Bayesian Inference analysis of jet quenching using inclusive jet and hadron suppression measurements

The JETSCAPE Collaboration employs Bayesian Inference with Active Learning to determine the jet transport parameter $\hat{q}$ in the Quark-Gluon Plasma by simultaneously analyzing inclusive hadron and jet suppression data from RHIC and LHC, revealing systematic tensions that offer new insights into jet transport physics.

The Gist

Imagine you are trying to figure out how thick and sticky a giant, invisible bowl of soup is. But you can't touch the soup, and you can't see inside it. The only way to learn about it is to throw tiny, super-fast marbles into the bowl and watch how they slow down, bounce around, or break apart as they travel through.

This is essentially what the JETSCAPE Collaboration did in this paper, but instead of a bowl of soup, they are studying the Quark-Gluon Plasma (QGP).

Here is the breakdown of their work using simple analogies:

1. The "Soup" (The Quark-Gluon Plasma)

Right after the Big Bang, the universe was a hot, dense soup of fundamental particles called quarks and gluons. Today, scientists recreate this "soup" by smashing heavy atoms (like gold or lead) together at nearly the speed of light in giant machines called particle colliders (the LHC and RHIC).

This soup is weird. It flows like a perfect liquid with almost no friction, but it's also incredibly dense. When a high-energy particle (a "jet") tries to fly through it, the soup resists, slowing the particle down. This slowing down is called "Jet Quenching."

2. The "Marbles" (Jets and Hadrons)

To measure the soup's thickness, scientists shoot high-energy particles into the collision.

• Jets: These are like a shotgun blast of particles. When a high-energy particle hits the soup, it sprays out a cone of other particles.
• Hadrons: These are the individual particles (like pions) that make up that spray.

Scientists measure how many of these particles make it out of the soup compared to how many would make it out if there were no soup (just empty space). If the soup is thick, fewer particles escape. This ratio is called $R_{AA}$.

3. The Mystery: How "Thick" is the Soup?

The key number scientists want to find is called $\hat{q}$ (pronounced "q-hat"). Think of $\hat{q}$ as the "drag coefficient" or the "stickiness" of the soup.

• A high $\hat{q}$ means the soup is very thick and stops particles quickly.
• A low $\hat{q}$ means the soup is thin and particles fly through easily.

The problem? Different scientists have been throwing different types of "marbles" (some looked at the whole spray, some looked at single particles) and using different theories to calculate the stickiness. They kept getting slightly different answers, like a group of people trying to guess the weight of a cow by looking at different parts of it and getting confused.

4. The Solution: A "Super-Computer" Detective (Bayesian Inference)

This paper introduces a new, smarter way to solve the mystery. Instead of guessing, they used a method called Bayesian Inference.

Think of this like a detective solving a case with a massive database:

1. The Theory: They built a complex computer simulation (the JETSCAPE framework) that acts like a virtual universe. It simulates the soup, the collision, and the particles flying through it.
2. The Clues: They gathered every piece of experimental data available from the last few years (729 different measurements of how particles were suppressed).
3. The Guessing Game: The computer starts by guessing a value for the "stickiness" ($\hat{q}$). It runs the simulation.
4. The Comparison: It compares its simulation to the real experimental data.
   • If the simulation matches the data, the guess was good.
   • If it doesn't match, the computer tweaks the guess and tries again.
5. Active Learning: Because running these simulations takes a huge amount of computer power (like running a supercomputer for years), they used a special AI trick called Active Learning. This is like a smart student who knows exactly which questions to ask to learn the most in the shortest time. The computer only runs the simulations that are most likely to teach it something new, saving massive amounts of time.

5. The Big Discovery: It Depends on How You Look

After crunching the numbers, the team found something fascinating.

• The Good News: When they looked at the "stickiness" ($\hat{q}$) using both the full spray (jets) and the individual particles (hadrons) together, they got a consistent picture. The soup behaves like a universal fluid with a specific stickiness that changes depending on the temperature.
• The Tension (The Plot Twist): When they looked only at low-energy particles or only at high-energy particles, the answers started to drift apart.
  • It's as if the soup feels "thicker" to a slow-moving marble than to a fast-moving one, or vice versa.
  • This suggests that our current theory of how the soup interacts with particles might be missing a piece of the puzzle. The "stickiness" might not be a single, simple number; it might depend on the speed and energy of the particle hitting it.

6. Why This Matters

This paper is a huge step forward because it's the first time scientists have combined all the available data into one giant, unified analysis.

• Before: Scientists were arguing over which data to trust.
• Now: They have a single, best-estimate map of the soup's properties.
• The Future: The fact that there are still small disagreements (tensions) between different types of data tells scientists exactly where to look next. It's like finding a crack in a wall; it tells you exactly where to dig to find the treasure (a deeper understanding of the universe).

In summary: The JETSCAPE team used a super-smart, AI-assisted computer detective to analyze every possible clue about how particles get stuck in the early universe's "soup." They found a consistent picture of the soup's thickness, but also discovered that the rules of the game might be more complex than we thought, pointing the way to future discoveries.

Technical Summary

Here is a detailed technical summary of the paper "Bayesian Inference analysis of jet quenching using inclusive jet and hadron suppression measurements" by the JETSCAPE Collaboration.

1. Problem Statement

The primary goal of this research is to precisely determine the jet transport coefficient ($\hat{q}$) of the Quark-Gluon Plasma (QGP). $\hat{q}$ characterizes the average transverse momentum squared transferred from the medium to a high-energy parton per unit path length.

Key challenges addressed in this work include:

• Inconsistency in Previous Constraints: Previous determinations of $\hat{q}$ (often expressed as $\hat{q}/T^3$ to factor out temperature dependence) have yielded inconsistent results depending on whether they used hadron suppression data, jet suppression data, or specific kinematic ranges.
• Theoretical Limitations: Existing models often rely on simplified assumptions about how $\hat{q}$ depends on parton energy ($E$), virtuality ($\mu$), and medium temperature ($T$). Specifically, the convergence of perturbative expansions for $\hat{q}$ is poor, and the treatment of "coherence effects" (how the medium interacts with a parton shower of finite size) varies between models.
• Data Scope: Previous Bayesian analyses by the JETSCAPE collaboration were limited to a subset of inclusive hadron data. This work aims to incorporate all available inclusive hadron and jet $R_{AA}$ (nuclear modification factor) data from RHIC and the LHC to test the universality of $\hat{q}$.

2. Methodology

A. The JETSCAPE Framework

The analysis utilizes the JETSCAPE modular framework, which simulates the full lifecycle of heavy-ion collisions:

1. Bulk Medium Evolution: Uses a 2+1D viscous hydrodynamic model (VISHNU) with initial conditions from the Trento model. Parameters for the bulk medium were previously calibrated using soft-sector observables.
2. Jet Production: Hard scattering and initial state radiation are modeled using Pythia 8.
3. Jet Quenching (Multi-Stage Approach): The energy loss of the parton shower is modeled in two stages based on parton virtuality ($\mu$):
   • High-Virtuality Stage ($\mu > Q_0$): Modeled by the Matter generator. It incorporates coherence effects, where the interaction with the medium is suppressed as virtuality increases.
   • Low-Virtuality Stage ($\mu \le Q_0$): Modeled by the Lbt (Linear Boltzmann Transport) generator, treating the parton as a collection of on-shell particles undergoing multiple scatterings.

B. Theoretical Formulation of $\hat{q}$

The paper introduces a new functional form for $\hat{q}$ that differs from previous JETSCAPE calibrations:

• Virtuality Dependence: Unlike previous models where $\hat{q}/T^3$ increased with virtuality in the high-virtuality stage, this analysis implements a continuous decrease of $\hat{q}$ with increasing virtuality due to coherence effects.
• Equation: The transport coefficient is defined as:
  $$\hat{q}(\mu^2) = f(\mu^2) \hat{q}_{HTL}$$
  where $\hat{q}_{HTL}$ is the Leading Order Hard Thermal Loop result, and $f(\mu^2)$ is a suppression factor that reduces the interaction strength at high virtuality.
• Parameters: The Bayesian inference calibrates the strong coupling constant ($\alpha_s$), the switching scale between stages ($Q_0$), the start time of modification ($\tau_0$), and coherence parameters ($c_1, c_2, c_3$).

C. Bayesian Inference and Active Learning

• Data: The analysis incorporates 729 data points of inclusive hadron and jet $R_{AA}$ from RHIC ($\sqrt{s_{NN}} = 0.2$ TeV) and LHC ($\sqrt{s_{NN}} = 2.76, 5.02$ TeV), covering centralities $0-50%$and$p_T > 9$ GeV/c.
• Active Learning: Due to the computational cost of running millions of simulations, the team employed Active Learning (a machine learning technique). A Gaussian Process Emulator (GPE) was trained on a strategic set of "design points" to interpolate the parameter space efficiently.
• Process: The algorithm iteratively selected new simulation points to maximize information gain (reducing emulator uncertainty) while exploring the parameter space, eventually converging on the posterior distributions of the model parameters.

3. Key Contributions

1. Multi-Observable Calibration: This is the first comprehensive Bayesian calibration of $\hat{q}$ using both inclusive hadron and inclusive jet suppression data simultaneously, rather than relying on hadrons alone.
2. Virtuality-Dependent Coherence: Implementation of a physically motivated, continuous suppression of $\hat{q}$ at high virtuality, moving beyond the "sudden switch" approximation used in earlier models.
3. Systematic Sensitivity Analysis: The authors systematically tested the robustness of the extracted $\hat{q}$ by varying:
   • Observable type (Hadron-only vs. Jet-only).
   • Kinematic ranges (Low-$p_T$ vs. High-$p_T$).
   • Collision centrality (Central vs. Semi-central).
4. Resolution of "Jet Puzzle": The study investigates why different probes (hadrons vs. jets) previously yielded different constraints on $\hat{q}$.

4. Key Results

A. Parameter Constraints

The combined calibration yields the following constraints:

• Coupling ($\alpha_s$): Constrained to $0.3 - 0.5$, peaking near$0.4$.
• Switching Scale ($Q_0$): Predominantly $1 - 2$GeV, with a tail up to$6$ GeV.
• Coherence ($c_3$): A preference for large values of $c_3$ is observed, indicating strong coherence effects at high virtuality.
• Correlations: An anti-correlation is observed between $\alpha_s$ and $Q_0$; a larger $Q_0$ (longer high-virtuality stage) requires a lower $\alpha_s$ to match the data, as the Matter stage emits less radiation than the Lbt stage for the same coupling.

B. $\hat{q}/T^3$ Behavior

• The extracted $\hat{q}/T^3$ increases as temperature ($T$) decreases. This trend is driven by both the data and the underlying physical model.
• The distribution is consistent with previous JETSCAPE and JET collaboration results when evolved to the same virtuality scale, despite different theoretical formulations.

C. Tensions and Inconsistencies

• Hadron vs. Jet: While the "Combined" calibration fits the data well overall, there is a tension between calibrations using hadron-only data versus jet-only data.
  • Hadron-only (dominated by low-$p_T$ data) prefers stronger quenching (higher $\hat{q}/T^3$).
  • Jet-only (and high-$p_T$ hadrons, $p_T > 30$ GeV/c) prefers weaker quenching.
• Kinematic Dependence: The inconsistency suggests that the current theoretical model's dependence of $\hat{q}$ on parton energy ($E$) and virtuality ($\mu$) is not fully accurate. The model struggles to simultaneously describe the suppression of low-$p_T$ hadrons and high-$p_T$ jets with a single universal $\hat{q}$ formulation.
• Centrality: The extracted $\hat{q}/T^3$ shows approximate independence from collision centrality when using jet-only or hadron-only data, suggesting the tension arises from kinematic coverage rather than temperature profile differences.

5. Significance and Conclusion

• Validation of Multi-Stage Models: The study confirms that multi-stage models with virtuality-dependent coherence are necessary to describe the full spectrum of jet quenching data.
• Identification of Theoretical Gaps: The observed tension between low-$p_T$ hadrons and high-$p_T$ jets indicates that the functional dependence of $\hat{q}$ on parton energy requires refinement. The current Leading Order (LO) HTL formulation may be insufficient, and higher-order corrections or alternative non-perturbative approaches are needed.
• Future Directions: The paper highlights the need for:
  • Incorporating higher-order corrections to the $\hat{q}$ calculation.
  • Developing better methods to quantify theoretical uncertainties within the Bayesian likelihood.
  • Expanding the analysis to include more differential observables (e.g., jet substructure, $\gamma$-hadron correlations).

In summary, this work represents a significant step toward a comprehensive, data-driven understanding of QGP transport properties, successfully integrating a vast array of experimental data while pinpointing specific areas where theoretical models of jet-medium interactions must be improved.