Fluid-Dynamic Heavy-Quark Diffusion in QGP

• The paper demonstrates that the fluid-dynamic model integrates kinetic theory and lattice QCD to derive transport coefficients for heavy-quark diffusion.
• It employs a second-order Israel–Stewart evolution equation to capture out-of-equilibrium charm quark dynamics, aligning with experimental D-meson spectra.
• The approach validates hydrodynamic approximations across various system sizes by solving the conservation laws for energy–momentum and heavy-quark current.

A fluid-dynamic description of heavy-quark diffusion provides a macroscopic, causal, and coupled framework to model the evolution and transport of heavy quarks—most notably charm and bottom—in the quark–gluon plasma (QGP) created in relativistic nuclear collisions. In this approach, heavy quarks, produced primarily in the earliest stages of the collision, are treated as almost conserved charges that diffuse through, and thermalize with, the rapidly expanding QGP. This diffusion is characterized by transport coefficients, such as the spatial diffusion constant and associated relaxation times, which are determined via microscopic QCD processes and lattice calculations. By integrating kinetic theory and hydrodynamics, this framework allows the computation of experimentally accessible observables, such as the momentum distributions of heavy-flavor hadrons, across both large and small collision systems (Capellino et al., 29 Aug 2025).

1. Fluid-Dynamic Model: Mapping Transport Theory to Hydrodynamics

The fluid-dynamic model is built upon the conservation of the energy–momentum tensor $T^{\mu\nu}$ and the net heavy-quark current $N^{\mu}$: $$\begin{align} \nabla_\mu T^{\mu\nu} &= 0, \ \nabla_\mu N^{\mu} &= 0, \end{align}$$ where $T^{\mu\nu}$ represents the medium's energy and momentum evolution and $N^{\mu}$ records the transport and diffusion of heavy quarks (charm, in the context of recent applications (Capellino et al., 29 Aug 2025, Capellino et al., 2023, Capellino et al., 2023)). In the Landau frame, $T^{\mu\nu}$ and $N^{\mu}$ decompose as: $$T^{\mu\nu} = (\epsilon + p) u^\mu u^\nu + \Delta^{\mu\nu}(p + \Pi) + \pi^{\mu\nu}, \quad N^{\mu} = n u^{\mu} + \nu^{\mu},$$ where $u^\mu$ is the local flow four-velocity, $n$ denotes the charm density, and $\nu^{\mu}$ is the dissipative diffusion current (orthogonal to $u^\mu$). The tensors $\pi^{\mu\nu}$ and $\Pi$ encode shear and bulk viscous corrections, respectively, and $\Delta^{\mu\nu} = g^{\mu\nu} + u^\mu u^\nu$ projects onto the spatial directions orthogonal to $u^\mu$.

This construction amounts to a hydrodynamic extension where the heavy-quark sector is treated analogously to baryon number or electric charge, thereby allowing a coupled, macroscopic evolution of heavy-flavor observables (Capellino et al., 2022).

2. Heavy-Quark Diffusion: Second-Order Evolution Equation

To capture the out-of-equilibrium evolution of the heavy-quark sector, a second-order hydrodynamic evolution equation of the Israel–Stewart type is used for the diffusion current: $$\tau_n \, \Delta^{\alpha}_{\ \beta} u^\mu \nabla_\mu \nu^\beta + \nu^\alpha + \kappa_n \Delta^{\alpha\beta} \partial_\beta \alpha = 0,$$ where:

• $\tau_n$ is the heavy-quark diffusion relaxation time,
• $\kappa_n$ is a diffusion coefficient proportional to the spatial diffusion constant $D_s$,
• $\alpha$ is the ratio of chemical potential to temperature, $\alpha = \mu/T$,
• The equation is projected to ensure $\nu^\mu u_\mu = 0$.

This equation is solved alongside the conservation laws for $T^{\mu\nu}$ and $N^\mu$ within a relativistic fluid-dynamic code (often employing boost-invariant, azimuthally symmetric coordinates for heavy-ion collisions). The underlying assumption, justified by microscopic and lattice studies, is that $|\nu^r| \ll n$ everywhere, so the heavy-quark diffusion current remains a small correction to the convective flow (Capellino et al., 29 Aug 2025).

3. Microscopic Foundations and Input Parameters

The transport coefficients entering the fluid-dynamic equation, notably the heavy-quark spatial diffusion constant $D_s$ and relaxation time $\tau_n$, are determined via kinetic theory, lattice QCD, or Bayesian extraction from experimental data:

• Kinetic theory relates $D_s$ and $\tau_n$ to drag and diffusion coefficients computed from the Fokker–Planck equation under many-soft-collision approximations (Capellino et al., 2022).
• Lattice QCD provides nonperturbative calculations of the heavy-quark momentum diffusion parameter $\kappa$, which determines $D_s = 2T^2/\kappa$. Recent results for $D_s$ are lower than previous quenched or purely perturbative estimates, supporting fast hydrodynamization of charm (Altenkort et al., 2023).
• Bayesian analyses of charm-hadron $p_T$ spectra can further constrain $D_s$ in phenomenological fluids (Capellino et al., 29 Aug 2025).

A typical implementation sets $D_s$ based on these theoretical and phenomenological inputs, with $\kappa_n$ and $\tau_n$ determined consistently.

4. Results: Momentum Distributions, Relaxation, and System Size

Solving the coupled equations for $T^{\mu\nu}$ and $N^\mu$ yields space-time evolution for temperature, flow, equilibrium charm density $n$, and the diffusion field $\nu^\mu$ in heavy-ion collisions of varying system size (Au–Au at RHIC, O–O and Ne–Ne at LHC, etc.):

• For Au–Au at $\sqrt{s_{NN}} = 200$ GeV, the model successfully reproduces D-meson transverse momentum distributions up to $p_T \sim 3$ GeV using experimentally constrained $D_s$.
• Even with small absolute numbers of charm pairs ($\sim 2$–3 around midrapidity), the relaxation dynamics ensure $|\nu^r| \ll n$ at all times, validating the hydrodynamic approximation (Capellino et al., 29 Aug 2025).
• In small systems (O–O, Ne–Ne), key fluid observables ($p$, $\pi^\phi_\phi$) demonstrate the dominance of equilibrium pressure, and the heavy-quark sector continues to satisfy $|\nu^r| \ll n$ throughout the entire QGP phase.

The coupled evolution code therefore captures both the collective bulk responses and the subtle diffusion of heavy-flavor observables across rapidly expanding, radially inhomogeneous collision systems.

5. Coupling and Conservation

A central aspect is the self-consistent, coupled evolution of both the bulk and heavy-quark sectors, with charm current conservation ($\nabla_\mu N^{\mu} = 0$) solved in tandem with the energy–momentum equations. Local temperature and flow fields from the bulk directly steer the diffusion and spatial evolution of charm density, while gradients of the chemical potential $\alpha$ (tied to the charm EoS) set the driving force for $\nu^\mu$. Nonequilibrium dynamics (encoded in $\nu^\mu$) are dynamically regulated by the competition between the diffusion coefficient $\kappa_n$ and relaxation time $\tau_n$.

Initial conditions reflect hard-process charm production profiles and are normalized via the Glauber model; subsequent evolution preserves net charm number and propagates the spectrum through the QGP until hadronization. The approach is robust in settings with few or many charm pairs, and even as system size or temperature varies, provided the near-equilibrium condition persists (Capellino et al., 2023, Capellino et al., 29 Aug 2025).

6. Physical Implications and Applicability

This fluid-dynamical methodology has several far-reaching consequences:

• Rapid Hydrodynamization: The model shows that charm quarks can approach local kinetic equilibrium (hydrodynamize) on timescales short compared to QGP evolution, supported quantitatively by both lattice QCD and $p_T$ spectra analyses (Altenkort et al., 2023, Capellino et al., 2023, Capellino et al., 29 Aug 2025).
• Range of Validity: Applicability extends across large and small system sizes (including as few as $\sim$two charm pairs per rapidity unit) and over a range of collision energies. The key requirement is a sufficiently small ratio $|\nu^r|/n$, verified in all tested configurations.
• Bridging Micro and Macro: The method connects kinetic theory and lattice QCD inputs (such as $D_s$ and $\kappa$) to hydrodynamical observables (charm spectra, yields, potentially also heavy-quark flow coefficients), enabling first-principles constraints on transport properties from experimental heavy-ion data (Capellino et al., 29 Aug 2025).
• Predictions and Experimental Relevance: The accurate reproduction of low- to moderate-$p_T$ D-meson spectra strengthens the case for a hydrodynamic treatment of heavy flavor, and motivates future studies of beauty quarks and further experimental campaigns in small collision systems.

7. Summary Table: Key Equations and Quantities

Equation / Quantity	Definition / Role	Reference
$\nabla_\mu T^{\mu\nu} = 0$	Bulk energy–momentum conservation equation	(Capellino et al., 29 Aug 2025)
$\nabla_\mu N^{\mu} = 0$	Heavy-quark current conservation	(Capellino et al., 29 Aug 2025)
$N^\mu = n u^\mu + \nu^\mu$	Decomposition of heavy-quark current	(Capellino et al., 2023)
$$\tau_n \Delta^{\alpha}_{\ \beta} u^\mu\nabla_\mu\nu^\beta + \nu^\alpha + \kappa_n \Delta^{\alpha\beta}\partial_\beta\alpha = 0$$	Second-order diffusion evolution	(Capellino et al., 29 Aug 2025)
$D_s = 2T^2/\kappa$	Relation of spatial diffusion to momentum diffusion coefficient	(Altenkort et al., 2023)
$|\nu^r| \ll n$	Condition for validity of diffusion approximation	(Capellino et al., 29 Aug 2025)

8. Outlook

The presented framework unifies fluid-dynamic modeling of the QGP bulk with a consistent treatment of heavy-quark diffusion, making it well suited to bridge theoretical computations with experimental measurements of heavy-flavor observables. The success in reproducing D-meson spectra in both large and small systems, the robustness of the diffusion approximation for low charm density, and the direct mapping to both kinetic-theory and lattice-QCD derived transport coefficients all point to a mature, quantitative toolset for studying the collective and transport properties of heavy quarks in hot QCD matter (Capellino et al., 29 Aug 2025, Capellino et al., 2023, Capellino et al., 2023).