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Treating the Polyakov loop as a temperature-dependent background field suppresses collisional energy loss and momentum diffusion of charm and bottom quarks in a thermal SU(3) gluon plasma, with the strongest effect near the critical tempera

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2026-08-04 10:18 UTC pith:K2RG6EFB

load-bearing objection Plausible qualitative semi-QGP suppression of heavy-quark transport, but the quantitative curves are not controlled because the soft-hard split is run far outside its formal domain; worth refereeing with a demand for sensitivity scans. the 2 major comments →

arxiv 2510.10294 v2 pith:K2RG6EFB submitted 2025-10-11 hep-ph

Perturbative and nonperturbative properties of heavy quark transport in a thermal SU(3) gluon plasma

Jiazhen Peng , Yage Zhen , Jiale Lou , Fei Sun , Kejun Wu , Wei Xie , Zuman Zhang , Shuang Li
show 1 more author
Sa Wang
This is my paper
classification hep-ph PACS 12.38.Mh25.75.-q
keywords heavy quark transportcollisional energy lossmomentum diffusion coefficientPolyakov loop backgroundsemi-QGPgluon plasmasoft-hard factorizationcharm and bottom quarks
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper claims that in a thermal SU(3) gluon plasma, the collisional energy loss and momentum diffusion coefficients of charm and bottom quarks are substantially smaller than standard perturbative QCD estimates once a temperature-dependent Polyakov-loop background field is included. The suppression is strongest just above the critical temperature Tc, and weakens as the plasma becomes more deconfined. The effect comes from the background field acting as an imaginary chemical potential that reduces the occupation of low-momentum gluon modes, thinning the medium's density of color scatterers. This matters because heavy quarks are the main experimental probes of the quark-gluon plasma, and a temperature-dependent suppression changes the interpretation of measured heavy-flavor observables at RHIC and the LHC.

Core claim

Extending the soft-hard factorized model to the semi-QGP regime by coupling it to a temperature-dependent Polyakov-loop background Q(T), the authors compute the scattering rate, collisional energy loss -dE/dz, and transverse/longitudinal momentum diffusion coefficients kappa_T and kappa_L for charm and bottom quarks scattering off thermal gluons. Compared to the same model with Q=0 (the pure perturbative baseline), the background field suppresses all transport coefficients across the temperature range from Tc to about 3 Tc, with the suppression most pronounced near Tc. The mechanism is not a change in the scattering kinematics but a reduction of the effective density of color charges: the ba

What carries the argument

The central object is the temperature-dependent Polyakov-loop background field Q(T), represented as a constant diagonal temporal gluon field, which shifts the Matsubara energies of gluons and quarks by imaginary chemical-potential-like factors iQ. In the double-line color basis this shifts the thermal Bose distribution n_B(E - iQ_ab) and alters the HTL-resummed gluon propagator through Q-dependent Debye masses. The mechanism that carries the argument is the suppression of low-momentum bosonic occupation numbers: fewer soft gluons means fewer scattering partners, and the modified screening reduces the strength of color interactions, producing the computed suppression.

Load-bearing premise

The result rests on the soft-hard factorization of the scattering rate staying valid near Tc even though the formal condition m_D^2 << T^2 fails there; if the factorization breaks, the computed suppression factors are artifacts of the model.

What would settle it

A lattice QCD computation of the heavy-quark momentum diffusion coefficient kappa_T in pure SU(3) gauge theory at T about 1.1 to 1.5 Tc, evaluated with the same alpha_s used here, would falsify the central claim if the result does not lie below the corresponding leading-order perturbative value.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the calculation is right, heavy-quark transport coefficients in a gluonic plasma near Tc are a T-dependent factor below leading-order pQCD values with the same alpha_s.
  • Because the suppression is approximately energy-independent, it acts as a multiplicative medium property rather than a probe-dependent correction, so it should affect all heavy-flavor observables similarly for a given temperature.
  • The stronger reduction of kappa_T than kappa_L modifies the velocity dependence of kappa_T/kappa_L, a quantity that can be compared directly with lattice and other nonperturbative calculations.
  • The framework bridges the high-temperature perturbative regime and the near-critical semi-QGP, giving a single description of energy loss from large to small momentum transfers.
  • The computed coefficients can be fed into Langevin transport models to obtain nuclear modification factors and elliptic flow for heavy-flavor mesons at RHIC and LHC energies.

Where Pith is reading between the lines

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

  • If the suppression is real, the same background field should also reduce the heavy-quark drag coefficient in Langevin simulations, though that coefficient is not explicitly reported here; extracting it from -dE/dz and kappa would make the prediction testable against experimental R_AA and v_2 data.
  • The direction of the effect runs opposite to holographic strong-coupling drag predictions, so comparing this framework with lattice data on kappa near Tc would indicate which nonperturbative mechanism (semi-QGP occupancy suppression vs strong-coupling enhancement) dominates.
  • Since the paper includes only gluonic contributions, adding thermal quarks (a caveat the authors list) could partially compensate the suppression; an extension to full QCD would test how robust the effect is.
  • The Q-dependent Debye mass distinguishes off-diagonal and diagonal gluons, so a lattice measurement of kappa in pure gauge theory could potentially constrain the functional form of Q(T) beyond the specific model used here.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The manuscript extends the soft-hard factorized model (SHFM) for heavy-quark scattering in a thermal gluon plasma to the near-critical semi-QGP region by coupling it to a temperature-dependent SU(3) Polyakov-loop background. It derives Q-modified thermal distributions, modified Debye screening masses, and double-line-basis color factors, and uses these to compute the collisional energy loss, transverse momentum diffusion, and longitudinal momentum diffusion for charm and bottom quarks. The central numerical claim is that near Tc the background field suppresses all transport coefficients relative to the Q=0 perturbative baseline, with the suppression strongest at low T and weak probe-energy dependence, and that the ratio κT/κL is reduced. The analytic Q=0 limits of the background-field formulas are explicitly shown to reduce to the perturbative results, and the color algebra in App. B is presented in detail.

Significance. If the central claim is correct, the paper provides a concrete, tractable model connecting perturbative heavy-quark transport at high temperature with the semi-QGP regime, and makes a falsifiable prediction that near-Tc heavy-quark drag and diffusion in pure SU(3) gauge theory are smaller than pQCD estimates by a temperature-dependent factor. This is of phenomenological interest for heavy-flavor observables and for comparison with lattice computations of heavy-quark transport in gluonic plasma. Strengths of the manuscript include the transparent derivation in the double-line basis, the explicit reduction of the Q≠0 formulas to the Q=0 limit, the absence of any fitting of the output transport coefficients, and the candid statement of limitations (no thermal quarks, no radiative processes). These features make the underlying framework reproducible even though no numerical code is provided.

major comments (2)
  1. [Sec. II.B and Sec. III, Eqs. (19)-(21), Figs. 3-5] The numerical application violates the formal condition for the soft-hard split stated in Sec. II.B: the text requires m_D^2 << -t* << T^2, but the numerics use α_s=0.3 and -t*=4m_D^2. With N_c=3, Eq. (9) gives m_D^2 = 4π α_s T^2 ≈ 3.77 T^2, so -t* ≈ 15.1 T^2. Thus the upper inequality -t* << T^2 fails badly, and the lower inequality m_D^2 << -t* holds only by a factor 4. The soft region therefore extends to momentum transfers of order sqrt(-t*) ≈ 3.9T, where HTL-resummed propagators are not parametrically justified, while the hard region begins at the same scale, leaving no controlled intermediate window. Because the background-field extension (Sec. II.C) changes only the Debye mass and distribution functions and keeps the same t* and kinematics, the Q≠0/Q=0 suppression ratios in Figs. 3-5 could depend on the arbitrary choices of t* and the large α_s. No sensitivity study is presented.
  2. [Sec. II.C, Eqs. (50) and (52)] The nonperturbative Debye-mass modification contains the model input (30/81)(Tc/T)^2 taken from Refs. [41,50]. This coefficient controls the T-dependence of the screening masses in the soft sector and therefore feeds directly into the suppression ratios plotted in Figs. 3-5. The manuscript does not discuss the sensitivity of the results to this coefficient or to the associated mass scale Mg/Tc. Since the central prediction is the magnitude of the near-Tc suppression, the authors should either show that the suppression is robust to plausible variations of this input (e.g., O(1) changes to 30/81) or state the range of validity over which the prediction should be trusted.
minor comments (4)
  1. [Sec. II.C, Eq. (36) and text below Eq. (39)] The statement that at T≈Tc Eq. (36) gives Q≈1/4 and hence ℓ≈1/3 is numerically inconsistent with the displayed formula. Setting T=Tc in Eq. (36) gives Q=2/9≈0.222 and Eq. (39) gives ℓ≈0.449, not ℓ≈1/3. Please correct the text or clarify which approximation is intended.
  2. [Throughout] There are numerous typographical errors that should be corrected: 'Subsituting', 'auther', 'theroy', 'depedent', 'distribuion', 'Combinging', 'bolb' for 'blob', and 'Equantion'. Also, the axis labels in Figs. 2-5 are typeset awkwardly (e.g., 'n Q≠0 Avg,B / n Q=0 B'); please reformat for readability.
  3. [Abstract and Sec. I] The abstract opens with 'thermal QCD medium' and later says 'unified theoretical framework applicable across both high- and low-momentum regimes'. Since the paper explicitly treats only a gluonic plasma and omits radiative processes (as acknowledged in Sec. IV), the wording should be narrowed to 'gluonic medium' and to elastic energy loss, to avoid overstating the scope.
  4. [Sec. II.C.3, after Eq. (65)] The hard-region extension assumes that the background field modifies only distributions and color factors and leaves the vacuum matrix elements unchanged. This is stated and is a reasonable leading-order assumption, but the hard region here begins at sqrt(-t*)≈3.9T, where corrections to this leading-order treatment need not be numerically small. A brief comment discussing the size of expected O(g^2) corrections in this regime would help.

Circularity Check

0 steps flagged

No significant circularity: the suppression is computed from external background-field inputs, not fitted or defined into the transport coefficients.

full rationale

The derivation chain is not circular. The transport coefficients are obtained by inserting the scattering rate (Eq. 1) into Eqs. (2), (4), and (5), with separate soft and hard contributions (Eqs. 19-21, 27-33, 58-60, 66-71). The Q≠0 suppression is not a fitted output: the background field Q(T) (Eq. 36), the Polyakov-loop-modified distributions (Eqs. 44-47), and the modified Debye masses (Eqs. 50, 52) are taken from previous effective-theory inputs (Refs. 40, 41, 50), none of which contain the heavy-quark transport coefficients being computed. The same α_s=0.3, t*=-4m_D^2, and vacuum matrix elements are used in both Q=0 and Q≠0 calculations, so the reported ratios isolate the effect of the modified distributions and screening rather than importing the result. Although Eq. (48) already shows the color-averaged modified distribution is below unity, the energy-loss and diffusion ratios require nontrivial phase-space and spectral-function integrals; they are derived, not assumed. The self-citations to the authors' SHFM papers [18-20] supply standard formulas and matrix elements but are not the source of the central suppression claim. Section IV's caveats (gluon-only plasma, no radiative/LPM processes) and the skeptical concern about the soft-hard hierarchy at α_s=0.3 concern model validity, not circularity: no equation reduces a transport coefficient to a fitted parameter or to the distribution ratio by definition.

Axiom & Free-Parameter Ledger

4 free parameters · 5 axioms · 0 invented entities

The central inputs Q(T), lambda_de, and lambda_h are imported from prior work; no parameter is fitted to the transport coefficients in this paper. The main carried parameters are alpha_s, -t*, and the nonperturbative screening coefficient.

free parameters (4)
  • QCD coupling alpha_s = 0.3
    Fixed input for all numerical results; with this value m_D ~ 1.9T, so the weak-coupling hierarchy m_D^2 << T^2 is violated (Figs. 3-5).
  • soft-hard factorization cutoff -t* = 4 m_D^2
    Chosen by hand to separate soft and hard scattering; no sensitivity study provided and it violates the formal -t* << T^2 condition at alpha_s = 0.3.
  • nonperturbative Debye mass coefficient (30/81)(Tc/T)^2 = 30/81 ~ 0.370
    Carried from Guo-Kuang (Ref. 50) into Eqs. (50) and (52); sets the size of the nonperturbative screening modification near Tc and is not derived in this paper.
  • mass scale Mg/Tc = 2*sqrt(10*pi)/9 ~ 3.73
    Carried from Ref. 41 into the effective potential Eq. (43); determines Q(T) via Eq. (36) and was originally fixed to reproduce lattice data.
axioms (5)
  • standard math HTL resummation and Weldon's relation between ImSigma and scattering rate
    Used in Eqs. (6), (11), and (55) to convert the heavy-quark self-energy imaginary part into the soft scattering rate; standard thermal field theory.
  • domain assumption Pure gluonic plasma: fermionic contributions to screening and scattering are neglected
    Stated in the Introduction and Sec. IV; all distribution functions and propagators are bosonic and thermal quarks are omitted.
  • domain assumption Background field affects transport only through modified thermal distribution functions and Debye masses at leading order; vacuum matrix elements and on-shell dispersion remain unchanged
    Sec. II C, paragraphs following Eq. (43); taken from Refs. 40 and 42. If false, hard-scattering contributions are mis-modeled near Tc.
  • domain assumption The effective potential Eq. (43) with fixed Mg/Tc gives the correct Q(T), Eq. (36), and hence the Polyakov loop
    Taken from Refs. 41, 45, 46; if this potential is wrong, the temperature dependence of the suppression changes.
  • domain assumption Guo-Kuang modified HTL propagators (Eqs. 49-52) with lambda_de and lambda_h remain valid near Tc
    Adopted from Ref. 50; central to the soft-region results and not re-derived here.

pith-pipeline@v1.3.0-alltime-deepseek · 25187 in / 18781 out tokens · 150828 ms · 2026-08-04T10:18:25.068462+00:00 · methodology

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read the original abstract

We investigate the perturbative and nonperturbative aspects of heavy quark transport in a thermal SU(3) gluon plasma. Based on the soft-hard factorized model, we extend the original perturbative framework to the near-critical temperature region, where nonperturbative effects become significant. The transition behavior of the semi-quark-gluon-plasma (semi-QGP) is described via a temperature-dependent background field incorporated in the background field effective theory. By implementing this approach, we quantitatively evaluate the collisional energy loss and momentum diffusion coefficients of charm and bottom quarks as functions of the incoming energy and medium temperature. Our results show a distinct suppression of both the energy loss and the diffusion coefficients relative to conventional perturbative estimates, especially near the critical temperature. This suppression originates from the emergence of a temperature-dependent color background field, which effectively reduces the color charge screening of the medium. These findings provide important theoretical insight into the phenomenology of heavy-flavor probes, offering a unified theoretical framework applicable across both high- and low-momentum regimes.

Figures

Figures reproduced from arXiv: 2510.10294 by Fei Sun, Jiale Lou, Jiazhen Peng, Kejun Wu, Sa Wang, Shuang Li, Wei Xie, Yage Zhen, Zuman Zhang.

Figure 1
Figure 1. Figure 1: FIG. 1. Tree-level Feynman diagrams for the scattering processes [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (Color online) Left (a): comparison of the temperature-dependence of the non-perturbative suppression of the bosonic [PITH_FULL_IMAGE:figures/full_fig_p013_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (Color online) Left (a): comparison of the energy loss [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: displays the ratio RT L = κT /κL as a function of the heavy-quark velocity v for both vanishing (Q = 0) and non-vanishing (Q ̸= 0) background color fields. In the static limit v → 0 the ratio approaches unity, RT L → 1, consistent with isotropic momentum diffusion when longitudinal and transverse fluctuations are indistinguishable. As the velocity increases RT L decreases monotonically in both cases, refle… view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Feynman diagram for the quark self-energy with the gluon HTL-resummed propagator (bolb) in the double line basis. [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: shows the Feynman diagrams for tree-level quark-gluon scattering in different channels in the double line basis. The color factor for the scattering in t-channel (panel-a in [PITH_FULL_IMAGE:figures/full_fig_p020_7.png] view at source ↗

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Reference graph

Works this paper leans on

71 extracted references · 53 linked inside Pith · cited by 1 Pith paper

  1. [1]

    The interaction rate for two-body scattering With the soft-hard factorization model, the scattering rate arises from three kinematic regimes: (1) Soft Region ( −t <−t∗): heavy quark scattering off medium partons via t-channel exchange with small momen- tum transfer; the interference with s/u-channels is neglected; (2) Hard Region ( −t > −t∗): heavy quark ...

  2. [2]

    The total energy loss in the soft-hard factorized approach is given by inserting Eq

    The collisional energy loss and transport coefficients of heavy quarks The heavy quark energy loss per distance traveled is written as − dE dz = Z d3⃗ qdΓ d3⃗ q ω v1 , (2) where, v1 is the velocity of the heavy quark, dΓ/d3⃗ qis the differential scattering rate with respect to the three- momentum transfer ⃗ qand the energy transfer ω. The total energy los...

  3. [3]

    The results in soft region −t <−t∗ It is argued [28, 39] that, a heavy fermion propagating through a hot QED/QCD plasma, the scattering rate can be obtained from the imaginary part of the particle’s self-energy ImΣ, Γsof t (t) (E1, T) = − 1 2E1 ¯nF (E1)T r ( /P 1 + m1) · ImΣ(E1 + iϵ, ⃗ p1) , (6) where, m1 is the mass of the injected fermion. For the heavy...

  4. [4]

    The results in hard region −t >−t∗ In hard collisions, as shown in Fig. 1, the heavy quark transition rate is defined as the rate of collisions with thermal gluon, which changes the heavy quark (gluon) momentum from incoming⃗ p1 (⃗ p2) to outgoing ⃗ p3 = ⃗ p1 −⃗ q(⃗ p4 = ⃗ p2 +⃗ q), ω(⃗ p1, ⃗ q, T) = Z p2 nB(E2)¯nF (E3)¯nB(E4) · vreldσ. (24) We neglect th...

  5. [5]

    (30g) 7 The contributions from s- and u-channels (panel-b and c in Fig

    − E2(s + m2 1) , (30e) c = − t p2 1 t (E1 + E2)2 − s + 4p2 1p2 2sin2ψ , (30f) D = b2 + 4a2c = −t ts + (s − m2 1)2 · 4E2sinψ p1 2 . (30g) 7 The contributions from s- and u-channels (panel-b and c in Fig. 1) are not divergent for small momentum transfers due to the finite heavy quark mass ( mc = 1.5 GeV and mb = 4.75 GeV). Therefore, the su-channels contrib...

  6. [6]

    As discussed in Ref

    Thermal distribution functions and effective propagators in a background field In this part we introduce some useful tricks for computing scattering rate. As discussed in Ref. [40], by expanding the propagators in a mixed representation [47, 48] in the presence of a background field Q ̸= 0, and then comparing with the ones for Q = 0, it is realized that t...

  7. [7]

    By extending Weldon’s model [28, 39] close to the critical temperature, the result with a given color a can be obtained from Eq

    The results in soft region In soft collisions the color-averaged scattering rate of heavy quark becomes ΓQ̸=0;sof t (t) (E1, T) = 1 Nc NcX a=1 ΓQ̸=0;sof t (t) a (E1, T), (54) where, Nc is the color factor of the incoming heavy quark. By extending Weldon’s model [28, 39] close to the critical temperature, the result with a given color a can be obtained fro...

  8. [8]

    In this case Q acts as an imaginary color-dependent chemical potential, its presence alters the thermal weights of both gluons [Eq

    The results in hard region In the hard momentum regime, where all relevant momenta are of order temperature, |⃗ p| ≳ T ≫ gT , the effect of the background temporal gluon field Q ∼T , at leading order in the QCD coupling g, manifests primarily through modifications of the statistical distribution functions. In this case Q acts as an imaginary color-depende...

  9. [9]

    G. D. Moore and D. Teaney, Phys. Rev. C 71, 064904 (2005), hep-ph/0412346

    Pith/arXiv arXiv 2005

  10. [10]

    Haque and M

    N. Haque and M. G. Mustafa, Progress in Particle and Nuclear Physics 140, 104136 (2025)

    2025

  11. [11]

    Caron-Huot and G

    S. Caron-Huot and G. D. Moore, JHEP 02, 081 (2008), arXiv:0801.2173 [hep-ph]

    Pith/arXiv arXiv 2008

  12. [12]

    Banerjee, S

    D. Banerjee, S. Datta, R. Gavai, and P. Majumdar, Phys. Rev. D 85, 014510 (2012), arXiv:1109.5738 [hep-lat]

    Pith/arXiv arXiv 2012

  13. [13]

    H.-T. Ding, A. Francis, O. Kaczmarek, F. Karsch, H. Satz, and W. Soeldner, Phys. Rev. D 86, 014509 (2012), arXiv:1204.4945 [hep-lat]

    Pith/arXiv arXiv 2012

  14. [14]

    Altenkort, D

    L. Altenkort, D. de la Cruz, O. Kaczmarek, R. Larsen, G. D. Moore, S. Mukherjee, P. Petreczky, H.-T. Shu, and S. Stendebach (HotQCD), Phys. Rev. Lett. 132, 051902 (2024), arXiv:2311.01525 [hep-lat]

    Pith/arXiv arXiv 2024

  15. [15]

    Liao and E

    J. Liao and E. Shuryak, Phys. Rev. C 75, 054907 (2007), arXiv:hep-ph/0611131

    Pith/arXiv arXiv 2007

  16. [16]

    Liao and E

    J. Liao and E. Shuryak, Phys. Rev. Lett. 101, 162302 (2008), arXiv:0804.0255 [hep-ph]

    Pith/arXiv arXiv 2008

  17. [17]

    Liao and E

    J. Liao and E. Shuryak, Phys. Rev. Lett. 102, 202302 (2009), arXiv:0810.4116 [nucl-th]

    Pith/arXiv arXiv 2009

  18. [18]

    J. Xu, J. Liao, and M. Gyulassy, JHEP 02, 169 (2016), arXiv:1508.00552 [hep-ph]

    Pith/arXiv arXiv 2016

  19. [19]

    van Hees, M

    H. van Hees, M. Mannarelli, V. Greco, and R. Rapp, Phys. Rev. Lett. 100, 192301 (2008), arXiv:0709.2884 [hep-ph]

    Pith/arXiv arXiv 2008

  20. [20]

    Riek and R

    F. Riek and R. Rapp, Phys. Rev. C 82, 035201 (2010), arXiv:1005.0769 [hep-ph]

    Pith/arXiv arXiv 2010

  21. [21]

    M. He, H. van Hees, and R. Rapp, Prog. Part. Nucl. Phys. 130, 104020 (2023), arXiv:2204.09299 [hep-ph]

    Pith/arXiv arXiv 2023

  22. [22]

    C. P. Herzog, A. Karch, P. Kovtun, C. Kozcaz, and L. G. Yaffe, JHEP 07, 013 (2006), arXiv:hep-th/0605158

    Pith/arXiv arXiv 2006

  23. [23]

    S. S. Gubser, Phys. Rev. D 74, 126005 (2006), arXiv:hep-th/0605182

    Pith/arXiv arXiv 2006

  24. [24]

    B. Chen, X. Chen, X. Li, Z.-R. Zhu, and K. Zhou, Phys. Rev. D 111, 086033 (2025), arXiv:2404.18217 [hep-ph]

    Pith/arXiv arXiv 2025

  25. [25]

    B. Chen, L. Zhu, X. Chen, D. Hou, and X. Chen, (2025), arXiv:2508.16167 [hep-ph]

    arXiv 2025

  26. [26]

    J. Peng, K. Yu, S. Li, W. Xiong, F. Sun, and W. Xie, Phys. Rev. D 109, 096028 (2024), arXiv:2401.10644 [hep-ph]

    Pith/arXiv arXiv 2024

  27. [27]

    S. Li, F. Sun, W. Xie, and W. Xiong, Eur. Phys. J. C 81, 536 (2021), arXiv:2106.08491 [hep-ph]

    Pith/arXiv arXiv 2021

  28. [28]

    J. Lou, W. Wang, J. Peng, F. Sun, K. Wu, W. Xie, Z. Zhang, S. Li, and S. Wang, (2025), arXiv:2509.23872 [hep-ph]

    arXiv 2025

  29. [29]

    X. Dong, Y. Lee, and R. Rapp, Ann. Rev. Nucl. Part. Sci. 69, 417 (2019), arXiv:1903.07709 [nucl-ex]

    Pith/arXiv arXiv 2019

  30. [30]

    Tang, Z.-B

    Z. Tang, Z.-B. Tang, W. Zha, W.-M. Zha, Y. Zhang, and Y.-F. Zhang, Nucl. Sci. Tech. 31, 81 (2020), arXiv:2105.11656 [nucl-ex]

    Pith/arXiv arXiv 2020

  31. [31]

    Chen et al

    J. Chen et al. , Nucl. Sci. Tech. 35, 214 (2024), arXiv:2407.02935 [nucl-ex]

    Pith/arXiv arXiv 2024

  32. [32]

    Acharya et al

    S. Acharya et al. (ALICE), JHEP 01, 174 (2022), arXiv:2110.09420 [nucl-ex]

    Pith/arXiv arXiv 2022

  33. [33]

    Acharya et al

    S. Acharya et al. (ALICE), Eur. Phys. J. C 84, 813 (2024), arXiv:2211.04384 [nucl-ex]

    Pith/arXiv arXiv 2024

  34. [34]

    Braaten and T

    E. Braaten and T. C. Yuan, Phys. Rev. Lett. 2183, 66 (1991)

    1991

  35. [35]

    Blaizot and E

    J.-P. Blaizot and E. Iancu, Phys. Rept. 359, 355 (2002), arXiv:hep-ph/0101103. 22

    Pith/arXiv arXiv 2002

  36. [36]

    Braaten and M

    E. Braaten and M. H. Thoma, Phys. Rev. D 44, 1298 (1991)

    1991

  37. [37]

    Braaten and M

    E. Braaten and M. H. Thoma, Phys. Rev. D 44, R2625 (1991)

    1991

  38. [38]

    Romatschke and M

    P. Romatschke and M. Strickland, Phys. Rev. D 69, 065005 (2004), arXiv:hep-ph/0309093

    Pith/arXiv arXiv 2004

  39. [39]

    Romatschke and M

    P. Romatschke and M. Strickland, Phys. Rev. D 71, 125008 (2005), arXiv:hep-ph/0408275

    Pith/arXiv arXiv 2005

  40. [40]

    Djordjevic, Phys

    M. Djordjevic, Phys. Rev. C 74, 064907 (2006), arXiv:nucl-th/0603066

    Pith/arXiv arXiv 2006

  41. [41]

    Peigne and A

    S. Peigne and A. Peshier, Phys. Rev. D 77, 014015 (2008), arXiv:0710.1266 [hep-ph]

    Pith/arXiv arXiv 2008

  42. [42]

    Peigne and A

    S. Peigne and A. Peshier, Phys. Rev. D 77, 114017 (2008), arXiv:0802.4364 [hep-ph]

    Pith/arXiv arXiv 2008

  43. [43]

    W. M. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Monteno, M. Nardi, and F. Prino, Eur. Phys. J. C 71, 1666 (2011), arXiv:1101.6008 [hep-ph]

    Pith/arXiv arXiv 2011

  44. [44]

    P. B. Gossiaux and J. Aichelin, Phys. Rev. C 78, 014904 (2008), arXiv:0802.2525 [hep-ph]

    Pith/arXiv arXiv 2008

  45. [45]

    Rapp and H

    R. Rapp and H. van Hees, in Quark-Gluon Plasma 4 (World Scientific, Singapore, 2010) pp. 111–206

    2010

  46. [46]

    Li and J

    S. Li and J. Liao, Eur. Phys. J. C 80, 671 (2020), arXiv:1912.08965 [hep-ph]

    Pith/arXiv arXiv 2020

  47. [47]

    H. A. Weldon, Phys. Rev. D 28, 2007 (1983)

    2007

  48. [48]

    Hidaka and R

    Y. Hidaka and R. D. Pisarski, Phys. Rev. D 80, 036004 (2009), [Erratum: Phys.Rev.D 102, 059902 (2020)], arXiv:0906.1751 [hep-ph]

    Pith/arXiv arXiv 2009

  49. [49]

    Q. Du, M. Du, and Y. Guo, Phys. Rev. D 110, 034011 (2024), arXiv:2402.18004 [hep-ph]

    Pith/arXiv arXiv 2024

  50. [50]

    Hidaka and R

    Y. Hidaka and R. D. Pisarski, Phys. Rev. D 78, 071501 (2008), arXiv:0803.0453 [hep-ph]

    Pith/arXiv arXiv 2008

  51. [51]

    Hidaka, S

    Y. Hidaka, S. Lin, R. D. Pisarski, and D. Satow, JHEP 10, 005 (2015), arXiv:1504.01770 [hep-ph]

    Pith/arXiv arXiv 2015

  52. [52]

    Guo and Q

    Y. Guo and Q. Du, JHEP 05, 042 (2019), arXiv:1810.13090 [hep-ph]

    Pith/arXiv arXiv 2019

  53. [53]

    P. N. Meisinger, T. R. Miller, and M. C. Ogilvie, Phys. Rev. D 65, 034009 (2002), arXiv:hep-ph/0108009

    Pith/arXiv arXiv 2002

  54. [54]

    Hidaka and R

    Y. Hidaka and R. D. Pisarski, Phys. Rev. D 104, 074036 (2021), arXiv:2009.03903 [hep-ph]

    Pith/arXiv arXiv 2021

  55. [55]

    J. I. Kapusta and C. Gale, Finite-temperature field theory: Principles and applications , Cambridge Monographs on Math- ematical Physics (Cambridge University Press, 2011)

    2011

  56. [56]

    M. L. Bellac, Thermal Field Theory, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2011)

    2011

  57. [57]

    Singh, A

    B. Singh, A. Abhishek, S. K. Das, and H. Mishra, Phys. Rev. D 100, 114019 (2019), arXiv:1812.05263 [hep-ph]

    Pith/arXiv arXiv 2019

  58. [58]

    Guo and Z

    Y. Guo and Z. Kuang, Phys. Rev. D 104, 014015 (2021), arXiv:2009.01516 [hep-ph]

    Pith/arXiv arXiv 2021

  59. [59]

    W. Liu, L. Dong, and Y. Guo, Phys. Rev. D 111, 034016 (2025), arXiv:2412.16954 [hep-ph]

    Pith/arXiv arXiv 2025

  60. [60]

    Bazavov, N

    A. Bazavov, N. Brambilla, P. Petreczky, A. Vairo, and J. H. Weber (TUMQCD), Phys. Rev. D 98, 054511 (2018), arXiv:1804.10600 [hep-lat]

    Pith/arXiv arXiv 2018

  61. [61]

    S. S. Gubser, Nucl. Phys. B 790, 175 (2008), arXiv:hep-th/0612143

    Pith/arXiv arXiv 2008

  62. [62]

    Casalderrey-Solana and D

    J. Casalderrey-Solana and D. Teaney, JHEP 04, 039 (2007), arXiv:hep-th/0701123

    Pith/arXiv arXiv 2007

  63. [63]

    Caron-Huot and G

    S. Caron-Huot and G. D. Moore, Phys. Rev. Lett. 100, 052301 (2008), arXiv:0708.4232 [hep-ph]

    Pith/arXiv arXiv 2008

  64. [64]

    W. M. Alberico, A. Beraudo, A. De Pace, A. Molinari, M. Monteno, M. Nardi, F. Prino, and M. Sitta, Eur. Phys. J. C 73, 2481 (2013), arXiv:1305.7421 [hep-ph]

    Pith/arXiv arXiv 2013

  65. [65]

    Brambilla, V

    N. Brambilla, V. Leino, P. Petreczky, and A. Vairo, Phys. Rev. D 102, 074503 (2020), arXiv:2007.10078 [hep-lat]

    Pith/arXiv arXiv 2020

  66. [66]

    Gupta and R

    S. Gupta and R. Sharma, Phys. Rev. D 97, 036025 (2018), arXiv:1710.05345 [hep-ph]

    Pith/arXiv arXiv 2018

  67. [67]

    Berges, M

    J. Berges, M. P. Heller, A. Mazeliauskas, and R. Venugopalan, Rev. Mod. Phys. 93, 035003 (2021), arXiv:2005.12299 [hep-th]

    Pith/arXiv arXiv 2021

  68. [68]

    S. Li, C. Wang, R. Wan, and J. Liao, Phys. Rev. C99, 054909 (2019), arXiv:1901.04600 [hep-ph]

    Pith/arXiv arXiv 2019

  69. [69]

    S. Li, C. Wang, X. Yuan, and S. Feng, Phys. Rev. C 98, 014909 (2018), arXiv:1803.01508 [hep-ph]

    Pith/arXiv arXiv 2018

  70. [70]

    Li and C

    S. Li and C. Wang, Phys. Rev. C 98, 034914 (2018), arXiv:1805.05807 [hep-ph]

    Pith/arXiv arXiv 2018

  71. [71]

    S. Li, W. Xiong, and R. Wan, Eur. Phys. J. C 80, 1113 (2020), arXiv:2012.02489 [hep-ph]

    Pith/arXiv arXiv 2020