Bayesian Inference in Jet Quenching

• Bayesian inference in jet quenching is a rigorous statistical approach that combines theoretical models with high-precision heavy-ion collision data.
• It integrates multi-stage simulations, Gaussian Process emulators, and multi-observable analyses to extract key parameters like the jet transport coefficient.
• The methodology reduces uncertainties by addressing geometric biases, external fields, and error correlations in quantitative jet tomography.

Bayesian inference in jet quenching constitutes a rigorous statistical framework for extracting properties of the quark–gluon plasma (QGP) by systematically connecting theoretical models of parton energy loss and momentum broadening to high-precision experimental measurements from heavy-ion collisions. The field has evolved from parameter estimation in simple models to large-scale, multi-stage and multi-observable analyses that now constrain the temperature, energy, and flavor dependence of jet–medium interactions, quantify theory uncertainties, and refine the extraction of the jet transport coefficient ("$\hat{q}$")—a key parameter controlling the in-medium modification of energetic partons.

1. Theoretical Foundations and Jet Quenching Observables

Jet quenching is characterized by the suppression and modification of high-energy hadron and jet spectra in nucleus–nucleus versus proton–proton collisions. The fundamental observable is the nuclear modification factor,

$$R_{AA}(p_T) = \frac{dN_{AA}/dyd^{2}p_T}{T_{AA}(\vec{b})\,d\sigma_{pp}/dyd^{2}p_T},$$

which directly probes in-medium partonic energy loss mechanisms. Theoretical approaches relate $R_{AA}$ to the underlying probability distribution of parton energy loss via a convolution with the vacuum cross-section, as in

$$\sigma^{\rm med}(p_T) = \int d\epsilon \, D(\epsilon)\, \sigma^{\rm vac}(p_T+\epsilon),$$

where $D(\epsilon)$ is the "quenching weight" describing the probability for a parton to lose energy $\epsilon$ in the medium (Falcão et al., 21 Nov 2024).

In QCD, the jet transport coefficient $\hat{q}$ quantifies the average squared transverse momentum broadening per unit length and is computed within various theoretical frameworks, including perturbative QCD (pQCD) (e.g., higher-twist, SCET_G), nonperturbative lattice QCD, and holographic models based on AdS/CFT duality (Panero et al., 2013, Chien et al., 2015, Feal et al., 2019, Zhu et al., 31 May 2025).

2. Bayesian Inference Methodologies Applied to Jet Quenching

Bayesian inference provides a principled mechanism for updating prior beliefs about model parameters $\theta$, given new data $x$, through the posterior probability:

$P(\theta|x) = \frac{P(x|\theta)P(\theta)}{P(x)},$

where $P(\theta)$ is the prior, $P(x|\theta)$ the likelihood, and $P(x)$ the normalization or evidence. In jet quenching, $\theta$ can represent parameters governing $\hat{q}(T, E, Q)$, energy-loss distribution shapes, or geometric and soft-sector properties (Ehlers, 29 Jul 2025).

Key components found in state-of-the-art applications include:

• Modular jet quenching simulations (e.g., JETSCAPE), often with multi-stage approaches (MATTER for high-virtuality, LBT for low-virtuality regimes) (Cao et al., 2021, Mulligan et al., 2021, Ehlers et al., 2022, Ehlers et al., 8 Jan 2024).
• Gaussian Process (GP) emulators for surrogate modeling of expensive simulations, often trained on Latin hypercube or active learning design points (Xie et al., 2022, Ehlers et al., 15 Aug 2024, Falcão et al., 21 Nov 2024).
• Multivariate normal likelihoods that rigorously incorporate experimental systematic error covariance matrices, and careful paper of the impact of error correlations and data selection (Soltz et al., 4 Dec 2024).
• Full Markov Chain Monte Carlo (MCMC) posterior sampling, with advanced techniques such as parallel tempering and principal component analysis (PCA) on observable sets.

The flexibility in parameterization is critical: explicit functional forms for $\hat{q}(T)$ can restrict sensitivity to the actual, data-driven, temperature dependence. Non-parametric "information field" (IF) priors, modeling $F(x) = \log(\hat{q}/T^3)$ as a Gaussian random field, provide greater robustness and eliminate artificial long-range correlations (Xie et al., 2022, Xie et al., 2022).

3. Jet Transport Coefficient Extraction and Temperature Dependence

The extraction of $\hat{q}(T)$ is central to quantitative jet tomography. Analyses using both parametric forms,

$$\frac{\hat{q}(E,T)}{T^3} = 42\, C_R\, \frac{\zeta(3)}{\pi}\left(\frac{4\pi}{9}\right)^2 \left\{ \frac{A[\ln(E/\Lambda) - \ln B]}{[\ln(E/\Lambda)]^2} + \frac{C[\ln(E/T) - \ln D]}{[\ln(ET/\Lambda^2)]^2} \right\},$$

and nonparametric IF approaches now consistently find that $\hat{q}/T^3$ decreases with increasing temperature from $T_c$ to a few times $T_c$ (e.g., from $\sim5$ near $T_c$ to $\sim1$ at $3T_c$) (Mulligan et al., 2021, Xie et al., 2022, Xie et al., 2022). The posterior credible bands narrow with the inclusion of more central collision and higher energy data, reflecting the enhanced constraining power as the accessible $T$ range increases.

The IF prior approach ensures that constraints at low $T$ (from e.g. peripheral or low-$\sqrt{s}$ collisions) do not artificially inform unconstrained high-$T$ behavior, avoiding prior-induced bias (Xie et al., 2022, Xie et al., 2022).

4. Energy Loss Distributions, Flavor Hierarchies, and Universality

Modern Bayesian analyses factorize the heavy-ion cross section into a convolution of the energy-loss distribution with the vacuum reference, employing flexible parameterizations (gamma, log-normal, or normal distributions) for the quenching weight $D(\epsilon)$ (He et al., 2018, Falcão et al., 21 Nov 2024). Fits to inclusive jet, $\gamma$+jet, and $b$-jet data demonstrate extraction of flavor-dependent energy loss, supporting the expected hierarchy

$$\langle\Delta E_g\rangle > \langle\Delta E_q\rangle > \langle\Delta E_b\rangle,$$

where gluon-initiated jets lose more energy than light quarks, and bottom quark jets, due to their mass and color factors, lose the least (Zhang et al., 2023, Zhang et al., 2022). The width parameter of the loss distribution is related to the effective number of out-of-cone scatterings (He et al., 2018).

Universality of the quenching weight is supported: well-constrained $D(\epsilon)$ extracted from inclusive jet observables accurately predicts photon-tagged jet spectra, and vice versa, as demonstrated by posterior predictive checks and leave-one-out analyses (Falcão et al., 21 Nov 2024).

A significant result is the extraction of a color scaling ratio, $$C_R = \langle\epsilon_g\rangle/\langle\epsilon_q\rangle \approx 3.5$$, exceeding the naive Casimir factor ($N_c/C_F \approx 2.25$), pointing to the relevance of multi-parton quenching and nonlinear effects for realistic high-$p_T$ jets (Falcão et al., 21 Nov 2024).

5. Multi-Stage and Multi-Observable Bayesian Calibrations

Current methodologies emphasize simultaneous calibrations using inclusive hadron, reconstructed jet, and jet substructure data (e.g., fragmentation functions, groomed observables $z_g$ and $R_g$) (Ehlers et al., 8 Jan 2024, Ehlers et al., 15 Aug 2024). These "multi-messenger" analyses exploit the different sensitivities of observables to stages of the jet shower and geometric configurations in the QGP, allowing extraction of $\hat{q}$ as a function of $T$, $E$, and jet substructure variables.

Tensions have been identified: extractions using only low-$p_T$ hadron data tend to support larger $\hat{q}$, while those anchored to high-$p_T$ hadron or jet data yield lower values. The inclusion of jet substructure observables improves constraints on $\hat{q}$ and lessens sensitivity to such tensions. This suggests model deficiencies, such as incomplete treatment of nuclear shadowing, low-$p_T$ dynamics, or missing theory uncertainties, rather than a failure of the Bayesian methodology itself (Ehlers et al., 8 Jan 2024, Ehlers et al., 15 Aug 2024, Ehlers, 29 Jul 2025).

6. Incorporating Geometry, Magnetic Field, and Chemical Potential

Recent analyses extend Bayesian inference to incorporate bulk geometry and external fields. The use of event-by-event geometry from TRENTO models allows path-length dependence and geometric biases to be accounted for in calibrations (Soltz et al., 4 Dec 2024). Investigations of holographic energy loss models with a background magnetic field $B$ and baryon chemical potential $\mu_B$ reveal that $\hat{q}$ (or its analogs) exhibits a strong negative correlation with these external scales after data calibration, emphasizing that observations of jet quenching are sensitive to their combined effect (Zhu et al., 31 May 2025).

7. Computational and Methodological Developments

The high computational cost of jet quenching simulations has driven the adoption of active learning—where new simulation points are selected based on maximal emulator variance— and transfer learning to accelerate Gaussian process construction (Ehlers et al., 15 Aug 2024, Ehlers, 29 Jul 2025). Treatment of error correlations in the experimental data is crucial: fully covariant error matrices can bias Monte Carlo fits and the resulting best-fit functions unless handled appropriately (addressing, e.g., Peele’s Pertinent Puzzle). Strategies such as log-transformation of data and the model can mitigate biases caused by correlated relative errors (Soltz et al., 4 Dec 2024).

8. Significance, Impact, and Open Questions

Bayesian inference has emerged as the standard for quantitative jet tomography, enabling extraction of $\hat{q}(T, E, Q)$ and energy-loss distributions with rigorous uncertainty quantification and direct data-model comparison. Lessons highlight the sensitivity of results to parametrization choices, evidence for universality of the energy-loss distribution, and the need for broader observable sets and improved theory uncertainty quantification (Ehlers, 29 Jul 2025). Future directions include joint calibrations of soft and hard sector observables, exploration of additional external medium properties (e.g., vorticity, electromagnetic fields), and systematic multi-model comparisons under uniform computational protocols.

The field continues to evolve toward a comprehensive, precision-oriented understanding of jet–medium interactions, integrating multiple theoretical frameworks, observables, and experimental data through Bayesian inference as a unifying statistical paradigm.