized αₛ(k_t²) expression with parameter p to ensure integrability and correct analyticity in the soft limit.
• Derives the eikonal function χ(b,s) from the average number of partonic collisions n(b,s) = 2 Im χ(b,s), with n_hard(b,s) ≈ σ_jet(s) A_hard(b,s).
• Evaluates the total cross-section via σ_tot ≈ 2π∫db²[1 − exp(−n_hard(b,s)/2)], analyzing its high-energy asymptotic behavior.

Experimental results

Research questions

• RQ1 How does the inclusion of soft gluon radiation affect the high-energy rise of the mini-jet cross-section in proton-proton scattering?
• RQ2 To what extent can a mini-jet model with perturbative QCD processes and soft gluon resummation reproduce the Froissart bound?
• RQ3 What is the role of the infrared behavior of αₛ(k_t²) in stabilizing the total cross-section at high energies?
• RQ4 How do different parton distribution functions (PDFs) influence the energy dependence of the mini-jet cross-section and the resulting total cross-section?
• RQ5 Can the model’s asymptotic growth rate be analytically linked to the parameters of the soft gluon emission and αₛ behavior?

Key findings

• The mini-jet cross-section σ_jet grows asymptotically as s^ε with ε ≈ 0.3–0.4, depending on the PDF used (GRV, MRST, CTEQ).
• The model’s total cross-section asymptotically grows as [ε ln(s/GeV²)]^(1/p), which respects the Froissart bound when 1/2 < p < 1.
• The infrared regularization of αₛ(k_t²) via a phenomenological form with p < 1 ensures the convergence of the soft gluon integral and prevents divergences.
• The overlap function A_hard(b,s) is modeled via a modified Bessel function integral, with h(b,q_max) depending on the maximum transverse momentum q_max(s) derived from the full partonic phase space.
• The model predicts a total cross-section that grows slower than s^ε due to the exponential suppression from the eikonal form, ensuring finiteness at high energies.
• Numerical results show good agreement with experimental data and other models, particularly when using realistic PDFs and p_tmin = 1.15 GeV.