Geometric bias and centrality dependence of jet quenching in high-energy nuclear collisions

1. Problem Statement

High-energy heavy-ion collisions (e.g., Pb+Pb at $\sqrt{s_{NN}} = 5.02$ TeV) create a Quark-Gluon Plasma (QGP), characterized by the suppression of high-transverse momentum ($p_T$) hadrons, known as jet quenching. This suppression is quantified by the nuclear modification factor, $R_{AA}$.

A persistent challenge in theoretical modeling is the accurate description of $R_{AA}$ across different centrality classes (collision impact parameters). While models tuned to reproduce data in central collisions (small impact parameter) often work well, they tend to overestimate $R_{AA}$ in peripheral collisions (large impact parameter). In highly peripheral collisions, the QGP is dilute, and jet-medium interactions should be negligible, yet experiments still observe significant suppression ($R_{AA} < 1$).

The paper posits that this discrepancy arises from a geometric bias effect. Standard models (like the Monte Carlo Glauber model) assume that every inelastic nucleon-nucleon (NN) collision produces a constant number of hard partonic scatterings. However, in reality, the number of hard scatterings per NN collision depends on the NN impact parameter ($b_{NN}$). In peripheral AA collisions, the average $b_{NN}$ is larger than in unbiased $pp$ collisions, leading to fewer hard scatterings per binary collision. This initial-state geometric bias suppresses the jet yield independently of QGP interactions, a factor often ignored in standard $R_{AA}$ calculations.

2. Methodology

The authors developed a comprehensive framework combining an improved initial condition model with an enhanced jet transport model.

A. HIJING-Based Initial Condition Model

To address the geometric bias, the authors developed a model based on the HIJING (Heavy-Ion Jet Interaction Generator) event generator, which accounts for the impact parameter dependence of NN scattering.

• Impact Parameter Dependence: Unlike the standard MC-Glauber model (which assumes a hard cutoff $b_{NN} < b_0$ and exactly one hard scattering per inelastic collision), the HIJING-based model calculates the probability of inelastic scattering and the number of hard scatterings ($N_{hard}$) as continuous functions of $b_{NN}$.
• Cross Sections: It utilizes a dipole form factor for the nucleon thickness function and separates soft and hard cross sections ($\sigma_{soft}$ and $\sigma_{hard}$).
• Geometric Bias Factor ($R^{bias}_{AA}$): The model calculates the ratio of the average number of hard scatterings in a centrality-biased AA collision to that in an unbiased NN collision, normalized by the number of binary collisions ($N_{coll}$).
  $$R^{bias}_{AA} = \frac{\langle N^{hard}_{AA} \rangle}{\langle N_{coll} \rangle \langle \langle N^{hard}_{NN} \rangle \rangle}$$
  This factor deviates from 1 in peripheral collisions, quantifying the initial-state suppression.

B. Improved Linear Boltzmann Transport (LBT) Model

The authors utilized the LBT model to simulate jet-QGP interactions but introduced critical improvements to handle peripheral (small system) collisions:

• "Fake" Parton Scheme: In the LBT model, negative partons (representing energy depletion in the QGP) are replaced by "fake" partons to allow hadronization in Pythia.
  • Previous Issue: Assigning low momentum to these fake partons caused unphysical enhancements in $R_{AA}$ ($>1$) in peripheral collisions due to altered string configurations in Pythia.
  • Improvement: The authors assigned a large longitudinal momentum ($p_z = 10^4$ GeV) to fake partons. This mimics the connection to beam remnants in $pp$ collisions, restoring the baseline $R_{AA} \approx 1$ when QGP effects are weak.
• Negative Parton Hadronization: Instead of connecting uncorrelated negative partons into strings (which led to over-suppression), the authors sampled negative partons from a thermal distribution and boosted them to the global frame, enforcing energy conservation more rigorously.

C. Simulation Framework

• Initial State: Hard partons are generated using Pythia 8, with spatial distributions determined by either the standard MC-Glauber or the new HIJING-based model.
• Evolution: Partons evolve via vacuum showers, interact with the QGP (generated by the 3+1D CLVisc hydrodynamic model) via the LBT model, and then undergo further vacuum showers and hadronization.
• Final Calculation: The total $R_{AA}$ is calculated as the product of the geometric bias factor and the medium modification factor:
  $$R_{AA} = R^{bias}_{AA} \times R^{med}_{AA}$$

3. Key Contributions

1. Quantification of Geometric Bias: The study explicitly demonstrates that the suppression of high-$p_T$ hadrons in peripheral Pb+Pb collisions is largely driven by the initial-state geometry (fewer hard scatterings per binary collision due to dilute nucleon overlap) rather than strong jet-medium interactions.
2. Model Development:
   • Creation of a HIJING-based initial condition model that captures the $b_{NN}$ dependence of hard scattering probabilities.
   • Improvement of the LBT model's "fake parton" scheme to eliminate unphysical $R_{AA} > 1$ artifacts in small systems.
3. Unified Description: The combination of the geometric bias correction and the improved jet transport model allows for a simultaneous, satisfactory description of charged hadron $R_{AA}$ from central (0-10%) to highly peripheral (70-90%) collisions.

4. Results

• Geometric Bias Factor: The calculated $R^{bias}_{AA}$ is close to 1 in central collisions but drops significantly in peripheral collisions (e.g., dropping below 0.8 in the 70-90% centrality class). This drop mirrors the experimental suppression observed in peripheral data.
• Peripheral Collisions (70-90%):
  • Without the geometric bias correction, the model overestimates $R_{AA}$ (predicting values closer to 1 or even $>1$ after fixing the fake parton issue).
  • Including $R^{bias}_{AA}$ brings the theoretical prediction into excellent agreement with CMS experimental data, showing that the observed suppression is primarily a geometric effect.
• Central Collisions (0-10%): The geometric bias is negligible ($R^{bias}_{AA} \approx 1$). The suppression here is correctly attributed to parton energy loss in the dense QGP.
• Initial Vertex Distribution: While the HIJING-based model produces a more sparse distribution of hard collision vertices in peripheral collisions compared to the standard Glauber model, this difference has a negligible impact on the final $R_{AA}$ because parton energy loss is already weak in these dilute systems.

5. Significance

• Resolving the Peripheral Discrepancy: The paper resolves a long-standing issue where theoretical models failed to describe the centrality dependence of jet quenching, specifically the "excess" suppression in peripheral collisions.
• Re-evaluating Small Systems: The findings suggest that the suppression of high-$p_T$ particles in small systems (peripheral AA, pA, dA) should not be immediately interpreted as evidence of QGP formation or strong jet quenching. Instead, a significant portion of this suppression is a geometric bias inherent to the collision topology.
• Methodological Advancement: By separating the initial-state geometric bias from the final-state medium modification, the authors provide a more rigorous framework for extracting QGP properties (like the jet transport coefficient $\hat{q}$) from experimental data.
• Future Directions: The work highlights the need for further refinements in handling cold nuclear matter effects and centrality selection biases, and provides public access to the updated LBT and HIJING-based initial condition codes for the community.

In conclusion, the study establishes that geometric bias is a dominant factor in the suppression of high-$p_T$ hadrons in peripheral heavy-ion collisions, necessitating its inclusion in any quantitative analysis of jet quenching to avoid misinterpreting the properties of the QGP in small collision systems.