Non-equilibrium Dynamical Attractors and Thermalisation of Charm Quarks in Nuclear Collisions at the LHC Energy

Abstract

We study the non-equilibrium dynamics, thermalisation and attractor behaviour of charm quarks in a longitudinally expanding Quark-Gluon Plasma within the Relativistic Boltzmann Transport approach in 1+1D Bjorken expansion. Considering both a strong AdS/CFT coupling scenario with constant $2πT D_s=1$ and a temperature-dependent diffusion coefficient $D_s^\text{lQCD}(T)$ from the recent unquenched lattice QCD data, we analyse the evolution of effective temperature, momentum moments and distribution functions for different initial conditions, including FONLL and EPOS4HQ spectra. We find that charm quarks exhibit dynamical attractors; however, the temperature dependence of $D_s^\text{lQCD}(T)$ leads to significantly longer relaxation times compared to the strong coupling limit. While dynamical attractors occur within $\sim 1-1.5 \rm \,fm$ for $2πT D_s=1$, they are delayed to $\sim 5 \rm \,fm$ for $D_s^\text{lQCD}(T)$, becoming comparable to the lifetime of the Quark-Gluon Plasma phase in ultra-relativistic collisions. This indicates that charm quarks may not fully thermalise, especially in small systems such as peripheral or light-ion collisions. We further show that, for $D_s^\text{lQCD}(T)$, the deviation from equilibrium becomes as large as $δf_{HQ}/f_{eq} \sim p_T^β\sim \mathcal{O}(1)$ already at $p_T\simeq 3\rm\, GeV$, rising with $β\sim 4.5$, thus questioning the applicability of viscous hydrodynamics to charm dynamics.

Figures (5)

• Figure 1: Blue circles are unquenched lQCD data HotQCD:2025fbd, blue solid line: fit to lQCD data by a tuned effective coupling $g(T)$ in a Quasi-particle model (see text), referred in the text as $D_s^\text{lQCD}(T)$. Red dot-dashed line and uncertainty band: $2\pi TD_s$ of extended version of Quasi Particle Model (QPMp) Sambataro:2024mkrSambataro:2025obe. Cyan band: AdS/CFT estimate Gubser:2006qhHorowitz:2015dtaCasalderrey-Solana:2006fio. Green band: bayesian analysis from Ref. Xu:2017obm. Symbols report the quenched lQCD data taken from Ref.s Banerjee:2011raKaczmarek:2014jgaFrancis:2015daaBrambilla:2019tptBrambilla:2020siz.
• Figure 2: The time evolution of the effective temperature of the charm sector with $2\pi TD_s=1$ (left) and $D^\text{lQCD}_s(T)$ (right) for different initial conditions (coloured dashed lines) compared to the bulk temperature (black solid line).
• Figure 3: Left panels: time evolution of moments for $2\pi TD_s=1$; right panels: time evolution of moments for $D_s^\text{lQCD}$. Top panels: The anisotropy of energy momentum tensor $2T_{zz}/(T_{xx}+T_{yy})$. Middle and bottom panels: time evolution of the normalised moments $\overline M^{12}$ and $\overline M^{22}$. Different initial conditions for the charm sector (coloured dash lines) are compared to the bulk one (black solid line).
• Figure 4: Evolution of the stress tensor anisotropy (left panel) and of the normalised moments $\overline M^{21}$ (middle panel) and $\overline M^{22}$ (right panel) of the charm sector in terms of the scaled time $\tau/\tau_{eq}$. Different colours refer to different interaction regimes, from $2\pi T D_s=1$ to $2\pi T D_s=5$, including also $D_s^\text{lQCD}(T)$; dashed lines refer to the 3D-th $T_0=0.5$ GeV initial condition; solid lines to the FONLL initial distribution. The blue solid line report the bulk moments' evolution, while the grey solid line the bulk attractor curve.
• Figure 5: $\delta f(p_T) /f_{eq}(p_T)$ for different initial charm distribution functions (FONLL, EPOS4HQ, Boltzmann) for $2\pi TD_s=1$ (left panels) and $D_s^\text{lQCD}(T)$ (right panels) at different time from top ($\tau=0.2$ fm), middle ($\tau=0.4$ fm, i.e. after the initial strong longitudinal expansion) to bottom ($\tau= 8\,\rm fm)$.