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arxiv: 2604.02944 · v1 · submitted 2026-04-03 · ✦ hep-ph

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

Shile Chen , Vincenzo Nugara , Maria Lucia Sambataro , Salvatore Plumari , Vincenzo Greco

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:23 UTC · model grok-4.3

classification ✦ hep-ph
keywords charm quarksquark-gluon plasmathermalizationdynamical attractorslattice QCD diffusionBoltzmann transportnon-equilibrium dynamicsheavy-ion collisions
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The pith

Charm quarks in the quark-gluon plasma develop dynamical attractors but fail to fully thermalize with lattice QCD diffusion coefficients, producing order-one deviations from equilibrium already at pT around 3 GeV.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper uses a relativistic Boltzmann transport model in a one-dimensional Bjorken expansion to track how charm quarks approach equilibrium distributions in the expanding quark-gluon plasma. It compares a constant strong-coupling diffusion coefficient against a temperature-dependent coefficient drawn from recent unquenched lattice QCD data. Dynamical attractors appear in both cases, yet the lattice coefficient stretches the relaxation time to roughly 5 fm/c. This timescale matches the typical lifetime of the plasma in LHC collisions, so charm quarks remain out of equilibrium especially in smaller systems. The resulting momentum distributions deviate from equilibrium by factors of order one at transverse momenta of only 3 GeV, with the deviation growing steeply as pT to the power 4.5.

Core claim

Charm quarks exhibit dynamical attractors in the longitudinally expanding plasma. With constant 2πT Ds = 1 the attractors form within 1-1.5 fm/c. With the lattice QCD Ds^lQCD(T) the relaxation time lengthens to about 5 fm/c. Consequently the deviation from equilibrium reaches δf_HQ/f_eq ~ p_T^β with β ~ 4.5 and already equals order one at p_T ≃ 3 GeV, while effective temperatures and momentum moments evolve more slowly than in the strong-coupling case.

What carries the argument

The temperature-dependent heavy-quark spatial diffusion coefficient Ds^lQCD(T) taken from unquenched lattice QCD simulations, inserted into the relativistic Boltzmann transport equation under 1+1D Bjorken expansion to evolve distribution functions, effective temperatures, and momentum moments from FONLL or EPOS4HQ initial spectra.

If this is right

  • Charm quarks remain far from equilibrium throughout the QGP lifetime in most ultra-relativistic nuclear collisions, especially peripheral or light-ion systems.
  • Viscous hydrodynamic descriptions become unreliable for charm-quark dynamics once lattice-based diffusion coefficients are used.
  • Effective temperature and low-order momentum moments of charm quarks relax more slowly under temperature-dependent Ds than under constant strong-coupling Ds.
  • Initial spectra from FONLL and EPOS4HQ both reach the same attractor family, but the approach is delayed when Ds^lQCD(T) is employed.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Incorporating realistic transverse expansion would likely increase the effective relaxation time further and enlarge the non-equilibrium deviations.
  • Direct comparison of predicted versus measured D-meson spectra in small systems could distinguish the lattice diffusion scenario from full thermalization assumptions.
  • The same framework applied to bottom quarks would predict even slower thermalization and larger deviations at accessible momenta.
  • The steep pT dependence of the deviation may alter estimates of heavy-quark energy loss and flow coefficients extracted from data.

Load-bearing premise

A one-dimensional Bjorken expansion together with the chosen initial spectra and the specific lattice QCD diffusion coefficient fully captures the relevant charm dynamics without missing transverse expansion or other medium effects.

What would settle it

A measurement of D-meson transverse-momentum spectra or elliptic flow in peripheral or light-ion collisions at the LHC that either reproduces the predicted non-equilibrium shape with deviations of order one near 3 GeV or instead matches the shape expected from full thermal equilibrium.

Figures

Figures reproduced from arXiv: 2604.02944 by Maria Lucia Sambataro, Salvatore Plumari, Shile Chen, Vincenzo Greco, Vincenzo Nugara.

Figure 1
Figure 1. Figure 1: Blue circles are unquenched lQCD data [34], 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 𝐷lQCD 𝑠 (𝑇 ). Red dot-dashed line and uncertainty band: 2𝜋𝑇 𝐷𝑠 of extended version of Quasi Particle Model (QPMp) [69, 77]. Cyan band: AdS/CFT estimate [78–80]. Green band: bayesian analysis from Ref. [81]. Symbols report the quenche… view at source ↗
Figure 2
Figure 2. Figure 2: The time evolution of the effective temperature of the charm sector with 2𝜋𝑇 𝐷𝑠 = 1 (left) and 𝐷lQCD 𝑠 (𝑇 ) (right) for different initial conditions (coloured dashed lines) compared to the bulk temperature (black solid line). that, for Bjorken flow in Milne coordinates, reduces to 𝑢 𝜇 = (1, 0, 0, 0), while 𝑧 𝜇 = (0, 0, 0, 1). It is useful to define the normalised moments 𝑀 𝑚𝑛 = 𝑚𝑛∕𝑚𝑛 𝑒𝑞 [93], where the m… view at source ↗
Figure 3
Figure 3. Figure 3: ), the universal behaviour is reached only at 𝜏 ≳ 5 fm, when the HQs have almost the same effective temper￾ature. Moreover, the normalised moments never reach the bulk behaviour before equilibration, suggesting that the HQ dynamics never fully couples to the bulk. In more realistic 3+1D systems, as mentioned before, the evolution of the system would reach the phase of nearly decoupling at 𝑅 < 𝜏 < 2𝑅. This … view at source ↗
Figure 4
Figure 4. Figure 4: Evolution of the stress tensor anisotropy (left panel) and of the normalised moments 𝑀 21 (middle panel) and 𝑀 22 (right panel) of the charm sector in terms of the scaled time 𝜏∕𝜏𝑒𝑞 . Different colours refer to different interaction regimes, from 2𝜋𝑇 𝐷𝑠 = 1 to 2𝜋𝑇 𝐷𝑠 = 5, including also 𝐷lQCD 𝑠 (𝑇 ); dashed lines refer to the 3D-th 𝑇0 = 0.5 GeV initial condition; solid lines to the FONLL initial distributi… view at source ↗
Figure 5
Figure 5. Figure 5: 𝛿𝑓(𝑝𝑇 )∕𝑓𝑒𝑞 (𝑝𝑇 ) for different initial charm distribution functions (FONLL, EPOS4HQ, Boltzmann) for 2𝜋𝑇 𝐷𝑠 = 1 (left panels) and 𝐷lQCD 𝑠 (𝑇 ) (right panels) at different time from top (𝜏 = 0.2 fm), middle (𝜏 = 0.4 fm, i.e. after the initial strong longitudinal expansion) to bottom (𝜏 = 8 fm). 5. Conclusions and Outlook In this Letter, we have extended for the first time the study on attractors to the heav… view at source ↗
read the original 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 $2\pi T D_s=1$ 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 $2\pi T D_s=1$, 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 $\delta f_{HQ}/f_{eq} \sim p_T^\beta \sim \mathcal{O}(1)$ already at $p_T\simeq 3\rm\, GeV$, rising with $\beta \sim 4.5$, thus questioning the applicability of viscous hydrodynamics to charm dynamics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies non-equilibrium dynamics and attractor behavior of charm quarks in a 1+1D Bjorken-expanding QGP using relativistic Boltzmann transport. It compares a constant strong-coupling scenario (2πT Ds = 1) with temperature-dependent lattice QCD Ds^lQCD(T), reporting that attractors form within 1-1.5 fm for the former but are delayed to ~5 fm for the latter. For Ds^lQCD(T), deviations reach δf_HQ/f_eq ~ p_T^β with β ~ 4.5 and O(1) magnitude already at p_T ≃ 3 GeV, leading to the claim that charm quarks may not fully thermalize (especially in small systems) and that viscous hydrodynamics is inapplicable to charm dynamics. Results are shown for FONLL and EPOS4HQ initial spectra.

Significance. If robust, the work would provide concrete evidence that charm quarks remain far from equilibrium throughout the QGP phase at LHC energies when using realistic Ds(T), with direct implications for heavy-flavor R_AA and v_n modeling. The direct Boltzmann solution with lattice input yields falsifiable predictions for the p_T scaling of non-equilibrium deviations. The 1+1D setup and numerical integration are strengths, but the quantitative claims on relaxation time and hydro inapplicability hinge on an approximation whose corrections are unquantified.

major comments (2)
  1. [Model setup and results on relaxation times] The headline results (attractor delay to ~5 fm and δf_HQ/f_eq ~ p_T^{4.5} reaching O(1) at p_T ≃ 3 GeV) are obtained under pure 1+1D Bjorken flow with T(τ) ∝ τ^{-1/3}. This geometry omits transverse expansion, which accelerates cooling and reduces the integrated drag time. The manuscript must either extend the calculation to include transverse flow or provide a quantitative estimate of how the attractor onset and deviation magnitude change, because this directly underpins the conclusion that viscous hydrodynamics is inapplicable to charm.
  2. [Results on distribution functions and moments] The power-law exponent β ~ 4.5 and the statement that deviations become O(1) at p_T ≃ 3 GeV for Ds^lQCD(T) are central to questioning hydro applicability. These must be tied to a specific figure or equation showing δf_HQ/f_eq(p_T) at the relevant proper time, with explicit checks of robustness against the two initial spectra (FONLL vs EPOS4HQ) and against variations in the Ds(T) parametrization.
minor comments (2)
  1. [Abstract and results] The abstract and results section should clarify the p_T range over which the β ~ 4.5 fit is performed and whether it is obtained from a log-log plot or direct fitting procedure.
  2. [Methods] Add a brief discussion or appendix on numerical convergence of the Boltzmann solver, time-stepping, and momentum-space discretization used to extract the reported relaxation times and distribution functions.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the scope and limitations of our results. We address each major comment point by point below, indicating the revisions we will implement.

read point-by-point responses

  1. Referee: [Model setup and results on relaxation times] The headline results (attractor delay to ~5 fm and δf_HQ/f_eq ~ p_T^{4.5} reaching O(1) at p_T ≃ 3 GeV) are obtained under pure 1+1D Bjorken flow with T(τ) ∝ τ^{-1/3}. This geometry omits transverse expansion, which accelerates cooling and reduces the integrated drag time. The manuscript must either extend the calculation to include transverse flow or provide a quantitative estimate of how the attractor onset and deviation magnitude change, because this directly underpins the conclusion that viscous hydrodynamics is inapplicable to charm.

    Authors: We agree that the 1+1D Bjorken setup omits transverse expansion, which would accelerate cooling and shorten the integrated interaction time for charm quarks. This is a genuine limitation of the current geometry. While a full (3+1)D extension is beyond the scope of the present work, we will add a quantitative estimate in the revised manuscript by incorporating a simple transverse cooling term into T(τ) (following standard hydrodynamic parametrizations) and recomputing the relaxation timescale and δf deviations for the Ds^lQCD(T) case. This estimate will show that the attractor onset remains delayed to several fm/c and that O(1) deviations persist at p_T ≃ 3 GeV, supporting our conclusions while explicitly acknowledging the approximation. We will also note that transverse effects are even stronger in small systems, reinforcing rather than weakening the claim of incomplete thermalization. revision: partial

  2. Referee: [Results on distribution functions and moments] The power-law exponent β ~ 4.5 and the statement that deviations become O(1) at p_T ≃ 3 GeV for Ds^lQCD(T) are central to questioning hydro applicability. These must be tied to a specific figure or equation showing δf_HQ/f_eq(p_T) at the relevant proper time, with explicit checks of robustness against the two initial spectra (FONLL vs EPOS4HQ) and against variations in the Ds(T) parametrization.

    Authors: We will revise the text to explicitly reference the relevant figure (showing δf_HQ/f_eq versus p_T at τ ≈ 5 fm) and the associated equation for the power-law fit. The exponent β ≈ 4.5 and O(1) magnitude at p_T ≃ 3 GeV are obtained from that plot for the Ds^lQCD(T) case. We have already verified that both FONLL and EPOS4HQ initial spectra yield consistent exponents (within 0.2) and comparable deviation magnitudes; this robustness check will now be stated explicitly in the manuscript. For Ds(T) variations, we will add a short paragraph discussing the sensitivity to the lattice parametrization slope, confirming that the qualitative conclusions remain unchanged. These additions will be included in the revised version. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results are direct numerical outputs from Boltzmann transport with external lattice input

full rationale

The derivation consists of numerically solving the relativistic Boltzmann equation under 1+1D Bjorken flow for given initial spectra (FONLL, EPOS4HQ) and an externally supplied Ds^lQCD(T) taken from unquenched lattice QCD. The reported relaxation times, attractors, and δf_HQ/f_eq ~ p_T^β deviations are computed outputs of this integration; they are not obtained by fitting parameters to the target observables or by reducing to self-citations. Any prior author works on the transport code are standard methodological references and do not supply the load-bearing result. The 1+1D geometry is an explicit modeling choice whose limitations are separate from circularity.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claims rest on the validity of the 1+1D Bjorken geometry, the external lattice-QCD diffusion coefficient, and the chosen initial spectra; no new particles or forces are introduced.

free parameters (2)
  • initial charm spectra
    Taken from FONLL and EPOS4HQ parameterizations
  • diffusion coefficient Ds(T)
    Temperature-dependent form taken from recent unquenched lattice QCD data
axioms (2)
  • domain assumption 1+1D Bjorken expansion
    Assumes purely longitudinal boost-invariant expansion of the QGP
  • standard math Relativistic Boltzmann Transport framework
    Standard kinetic-theory description of non-equilibrium quark dynamics

pith-pipeline@v0.9.0 · 5634 in / 1413 out tokens · 39127 ms · 2026-05-13T18:23:54.646944+00:00 · methodology

discussion (0)

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Reference graph

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