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Soft-hard factorization of heavy-quark transport in QCD matter at finite chemical potential

Jiale Lou, Wu Wang, Jiazhen Peng, Fei Sun, Kejun Wu, Wei Xie, Zuman Zhang, Shuang Li, Sa Wang

TL;DR

This work addresses heavy-quark transport in hot and dense QCD matter at finite chemical potential by extending the soft-hard factorization model to include $\mu$-dependent Debye screening and fermionic distributions. It develops both full numerical formalism and high-energy analytic approximations (HEA), showing that the collisional energy loss $-\mathrm{d}E/\mathrm{d}z$ and momentum diffusion coefficients $\kappa_T$, $\kappa_L$ are enhanced as $\mu$ increases, with soft and hard channels contributing logarithmic $\mu$-dependent corrections. The leading μ-dependent terms are $\sim \mu^{2}\ln(|t^{*}|/M_D^{2})$ from soft gluonic exchanges and $\sim \mu^{2}\ln(E_1 T/|t^{*}|)$ from hard quark scatterings, and the $t^{*}$-dependence cancels in the full result, ensuring consistency. The findings highlight the importance of baryon-density effects for heavy-flavor observables in RHIC BES, FAIR, and NICA and provide a baseline for future refinements including lattice-informed screening masses and inelastic processes.

Abstract

We calculate the collisional energy loss and momentum diffusion coefficients of heavy quarks traversing a hot and dense QCD medium at finite quark chemical potential, $μ\neq0$. The analysis is performed within an extended soft-hard factorization model (SHFM) that consistently incorporates the $μ$-dependence of the Debye screening mass $M_D(μ)$ and of the fermionic thermal distribution functions. Both the energy loss and the diffusion coefficients are found to increase with $μ$, with the enhancement being most pronounced at low temperatures where the chemical potential effects dominate the medium response. To elucidate the origin of this dependence, we derive analytic high-energy approximations in which the leading $μ$-corrections appear as logarithmic terms: a soft logarithm $\simμ^{2}\ln(|t^{*}|/M_{D}^{2})$ from $t$-channel scattering off thermal gluonic excitations, and a hard logarithm $\simμ^{2}\ln(E_{1}T/|t^{*}|)$ from scattering off thermal quarks. In the complete result the dependence on the intermediate separation scale $t^{\ast}$ cancels, as required. We also confirm the expected mass hierarchy $-dE/dz(charm)<-dE/dz(bottom)$ at fixed velocity. Our findings demonstrate that finite chemical potential plays a significant role in heavy-quark transport and must be included in theoretical descriptions of heavy-flavor dynamics in baryon-rich environments, such as those probed in the RHIC Beam Energy Scan, and at FAIR and NICA.

Soft-hard factorization of heavy-quark transport in QCD matter at finite chemical potential

TL;DR

This work addresses heavy-quark transport in hot and dense QCD matter at finite chemical potential by extending the soft-hard factorization model to include -dependent Debye screening and fermionic distributions. It develops both full numerical formalism and high-energy analytic approximations (HEA), showing that the collisional energy loss and momentum diffusion coefficients , are enhanced as increases, with soft and hard channels contributing logarithmic -dependent corrections. The leading μ-dependent terms are from soft gluonic exchanges and from hard quark scatterings, and the -dependence cancels in the full result, ensuring consistency. The findings highlight the importance of baryon-density effects for heavy-flavor observables in RHIC BES, FAIR, and NICA and provide a baseline for future refinements including lattice-informed screening masses and inelastic processes.

Abstract

We calculate the collisional energy loss and momentum diffusion coefficients of heavy quarks traversing a hot and dense QCD medium at finite quark chemical potential, . The analysis is performed within an extended soft-hard factorization model (SHFM) that consistently incorporates the -dependence of the Debye screening mass and of the fermionic thermal distribution functions. Both the energy loss and the diffusion coefficients are found to increase with , with the enhancement being most pronounced at low temperatures where the chemical potential effects dominate the medium response. To elucidate the origin of this dependence, we derive analytic high-energy approximations in which the leading -corrections appear as logarithmic terms: a soft logarithm from -channel scattering off thermal gluonic excitations, and a hard logarithm from scattering off thermal quarks. In the complete result the dependence on the intermediate separation scale cancels, as required. We also confirm the expected mass hierarchy at fixed velocity. Our findings demonstrate that finite chemical potential plays a significant role in heavy-quark transport and must be included in theoretical descriptions of heavy-flavor dynamics in baryon-rich environments, such as those probed in the RHIC Beam Energy Scan, and at FAIR and NICA.

Paper Structure

This paper contains 10 sections, 74 equations, 7 figures.

Table of Contents

  1. INTRODUCTION
  2. Formalism
  3. Soft-hard factorization model at finite chemical potential
  4. Collisional energy loss and rransport coefficients of heavy quarks
  5. Soft region ($-t < -t^{\ast}$)
  6. Hard region ($-t > -t^{\ast}$)
  7. Energy loss in the High-Energy Approximation (HEA)
  8. Results and discussions
  9. Summary
  10. Derivation of the energy loss in soft region [Eq. (\ref{['eq:ELoss_Soft_vsX']})]

Figures (7)

  • Figure 1: (Color online) Left (a): comparison of the energy loss $dE/dz$ as a function of heavy-quark momentum, for charm quark with $\alpha_{s}=0.3$, $-t^{\ast}=8M_{D}^{2}$ and $\mu=0.3~\rm{GeV}$ at a given temperature $T=0.5~{\rm GeV}$, contributed by soft interactions in $t$-channel [dot-dashed cyan curve; Eq. (\ref{['eq:ELoss_Soft_vsX']})], and hard interactions in $t$-channel [long-dashed black curve; Eq. (\ref{['eq:ELoss_Hard']})], $su$-channels [dotted pink curve; Eq. (\ref{['eq:dEdz_Hard_su_Perturbative']})]. The combined results, i.e. the contributions from the soft and hard interactions in $t$-channels (dashed blue curve) and from the all components (solid red curve), are shown for comparison. Right (b): same as panel-a but for $dE/dz$ as a function of temperature at a given heavy-quark energy $E=80~{\rm GeV}$.
  • Figure 2: (Color online) Left (a): comparison of the total energy loss $dE/dz$ of a charm quark as a function of its momentum, displaying separately the results with different chemical potential values: $\mu=0$ (long-dashed black curve), 0.3 GeV (dotted blue curve) and 0.5 GeV (solid red curve). Right (b): comparison of $dE/dz$ obtained at finite chemical potential ($\mu\ne0$) with respect to the one at vanishing chemical potential ($\mu=0$).
  • Figure 3: Same as Fig. \ref{['fig:Charm_dEdz_AllChan_variousMu_vsP']} but as a function of temperature.
  • Figure 4: (Color online) Comparison of charm quark energy loss based on the full calculations ["Full"; thick curves; Eqs. (\ref{['eq:ELoss_Soft_vsX']}), (\ref{['eq:ELoss_Hard']}) and (\ref{['eq:dEdz_Hard_su_Perturbative']})] and the high-energy approximation ["HEA"; thin curves; Eqs. (\ref{['eq:ELoss_Soft_HEA']}), (\ref{['eq:ELoss_Hard_Qq_HEA']}), (\ref{['eq:ELoss_Hard_Qg_t_HEA']}) and (\ref{['eq:ELoss_Hard_suCh_Def']})] as a function of heavy-quark momentum. The associated results are shown as thick and thin curves, respectively. Various contributions to the energy loss are displayed separately as curves with different styles.
  • Figure 5: (Color online) Comparison of $dE/dz$ for charm and bottom quarks at fixed coupling $\alpha_{s}=0.3$, given temperature $T=0.5~{\rm GeV}$ and chemical potential $\mu=0.3~{\rm GeV}$.
  • ...and 2 more figures