⚛️ phenomenology

Out-of-equilibrium contributions to charm hadrons in a fluid-dynamic approach

This paper extends the fluid-dynamic description of charm quarks in the quark-gluon plasma by incorporating out-of-equilibrium corrections from both the initial free-streaming phase and the freeze-out surface, enabling precise calculations of charm hadron yields and momentum distributions to systematically determine transport coefficients while validating the model against experimental data and defining its kinematic limits.

Original authors: Rossana Facen, Federica Capellino, Eduardo Grossi, Andrea Dubla, Silvia Masciocchi

Published 2026-07-02

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CC0 1.0

Original authors: Rossana Facen, Federica Capellino, Eduardo Grossi, Andrea Dubla, Silvia Masciocchi

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a massive, ultra-hot soup made of the fundamental building blocks of matter, created when heavy atoms smash into each other at nearly the speed of light. Scientists call this soup the Quark-Gluon Plasma (QGP). Inside this soup, there are "heavy" particles called charm quarks. Because they are so heavy, they don't just float around randomly; they interact with the soup in a specific way, kind of like a bowling ball moving through a pool of water.

For a long time, scientists have tried to describe how these heavy particles move using fluid dynamics—the same math used to predict how water flows in a river or air moves over a wing. However, there's a catch: the standard fluid math assumes everything is perfectly balanced and calm (in "equilibrium"). But in a violent collision, things are messy and out of balance.

This paper is about fixing that math to account for the "messiness." Here is a breakdown of what the authors did, using simple analogies:

1. The Two "Messy" Moments

The authors realized that to get an accurate picture, they needed to fix the math at two specific times when things are most chaotic:

The Start (The "Free-Stream" Phase):
Imagine the charm quarks are like arrows shot from a bow. Before they hit the water (the soup), they fly through the air in a straight line. This is called "free-streaming."
- The Old Way: Previous models assumed the arrows instantly started swimming in the water as soon as they appeared.
- The New Way: This paper calculates exactly what happens during that flight through the air. They found that this flight creates a specific "push" or current before the quarks even hit the soup. It's like the arrows have built-up momentum that changes how they first hit the water.
The End (The "Freeze-Out" Phase):
Eventually, the hot soup cools down and turns back into solid particles (like steam turning into water droplets). This is called "freeze-out."
- The Old Way: Scientists assumed the particles just popped out of the soup perfectly balanced.
- The New Way: The authors added a correction factor for the "messiness" at the moment the soup freezes. They realized that if you ignore this messiness, your math predicts that the number of particles depends on how "thick" or "sticky" the soup is in a way that doesn't make physical sense. By adding the correction, the math finally makes sense: the total number of particles stays consistent, no matter how sticky the soup is.

2. The "Sticky" Soup (The Diffusion Coefficient)

A key part of this research is measuring how "sticky" the soup is. In physics, this is called the spatial diffusion coefficient ( $D_s$ ).

Think of it like honey vs. water. If the soup is like honey, the heavy charm quarks move slowly and struggle to get through. If it's like water, they move more freely.
The authors' new math allows them to calculate exactly how many particles come out of the soup for different levels of "stickiness." This is crucial because it lets scientists use real-world data to figure out exactly how sticky the QGP soup really is.

3. The "Negative" Problem (The Limit of the Model)

Here is the most interesting limitation the authors discovered. Their new math works great, but only up to a point.

The Analogy: Imagine you are trying to describe a crowd of people. You have a "base" number of people (the calm, normal crowd) and a "correction" number (the people running around wildly).
The Issue: If the crowd gets too wild (if the "stickiness" of the soup is too high), the number of people running around becomes so large that, when you subtract it from the calm crowd in your math, you end up with a negative number of people.
The Result: You can't have negative people! This means the math breaks down. The authors found that for very "sticky" soups, their model only works for particles moving at lower speeds. If the particles are moving too fast, the math predicts impossible results.

4. Checking Against Reality

The authors compared their new, corrected math with real data from the Large Hadron Collider (LHC).

The Good News: When they used their new corrections, their predictions for the number of particles and their speeds matched the real experimental data very well.
The One Glitch: They noticed a slight mismatch with a specific type of particle (a baryon called $\Lambda_c$ ). They suspect this is because their list of "possible particles" was missing some rare, excited states (like a car with extra parts attached). When they artificially boosted the weight of these missing states in their model, the match became perfect.

Summary

In short, this paper is about cleaning up the math used to describe heavy particles in a hot, chaotic soup.

They fixed the math for the start (accounting for the initial flight of the particles).
They fixed the math for the end (accounting for the chaotic freeze-out).
They proved that without these fixes, the math gives impossible answers.
They discovered that the math has a speed limit: if the soup is too sticky or the particles are too fast, the math breaks down and predicts "negative particles."

This work doesn't just give a better picture of the past; it provides the necessary tools for scientists to accurately measure the properties of the universe's hottest matter in future experiments.

Technical Summary: Out-of-equilibrium contributions to charm hadrons in a fluid-dynamic approach

Problem Statement
Charm quarks serve as unique probes for the Quark-Gluon Plasma (QGP) due to their large mass, which ensures they are produced primarily in initial hard scatterings and do not reach thermal equilibrium via thermal production within the fireball's lifetime. Previous fluid-dynamic frameworks successfully modeled charm quark dynamics by assuming proximity to local thermal equilibrium. However, these models neglected out-of-equilibrium corrections at two critical stages: the initial moment of the collision (after the free-streaming phase) and the freeze-out hypersurface. The omission of freeze-out corrections led to unphysical dependencies of charm hadron multiplicities on the spatial diffusion coefficient ( $D_s$ ), preventing a systematic determination of charm transport coefficients. Furthermore, the initial distribution was often assumed to be in local thermal equilibrium, ignoring the non-equilibrium nature of the initial charm production.

Methodology
The authors extend a previously developed fluid-dynamic framework for charm quarks in the QGP by explicitly computing out-of-equilibrium corrections at both the initial stage and the freeze-out surface.

Initial Stage Initialization:
- The model introduces a free-streaming phase between the charm production time and the hydrodynamic initialization time ( $\tau_0$ ).
- Using the method of characteristics, the authors analytically solve the kinetic equations for particles traveling in straight-line trajectories during this phase.
- This approach modifies the initial charm density and generates a non-zero diffusion current ( $\nu^\mu_q$ ) at $\tau_0$ , derived from the spatial dependence of the binary collision density ( $n_{coll}$ ) and the momentum distribution of the initial hard scatterings.
- Landau matching conditions are employed to fix the integral of the total charm distribution function in terms of the differential production cross-section.
Freeze-out Corrections (Multi-species Approach):
- The authors employ a multi-species approach within the Hadron Resonance Gas of charm (HRGc), treating the system as a fluid composed of all charm hadron species interacting with light QGP components.
- The out-of-equilibrium correction to the distribution function ( $\delta f_{i,p}$ ) for each hadron species $i$ is derived using the method of moments and the Navier-Stokes approximation.
- The correction is expanded in irreducible moments, truncated at rank $l=1$ (diffusion current), assuming bulk pressure and shear-stress contributions from charm hadrons are negligible compared to light components.
- A key analytical result (Eq. 23) expresses the correction as:
  $\delta f_{i,p} = f^{(0)}_{i,p} \frac{1}{P_i} \bar{\kappa}_i \nu^\mu_q p_{\langle \mu \rangle}$
  where $\bar{\kappa}_i$ is a normalized diffusion coefficient dependent on the species' charge and density, and $\nu^\mu_q$ is the total charm diffusion current. This formulation allows the evolution of only the total current rather than individual species currents.
Evolution and Spectra:
- The system is evolved using a 1+1D fluid-dynamic framework (FluiduM) with constant $D_s T$ .
- Invariant yields and transverse momentum ( $p_T$ ) spectra are computed via the Cooper-Frye procedure, incorporating the derived out-of-equilibrium corrections.
- The model is tested against experimental data from ALICE (Pb-Pb collisions at $\sqrt{s_{NN}} = 5.02$ TeV, 0–10% centrality) using both FONLL pQCD predictions and ALICE-measured charm cross-sections as inputs.

Key Contributions and Results

Resolution of Unphysical Dependencies: The inclusion of out-of-equilibrium corrections at freeze-out resolves the unphysical dependence of charm hadron multiplicities on the spatial diffusion coefficient $D_s$ . Without these corrections, the multiplicity varied significantly with $D_s$ ; with them, the integrated yields remain consistent with the non-diffusive case and the total number of initial charm quarks across the entire range of $D_s$ tested.
Initial Diffusion Current: The free-streaming initialization generates a diffusion current at $\tau_0$ that is an order of magnitude larger than currents produced at late times. However, the study finds that this initial condition has a negligible impact on the subsequent evolution of the charm fields and the final invariant yields compared to a zero-initial-current scenario.
Model-Data Comparison:
- Using the FONLL cross-section, the model reproduces the integrated yields of charm mesons within uncertainties.
- A tension is observed in the $\Lambda_c^+$ baryon yield, which is underpredicted. Following previous literature, the authors apply a "charm-baryon enhancement" by tripling the statistical weights of excited charm baryons. With this adjustment and using the ALICE-measured cross-section, the model achieves compatibility with experimental yields and $p_T$ spectra up to $p_T \approx 3$ GeV/c.
Limits of Applicability: The paper identifies a fundamental limit to the fluid-dynamic approach based on the positivity of the distribution function.
- The out-of-equilibrium correction $\delta f_{i,p}$ can become negative (due to the negative radial profile of the diffusion current).
- When $|\delta f_{i,p}| \approx f^{(0)}_{i,p}$ , the total distribution function becomes negative, rendering the physical interpretation invalid.
- This "zero-crossing" of the $p_T$ spectrum occurs at lower $p_T$ values as $D_s T$ increases. For $D_s T \gtrsim 0.2$ , the model yields negative spectra at $p_T \sim 2$ GeV/c.
- Consequently, the framework is reliable only for $D_s T \lesssim 0.1$ across the full $p_T$ range, or for higher $D_s T$ only within a restricted low- $p_T$ region (e.g., $p_T \lesssim 2.5$ GeV/c for $D_s T \approx 0.24$ ).

Significance
The paper establishes a consistent fluid-dynamic framework for charm hadrons that accounts for non-equilibrium effects at both the initial and final stages of the collision. By correcting the unphysical dependence of yields on transport coefficients, the work lays the necessary groundwork for future systematic extractions of the charm diffusion coefficient (e.g., via Bayesian analysis). However, the authors modestly conclude that the current linearized approach is limited by the breakdown of the perturbative assumption ( $|\delta f| \ll f^{(0)}$ ) at higher diffusion coefficients and transverse momenta. They suggest that future work must explore alternative methods, such as the maximum entropy approach, to guarantee a positive distribution function across the entire kinematic range.