[Paper Review] Diffraction, Saturation and pp Cross Sections at the LHC

K. Goulianos|arXiv (Cornell University)|May 25, 2011

Particle physics theoretical and experimental studies5 references3 citations

TL;DR

This paper proposes a phenomenological model based on saturation and unitarity constraints to predict total and inelastic proton-proton cross sections at the LHC. Using a saturated Froissart bound with experimentally determined parameters $s_F = 22$ GeV and $s_o$, the model predicts $\sigma_{\text{inel}} = 71 \pm 6$ mb at $\sqrt{s} = 7$ TeV, in excellent agreement with ATLAS and CMS measurements, offering a robust simulation framework for diffractive processes at the LHC.

ABSTRACT

Results from the large hadron collider (LHC) show that no available Monte Carlo simulation incorporates our pre-LHC knowledge of soft and hard diffraction in a way that could be reliably extrapolated to LHC energies. As a simulation is needed to establish triggers, perform underlying event corrections and calculate acceptances, the lack of a robust simulation affects all measurements at the LHC. Particularly affected are the measurements of processes with large diffractive rapidity gaps, which constitute about one quarter of the inelastic cross section. In this paper, a previously described phenomenological model based on a saturation effect observed in single diffraction dissociation in pre-LHC data, validated by its successful application to several diffractive processes, is used to predict the total and total-inelastic proton-proton cross sections at the LHC. The prediction for the total-inelastic cross section at a center of mass collision energy of 7 TeV is compared with recent results from ATLAS and CMS.

Motivation & Objective

• To address the lack of reliable Monte Carlo simulations for diffractive processes at the LHC, which affects trigger setup, underlying event corrections, and acceptance calculations.
• To provide a phenomenologically grounded, unitarized model that incorporates saturation effects observed in pre-LHC single diffraction data.
• To predict total and inelastic $pp$ cross sections at LHC energies using a saturated Froissart bound with experimentally determined parameters.
• To validate the model against emerging LHC data, particularly the $\sigma_{\text{inel}}$ measurements from ATLAS and CMS at $\sqrt{s} = 7$ TeV.

Proposed method

• The model employs a saturated Froissart bound with $\sigma_t \sim \ln^2(s/s_F)$, where $s_F = 22$ GeV is the saturation scale determined from pre-LHC data.
• The coefficient $C = \pi / s_o$ is derived from the pion mass scale, with $s_o$ determined from experimental fits to single-diffractive data.
• The total cross section is calculated via $\sigma_t^{\text{LHC}} = \sigma_t^{\text{CDF}} + \frac{\pi}{s_o} \left[ \left(\ln \frac{s^{\text{LHC}}}{s_F} \right)^2 - \left(\ln \frac{s^{\text{CDF}}}{s_F} \right)^2 \right]$.
• Elastic and inelastic cross sections are derived using the ratio $R_{\text{el/t}} = \sigma_{\text{el}} / \sigma_t$ from a global fit to existing data.
• The model avoids reliance on Regge theory or complex unitarization schemes by directly embedding saturation effects into the amplitude structure.
• The model is validated by comparing predictions to ATLAS and CMS measurements of $\sigma_{\text{inel}}$ at $\sqrt{s} = 7$ TeV.

Experimental results

Research questions

• RQ1 Can a saturation-based phenomenological model reliably predict total and inelastic $pp$ cross sections at LHC energies without relying on standard Monte Carlo simulations?
• RQ2 How does the inclusion of a saturated Froissart bound with $s_F = 22$ GeV improve the description of diffractive processes compared to Regge-based models?
• RQ3 To what extent do the model's predictions for $\sigma_{\text{inel}}$ at $\sqrt{s} = 7$ TeV agree with ATLAS and CMS measurements?
• RQ4 What is the impact of unitarity and saturation on the $s$-dependence of $\sigma_t$, $\sigma_{\text{el}}$, and $\sigma_{\text{inel}}$ at high energies?

Key findings

• The model predicts $\sigma_{\text{inel}} = 71 \pm 6$ mb at $\sqrt{s} = 7$ TeV, consistent with the ATLAS result of $69.4 \pm 2.4$ (exp.) $\pm 6.9$ (extr.) mb.
• The prediction for $\sigma_{\text{inel}}$ at $\sqrt{s} = 8$ TeV is $72 \pm 6$ mb, and at $\sqrt{s} = 14$ TeV it is $76 \pm 8$ mb.
• The total cross section at $\sqrt{s} = 14$ TeV is predicted as $109 \pm 12$ mb, falling within the range of other theoretical predictions.
• The model's prediction for $\sigma_{\text{inel}}$ at 7 TeV is in good agreement with the CMS result of $66.8 \leq \sigma_{\text{t}}^{\text{inel}} \leq 74.8$ mb.
• The model's $\sigma_t$ prediction at $\sqrt{s} = 14$ TeV is consistent with the global fit value of $114 \pm 5$ mb.
• The model successfully incorporates saturation effects and unitarity constraints without requiring additional tuning beyond two experimentally determined parameters: $s_F = 22$ GeV and $s_o$.