Dynamics of chiral phase transition in a $N_f=2+1$ soft-wall AdS/QCD model

Bin Dong; Danning Li; Han Tang; Nanxiang Wen; Yidian Chen

REVIEW 5 major objections 4 minor 71 references

Dynamics of chiral phase transition in a $N_f=2+1$ soft-wall AdS/QCD model

T0 review · 5 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Quark matter stalls, overshoots, or slows near the chiral transition when quenched close to it, and only then shows metastable plateaus, overshoot, and critical slowing down before reaching equilibrium.

desk verdict A solid but incremental 2+1-flavor extension of the group's two-flavor holographic thermalization study; the advertised strange-quark dynamics likely rest on a fixable typo in the initial condition for chi_s. read the letter →

arxiv 2507.21735 v1 pith:5O7SRVTF submitted 2025-07-29 hep-ph

Han Tang , Bin Dong , Nanxiang Wen , Yidian Chen , Danning Li This is my paper

classification hep-ph

keywords chiralphasetransitionthermalizationsoft-wallAdS/QCDcondensatenonequilibriumdynamicscriticalslowingdownquarkmassplaneprethermalization

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper attempts to establish that, in a holographic model of QCD with two light and one strange quark, the way quark matter reaches equilibrium depends strongly on where the quench temperature sits on the phase diagram. When the system is quenched close to the chiral transition, the chiral condensates do not relax monotonically: near the first-order line they get trapped in a metastable plateau, with larger initial amplitudes they overshoot and then relax slowly, and at the second-order line they show critical slowing down. Far from the transition, thermalization is fast, of order the inverse temperature. The result matters because it makes concrete predictions for the intermediate-time, out-of-equilibrium stage of matter that might be formed in heavy-ion collisions, and it extends the known prethermalization-like behavior from two flavors to the more realistic $N_f=2+1$ case.

What carries the argument

The machinery is the coupled set of real-time equations of motion for the two chiral condensate fields $\chi_l(\nu,z)$ and $\chi_s(\nu,z)$, derived from the 5D soft-wall action in the probe limit and written in Eddington-Finkelstein coordinates (Eqs. 22 and 23). The background is a fixed AdS-Schwarzschild black hole with temperature $T=1/(\pi z_h)$ and a modified dilaton profile that encodes confinement and chiral dynamics. The order parameters $\sigma_l(t)$ and $\sigma_s(t)$ are read off from the UV boundary expansion $\chi = m\zeta z + \cdots + (\sigma/\zeta) z^3 + \cdots$, with $\zeta = \sqrt{3}/(2\pi)$. The initial nonequilibrium states are parameterized by $\lambda_l$ and $\lambda_s$, and the evolution equations are solved by a pseudospectral method. This setup converts the question 'how does the chiral condensate thermalize after a quench' into a solvable initial-value problem in one holographic dimension.

What would settle it

Solve the full Einstein-scalar system with backreaction included for the same quenches (same masses, initial $\lambda_l$ and $\lambda_s$, and temperatures) and check whether the metastable plateau, the overshoot, and the critical slowing down persist; if the plateau disappears or the relaxation times shorten drastically, the probe limit is the load-bearing assumption.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the real-time evolution of the chiral order parameters in the $N_f=2+1$ soft-wall AdS/QCD model, solved from far-from-equilibrium initial states, shows universal late-time thermalization to the equilibrium condensates, but non-trivial intermediate-time behavior whenever the quench temperature is close to the transition region. In the first-order region ($m_l=0.01$ GeV, $m_s=0.1$ GeV, $T=0.18855$ GeV), a small initial condensate relaxes first to a metastable state on the lower branch of the multi-valued equilibrium curve and stays there until $t>700/(\pi T)$ before decaying to the true equilibrium; a larger initial condensate instead rises quickly, overshoots the equilibrium value, and then slowly decays. At the second-order critical point ($m_l=0.01$ GeV, $m_s=0.120105...$ GeV, $T=T_c=0.19123$ GeV), the relaxation time becomes enormous, a critical slowing down, with the condensates lingering near their equilibrium values for up to $10^6/(\pi T)$. At a crossover point ($m_s=0.15$ GeV, $T_p=0.194487$ GeV), there is no exact critical slowing down, but the intermediate-time evolution still shows structure. Far from the transition ($T=0.1$ GeV), the condensates thermalize within about $1/(\pi T)$ with damped oscillations.

Load-bearing premise

The load-bearing assumption is the probe limit: the quark condensate fields move on a fixed, static black-hole background that is not altered by the condensate dynamics, so if the backreaction of the matter on the geometry is significant, the metastable plateau, overshoot, and critical slowing down could be changed.

Editorial extensions

If this is right

Near the first-order transition, a small initial chiral condensate first relaxes to a metastable state and only leaves it after a long time ($t > 700/(\pi T)$), so thermalization is two-staged.
With a larger initial condensate in the same first-order region, the system overshoots its equilibrium value and then slowly relaxes, instead of passing through the metastable plateau.
At the $N_f=2+1$ second-order line, the relaxation time becomes extremely large ($\sim 10^6/(\pi T)$), a holographic critical slowing down, even though the equilibrium condensates are finite.
At the crossover line, the divergence of $d\sigma/dT$ is replaced by a peak, and correspondingly there is no exact critical slowing down, but nontrivial intermediate-time structure remains.
Far from the transition, thermalization is fast, of order $1/(\pi T)$, with oscillations in the intermediate region, confirming that the slow behavior is tied to proximity to the transition.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit a scaling law for the metastable lifetime as the quench temperature approaches the transition; extracting that scaling would be a direct test of whether the plateau is a genuine prethermalization phenomenon.
If the probe-limit behavior survives backreaction, the two-stage relaxation could be looked for in heavy-ion collisions as a delayed approach of chiral observables to equilibrium.
Because the equilibrium phase diagram matches the Columbia plot, the same critical slowing down could be relevant at the physical quark masses if they lie near a critical line; the paper only explores representative mass points, so locating the physical point on this map is left open.
The initial states are chosen by hand through $\lambda_l$ and $\lambda_s$; replacing these with a collision-like driving source would test whether the nontrivial intermediate-time behavior is generic.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Here's the short version: this is a clean extension of the two-flavor study [49] to the N_f=2+1 case, and the novel bit is the strange quark's role in the dynamics near the lower branch of the Columbia-plot critical line. The qualitative phenomena—metastable plateau, overshoot, critical slowing down—were already reported in the two-flavor paper, so this is incremental but real.

What the paper does well: it solves the real-time equations for the chiral condensates directly, uses a standard pseudospectral method, and includes far-from-transition control runs that show fast relaxation. The authors are transparent about the probe limit and about the difficulty of pinning down the exact critical temperature when the condensate doesn't vanish. The figures are readable and the conclusions track the numerics.

The one concrete technical problem is the initial condition for the strange scalar. In Sec. III and again in III.A, they write chi_s(nu=0) = m_l zeta z + ... + lambda_s pi^3 T^3 z^3, with the light-quark mass m_l in the leading term. The UV expansion (16) requires the coefficient of z to be m_s zeta. If the code used the printed formula, then the extracted sigma_s is not the strange condensate for the advertised m_s, and the N_f=2+1 interpretation is off. More likely this is a copy-paste typo—they just forgot to change m_l to m_s—but it must be fixed and the relevant figures regenerated. That's a referee-level request, not a killer flaw, because the two-flavor version behaves qualitatively the same.

Minor soft spots: no numerical convergence tests, no code release, and the initial-state amplitudes lambda are only taken at two values (0.01 and 0.1). A few more points would make the "non-trivial intermediate-time behavior" claim more solid. The phrase itself is vague, but the paper openly restricts to qualitative statements. The probe limit is a real limitation, but it's stated and standard in this line of work; the authors don't oversell it.

Who this is for: people working on holographic far-from-equilibrium dynamics, and the heavy-ion crowd curious about prethermalization in a more realistic flavor setting. It's a modest step, but a well-posed one.

Recommendation: send to peer review. The fix is straightforward, and the paper is a legitimate incremental contribution that deserves a careful referee rather than a desk rejection.

Referee Report

5 major / 4 minor

Summary. The paper studies the real-time thermalization of the chiral condensate in a soft-wall AdS/QCD model with N_f=2+1 non-degenerate flavors. The equilibrium phase diagram is taken from earlier work (Refs. [40-42]) and is claimed to be qualitatively consistent with the Columbia plot. The authors then solve the time-dependent scalar equations in the probe limit on a fixed AdS-Schwarzschild background, starting from simple non-equilibrium initial profiles, and extract the time-dependent condensates sigma_l(nu) and sigma_s(nu). The central claim, stated in the abstract and in Sec. III, is that when the final temperature is close to the transition region, the intermediate-time evolution shows non-trivial behavior: a metastable plateau in the first-order region (Fig. 3a), an overshoot followed by slow relaxation for larger initial amplitudes (Fig. 3b), and critical slowing down at the N_f=2+1 critical line (Fig. 5). The paper explicitly restricts itself to qualitative statements and leaves quantitative scaling analysis for future work.

Significance. If the results are correct, the paper makes a useful phenomenological contribution by extending holographic real-time studies of chiral dynamics from two flavors (Ref. [49]) to the more realistic N_f=2+1 case, and by making concrete qualitative predictions: long-lived intermediate plateaus, overshoot relaxation, and critical slowing down near the chiral transition. The study is directly aligned with ongoing interest in prethermalization and non-equilibrium dynamics in strong interactions. The numerical solutions of the coupled partial differential equations (22)-(23) are a non-trivial output, and the qualitative agreement of the equilibrium phase diagram with the Columbia plot is a point in favor of the model. However, the paper does not provide code, data, or convergence tests, and it relies on a probe-limit background and on arbitrary initial states; moreover, there are internal inconsistencies in the stated initial conditions and in the light-quark mass values that must be resolved before the central claim can be credited.

major comments (5)

[Sec. III and Sec. III.A, Eqs. (15)-(16), (22)-(23)] The initial data for the strange condensate are written as chi_s(nu=0) = m_l zeta z + ... + lambda_s pi^3 T^3 z^3, both in the general setup and again in Sec. III.A. According to the holographic dictionary in Eq. (16), the leading UV coefficient of chi_s must be m_s zeta, not m_l zeta. As written, the initial profile violates the Dirichlet boundary condition for the stated strange quark mass, so the extracted sigma_s(nu) is not the condensate for m_s=0.1 or 0.120105 GeV. This is an internal inconsistency that must be resolved: the authors should state clearly which source was actually imposed in the code and, if the printed formula was used, rerun the simulations with the correct m_s boundary condition.
[Sec. II, Fig. 1 vs. Sec. III and Fig. 2] There is an inconsistency in the light-quark mass value. The text at the end of Sec. II says 'We take m_l = 0.01 GeV' for the equilibrium results, but the Fig. 1 caption and the next sentence use m_l = 0.1 GeV, while Sec. III and Fig. 2 consistently use m_l = 0.01 GeV. This ambiguity affects which equilibrium branch structure (first-order vs. crossover) is actually used to define the green point and the first-order region. Please reconcile the caption and text and confirm that the equilibrium curves in Fig. 1 correspond to the same masses used in the dynamical runs.
[Sec. III, probe limit (Eq. (4) and text after Eq. (19))] The entire dynamical calculation is performed in the probe limit: the scalar fields chi_l and chi_s evolve on a fixed, static AdS-Schwarzschild background that does not respond to the condensate dynamics. With N_f=3 and N_c=3 the flavor sector is not parametrically suppressed, so the backreaction need not be small. Since the claimed intermediate-time plateau, overshoot, and critical slowing down are the central results, the authors should provide at least a quantitative estimate of the backreaction (for example, the ratio of the scalar energy density to the background energy density, or a comparison with a dynamical gravity computation for one representative case) to argue that these features are not artifacts of the probe approximation.
[Sec. III, initial-state ansatz and Figs. 3-7] The central qualitative claim rests on only two choices of the free initial-state parameters (lambda_l, lambda_s) = (0.01, 0.01) and (0.1, 0.1), for a single form of the initial profile chi(nu=0) = m zeta z + ... + lambda pi^3 T^3 z^3. Because these parameters are not fixed by the model, the paper should demonstrate that the intermediate-time phenomena persist over a range of amplitudes and are not sensitive to the truncation of the initial expansion. A systematic scan, even at coarse resolution, would materially strengthen the claim that the behavior is a property of the transition region rather than of a particular initial state.
[Sec. III.B, Figs. 5-6 and numerical method] The paper reports relaxation times as long as O(10^6)/(pi T) in Fig. 5, but it provides no convergence tests for the pseudospectral discretization or for the time integration, and no estimate of numerical error. Since the long-time plateaus and the critical slowing down are load-bearing for the paper's conclusions, please report convergence checks (for example, dependence on the number of grid points and on the time step, or a Richardson-type estimate) and, if possible, make the numerical code or data available so the results can be verified.

minor comments (4)

[Sec. III.A, text after Fig. 3] The text says the lambda_l = lambda_s = 0.1 results are 'shown in Fig. Fig. 3 (a)', but they are shown in Fig. 3(b); please correct the reference.
[Eq. (18)] Equation (18) is printed as chi'' = m zeta z + ... + sigma/zeta z^3; it should be chi = m zeta z + ... + sigma/zeta z^3, matching the structure of Eqs. (15)-(16).
[Sec. III.C] In the crossover section, the sentence 'we take m_u = 0.01,0.1 GeV' is ambiguous and inconsistent with the Fig. 7 caption (m_s = 0.15 GeV). Please specify the exact values of m_u and m_s used for Fig. 7 and for the quoted T_p = 0.194487 GeV.
[Sec. II, Fig. 2 caption] The caption of Fig. 2 says the tricritical point is at m_l = m_u = m_d = 0 GeV, m_s = 0.29 GeV, but the main text uses m_u = m_d for the light flavors; this notation should be unified for readability.

Circularity Check

0 steps flagged · score 1.0 of 10

Thermalization curves are emergent numerical outputs of Eqs. (22)-(23), not fits; the paper's self-citations supply the model framework but do not carry the central claim.

full rationale

The central claim — non-trivial intermediate-time thermalization near the transition region (metastable plateau in Fig. 3a, overshoot followed by slow relaxation in Fig. 3b, and critical slowing down at the N_f=2+1 critical line in Fig. 5) — is the direct numerical output of the initial-boundary value problem in Eqs. (22)-(23), evolved from the stated initial data χ_l(ν=0) = m_l ζ z + ... + λ_l π^3 T^3 z^3 and χ_s(ν=0) = m_s ζ z + ... + λ_s π^3 T^3 z^3. No quantity entering the claim is fitted or calibrated: λ_l and λ_s are free parameters, and the late-time values are checked against, not matched to, the static multi-branch solutions of Fig. 1; the ~0.02 GeV^3 plateau of Fig. 3a is emergent because the λ=0.01 initial condensate is an order of magnitude smaller. The model action, all parameters (µ0=0.43, µ1=0.83, µ2=0.176 GeV, v3=-3, v4=8), the equilibrium phase diagram, and the critical strange mass m_s=0.120105... GeV are inherited from the same group's earlier papers [40-42], and the probe limit and numerical method follow [49]; these self-citations are framework scaffolding whose content is background, not the derivation target, so the Columbia-plot agreement is a consistency check and not circular. Per the review rules I flag three non-circularity concerns. (i) Sec. III and III.A write the strange initial profile with the light mass, 'χ_s(ν= 0) = m_l ζ z+...+λ_s π^3 T^3 z^3', which conflicts with the UV dictionary Eq. (16) and with the stated m_s values; this is an internal-inconsistency defect that makes the reported N_f=2+1 evolution ill-defined as written, but it does not reduce the conclusion to its inputs. (ii) The probe limit is adopted by citation from [49] with no backreaction estimate, a limitation the text acknowledges. (iii) Footnote 3 concedes that the critical temperature and exponents are not determined accurately, so the critical-slowing-down claim is qualitative rather than a quantified scaling law. Since the central thermalization result is emergent from the solved PDEs with free parameters and is independently checkable against late-time equilibrium data, the paper is essentially self-contained for its main result; the residual self-citation pattern is minor, giving score 1.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or dimensions are introduced; the model ingredients (dilaton, scalar potential) are from prior literature. The free parameters are all inherited from earlier fits or chosen by hand for the initial conditions.

free parameters (6)

mu0 (dilaton IR scale) = 0.43 GeV
Fixed to reproduce the Regge slope of excited mesons in the soft-wall model; taken from earlier work [40-42].
mu1 = 0.83 GeV
Chosen together with mu2 to give T_c ~ 151.5 MeV and vacuum condensate ~0.035 GeV^3 for two flavors in [40-42].
mu2 = 0.176 GeV
Same as mu1: part of the dilaton profile fitted to chiral observables.
v3 = -3
Scalar potential coupling for the determinant term; fixed in [42] to reproduce the phase diagram.
v4 = 8
Scalar potential coupling for the quartic term; fixed in [42].
lambda_l, lambda_s (initial state amplitudes) = 0.01 or 0.1
Free coefficients setting the initial z^3 coefficient of the condensate fields. The results are shown for these two values only.

assumptions (5)

standard math AdS/CFT correspondence and the holographic dictionary relating 5D scalar sources to 4D chiral condensates
Used throughout to interpret UV expansions (Eqs. 15-16) and to extract condensates sigma_l, sigma_s.
domain assumption Probe limit with fixed AdS-Schwarzschild background
Introduced in Sec. III: the metric (Eq. 4) is static and unaffected by the condensate dynamics.
domain assumption Specific dilaton profile Phi(z) = -mu1^2 z^2 + (mu1^2+mu0^2) z^2 tanh(mu2^2 z^2)
Eq. (2), taken from [40-42]; the parameter values are fitted to hadron spectroscopy and chiral observables.
ad hoc to paper Initial state ansatz with free coefficients lambda_l, lambda_s
Sec. III: initial conditions are set as chi(mu)(0) = m zeta z + ... + lambda pi^3 T^3 z^3; the same form is used for chi_s with the strange mass (although the text accidentally writes m_l).
domain assumption N_f = 2+1 flavor symmetry with m_u = m_d != m_s
Used to reduce the three-field system to two fields chi_l, chi_s.

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Cite this review

Pith. "Pith review of Dynamics of chiral phase transition in a $N_f=2+1$ soft-wall AdS/QCD model." pith.science (2026). https://pith.science/paper/5O7SRVTF

@misc{pith2026250721735,
  author       = {Pith},
  title        = {Pith review of: Dynamics of chiral phase transition in a $N_f=2+1$ soft-wall AdS/QCD model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5O7SRVTF}},
  note         = {Machine review of arXiv:2507.21735}
}

read the original abstract

We investigate the real-time dynamics of the chiral phase transition in a soft-wall AdS/QCD model, of which the mass plane phase diagram from equilibrium calculation is qualitatively consistent with the so-called Columbia plot. By directly solving the non-equilibrium evolution of the order parameter of the chiral phase transition, i.e. the chiral condensate, we study the thermalization of the QCD matter in different regions of the quark mass plane. It is shown that, when the system is close to the transition region, the thermalization process will show non-trivial behavior in the intermediate time region.

Figures

Figures reproduced from arXiv: 2507.21735 by the authors.

**Figure 2.** Phase diagram of chiral phase transition in [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 4.** Thermalization with λl = λs = 0.1 at ml = 0.01GeV, ms = 0.1GeV in the first order region. The quenched temperature is taken as T = 0.1GeV, far from the transition region ( with the multiple solutions). In this section, we will consider the thermalization in the first order region. We take ml = 0.010GeV, ms = 0.1GeV as an example. As discussed above, we will simply take the initial states as χl(ν = 0) = mlζz + ... +… view at source ↗

Figures from the paper (5 more)

**Figure 1.** Figure 1: Fig.1. It seems the system relaxes to a metastable state [PITH_FULL_IMAGE:figures/full_fig_p006_1.png]

**Figure 3.** Figure 3: Thermalization with different initial conditions at [PITH_FULL_IMAGE:figures/full_fig_p006_3.png]

**Figure 5.** Figure 5: Relaxation in critical lines. The quark masses are [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 6.** Figure 6: Thermalization with λl = λs = 0.01 at ml = 0.01GeV, ms = 0.120105GeV in the critical line. The quenched temperature is taken as T = 0.1GeV, far from the critical temperature. Fig.7. From the results, we can see that since the divergence of − dσl/s dT becomes a peak, t…

**Figure 7.** Figure 7: Thermalization with λl = λs = 0.01 at ml = 0.01GeV, ms = 0.15GeV in the crossover region. The quenched temperature is taken as Tp = 0.194487GeV, at which − dσl/s dT reach their maximums. IV. CONCLUSIONS AND DISCUSSIONS We extend our previous non-equilibrium holographic…

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Works this paper leans on

71 extracted references · 22 canonical work pages

[49]

X. Cao, J. Chao, H. Liu, and D. Li, Phys. Rev. D107, 086001 (2023), arXiv:2204.11604 [hep-ph]

work page Pith review arXiv 2023
[1]

T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, J. Phys. B51, 112001 (2018), arXiv:1712.08790 [cond- mat.stat-mech]

arXiv 2018
[2]

Berges, S

J. Berges, S. Borsányi, and C. Wetterich, Phys. Rev. Lett.93, 142002 (2004)

work page 2004
[3]

Gagel, P

P. Gagel, P. P. Orth, and J. Schmalian, Phys. Rev. Lett. 113, 220401 (2014)

work page 2014
[4]

Chiocchetta, M

A. Chiocchetta, M. Tavora, A. Gambassi, and A. Mitra, Phys. Rev. B91, 220302 (2015)

2015
[5]

Chiocchetta, A

A. Chiocchetta, A. Gambassi, S. Diehl, and J. Marino, Phys. Rev. Lett.118, 135701 (2017)

work page 2017
[6]

Nicklas, M

E. Nicklas, M. Karl, M. Höfer, A. Johnson, W. Mues- sel, H. Strobel, J. Tomkovič, T. Gasenzer, and M. K. Oberthaler, Phys. Rev. Lett.115, 245301 (2015)

work page 2015
[7]

Arseneet al

I. Arseneet al. (BRAHMS), Nucl. Phys. A757, 1 (2005), arXiv:nucl-ex/0410020

arXiv 2005

Show all 71 references

[8]

Romatschke and U

P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium, Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2019) arXiv:1712.05815 [nucl-th]

2019 arXiv
[9]

Berges, M

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

2021 arXiv
[10]

Lappi and L

T. Lappi and L. McLerran, Nucl. Phys. A772, 200 (2006), arXiv:hep-ph/0602189

2006 arXiv
[11]

Gelis and R

F. Gelis and R. Venugopalan, Acta Phys. Polon. B37, 3253 (2006), arXiv:hep-ph/0611157

2006 arXiv
[12]

J. M. Maldacena, Adv. Theor. Math. Phys.2, 231 (1998), arXiv:hep-th/9711200

1998 arXiv
[13]

S. S. Gubser, I. R. Klebanov, and A. M. Polyakov, Phys. Lett. B428, 105 (1998), arXiv:hep-th/9802109

1998 arXiv
[14]

Witten, Adv

E. Witten, Adv. Theor. Math. Phys.2, 253 (1998), arXiv:hep-th/9802150

1998 arXiv
[15]

Policastro, D

G. Policastro, D. T. Son, and A. O. Starinets, Phys. Rev. Lett.87, 081601 (2001), arXiv:hep-th/0104066

2001 arXiv
[16]

Buchel and J

A. Buchel and J. T. Liu, Phys. Rev. Lett.93, 090602 (2004), arXiv:hep-th/0311175

2004 arXiv
[17]

Kovtun, D

P. Kovtun, D. T. Son, and A. O. Starinets, Phys. Rev. Lett.94, 111601 (2005), arXiv:hep-th/0405231

2005 arXiv
[18]

S. S. Gubser and A. Nellore, Phys. Rev. D78, 086007 (2008), arXiv:0804.0434 [hep-th]

2008 arXiv
[19]

Gursoy, E

U. Gursoy, E. Kiritsis, and F. Nitti, JHEP02, 019 (2008), arXiv:0707.1349 [hep-th]

2008 arXiv
[20]

Gursoy and E

U. Gursoy and E. Kiritsis, JHEP02, 032 (2008), arXiv:0707.1324 [hep-th]

2008 arXiv
[21]

Grefa, J

J. Grefa, J. Noronha, J. Noronha-Hostler, I. Portillo, C. Ratti, and R. Rougemont, Phys. Rev. D104, 034002 (2021), arXiv:2102.12042 [nucl-th]

2021 arXiv
[22]

R.-G. Cai, S. He, L. Li, and Y.-X. Wang, Phys. Rev. D 106, L121902 (2022), arXiv:2201.02004 [hep-th]

2022 arXiv
[23]

D. Li, S. He, M. Huang, and Q.-S. Yan, JHEP09, 041 (2011), arXiv:1103.5389 [hep-th]

2011 arXiv
[24]

Erlich, E

J. Erlich, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett.95, 261602 (2005), arXiv:hep- ph/0501128. 9

2005
[25]

Karch, E

A. Karch, E. Katz, D. T. Son, and M. A. Stephanov, Phys. Rev. D74, 015005 (2006), arXiv:hep-ph/0602229

2006 arXiv
[26]

G. F. de Teramond and S. J. Brodsky, Phys. Rev. Lett. 94, 201601 (2005), arXiv:hep-th/0501022

2005 arXiv
[27]

Adams, L

A. Adams, L. D. Carr, T. Schäfer, P. Steinberg, and J. E. Thomas, New J. Phys.14, 115009 (2012), arXiv:1205.5180 [hep-th]

2012 arXiv
[28]

Gao and M

F. Gao and M. Yamada, Phys. Rev. D106, 126003 (2022), arXiv:2202.13699 [hep-th]

2022 arXiv
[29]

P. M. Chesler and L. G. Yaffe, Phys. Rev. Lett.106, 021601 (2011), arXiv:1011.3562 [hep-th]

2011 arXiv
[30]

Grumiller and P

D. Grumiller and P. Romatschke, JHEP08, 027 (2008), arXiv:0803.3226 [hep-th]

2008 arXiv
[31]

M. P. Heller, R. A. Janik, and P. Witaszczyk, Phys. Rev. Lett.108, 201602 (2012), arXiv:1103.3452 [hep-th]

2012 arXiv
[32]

M. P. Heller, R. A. Janik, and P. Witaszczyk, Phys. Rev. D85, 126002 (2012), arXiv:1203.0755 [hep-th]

2012 arXiv
[33]

van der Schee, P

W. van der Schee, P. Romatschke, and S. Pratt, Phys. Rev. Lett.111, 222302 (2013), arXiv:1307.2539 [nucl-th]

2013 arXiv
[34]

Rajagopal, A

K. Rajagopal, A. V. Sadofyev, and W. van der Schee, Phys. Rev. Lett.116, 211603 (2016), arXiv:1602.04187 [nucl-th]

2016 arXiv
[35]

Critelli, R

R. Critelli, R. Rougemont, and J. Noronha, JHEP12, 029 (2017), arXiv:1709.03131 [hep-th]

2017 arXiv
[36]

Craps, E

B. Craps, E. Kiritsis, C. Rosen, A. Taliotis, J. Vanhoof, and H.-b. Zhang, JHEP02, 120 (2014), arXiv:1311.7560 [hep-th]

2014 arXiv
[37]

R. A. Janik, J. Jankowski, and H. Soltanpanahi, Phys. Rev. Lett.119, 261601 (2017), arXiv:1704.05387 [hep- th]

2017 arXiv
[38]

Bantilan, P

H. Bantilan, P. Figueras, and D. Mateos, Phys. Rev. Lett.124, 191601 (2020), arXiv:2001.05476 [hep-th]

2020 arXiv
[39]

Attems, Y

M. Attems, Y. Bea, J. Casalderrey-Solana, D. Mateos, and M. Zilhão, JHEP01, 106 (2020), arXiv:1905.12544 [hep-th]

2020 arXiv
[40]

Chelabi, Z

K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, Phys. Rev. D93, 101901 (2016), arXiv:1511.02721 [hep- ph]

2016 arXiv
[41]

Chelabi, Z

K. Chelabi, Z. Fang, M. Huang, D. Li, and Y.-L. Wu, JHEP04, 036 (2016), arXiv:1512.06493 [hep-ph]

2016 arXiv
[42]

J. Chen, S. He, M. Huang, and D. Li, JHEP01, 165 (2019), arXiv:1810.07019 [hep-ph]

2019 arXiv
[43]

Ballon-Bayona, T

A. Ballon-Bayona, T. Frederico, L. A. H. Mamani, and W. de Paula, Phys. Rev. D108, 106016 (2023), arXiv:2308.07503 [hep-ph]

2023 arXiv
[44]

Rinaldi, F

M. Rinaldi, F. A. Ceccopieri, and V. Vento, Eur. Phys. J. C82, 626 (2022), arXiv:2204.09974 [hep-ph]

2022 arXiv
[45]

X. Chen, D. Li, D. Hou, and M. Huang, JHEP03, 073 (2020), arXiv:1908.02000 [hep-ph]

2020 arXiv
[46]

D. M. Rodrigues, D. Li, E. Folco Capossoli, and H. Boschi-Filho, Phys. Rev. D103, 066022 (2021), arXiv:2010.06762 [hep-th]

2021 arXiv
[47]

M. Lv, D. Li, and S. He, JHEP11, 026 (2019), arXiv:1811.03828 [hep-ph]

2019 arXiv
[48]

X. Cao, H. Liu, D. Li, and G. Ou, Chin. Phys. C44, 083106 (2020), arXiv:2001.02888 [hep-ph]

2020 arXiv
[50]

Sui, Y.-L

Y.-Q. Sui, Y.-L. Wu, Z.-F. Xie, and Y.-B. Yang, Phys. Rev. D81, 014024 (2010), arXiv:0909.3887 [hep-ph]

2010 arXiv
[51]

de Paula, T

W. de Paula, T. Frederico, H. Forkel, and M. Beyer, Phys. Rev. D79, 075019 (2009), arXiv:0806.3830 [hep- ph]

2009 arXiv
[52]

Rinaldi and V

M. Rinaldi and V. Vento, Eur. Phys. J. A54, 151 (2018), arXiv:1710.09225 [hep-ph]

2018 arXiv
[53]

M. A. M. Contreras, E. F. Capossoli, D. Li, A. Vega, and H. Boschi-Filho, Phys. Lett. B822, 136638 (2021), arXiv:2108.05427 [hep-ph]

2021 arXiv
[54]

T. M. Kelley, S. P. Bartz, and J. I. Kapusta, Phys. Rev. D83, 016002 (2011), arXiv:1009.3009 [hep-ph]

2011 arXiv
[55]

R. Chen, D. Li, K. Bitaghsir Fadafan, and M. Huang, Chin. Phys. C47, 063106 (2023), arXiv:2212.10363 [hep- ph]

2023 arXiv
[56]

Folco Capossoli, D

E. Folco Capossoli, D. Li, and H. Boschi-Filho, Eur. Phys. J. C76, 320 (2016), arXiv:1604.01647 [hep-ph]

2016 arXiv
[57]

Fang, Y.-L

Z. Fang, Y.-L. Wu, and L. Zhang, Phys. Lett. B762, 86 (2016), arXiv:1604.02571 [hep-ph]

2016 arXiv
[58]

Li and M

D. Li and M. Huang, JHEP11, 088 (2013), arXiv:1303.6929 [hep-ph]

2013 arXiv
[59]

D. Li, M. Huang, and Q.-S. Yan, Eur. Phys. J. C73, 2615 (2013), arXiv:1206.2824 [hep-th]

2013 arXiv
[60]

Y. Aoki, S. Borsanyi, S. Durr, Z. Fodor, S. D. Katz, S. Krieg, and K. K. Szabo, JHEP06, 088 (2009), arXiv:0903.4155 [hep-lat]

2009 arXiv
[61]

Y. Aoki, Z. Fodor, S. D. Katz, and K. K. Szabo, Phys. Lett. B643, 46 (2006), arXiv:hep-lat/0609068

2006 arXiv
[62]

Li and M

D. Li and M. Huang, JHEP02, 042 (2017), arXiv:1610.09814 [hep-ph]

2017 arXiv
[63]

D. Li, M. Huang, Y. Yang, and P.-H. Yuan, JHEP02, 030 (2017), arXiv:1610.04618 [hep-th]

2017 arXiv
[64]

D. M. Rodrigues, D. Li, E. Folco Capossoli, and H. Boschi-Filho, Phys. Rev. D98, 106007 (2018), arXiv:1807.11822 [hep-th]

2018 arXiv
[65]

D. M. Rodrigues, D. Li, E. Folco Capossoli, and H. Boschi-Filho, EPL128, 61001 (2019), arXiv:1811.04117 [hep-ph]

2019 arXiv
[66]

Y. Chen, D. Li, and M. Huang, Phys. Rev. D106, 106002 (2022), arXiv:2208.05668 [hep-ph]

2022 arXiv
[67]

Y. Chen, D. Li, and M. Huang, JHEP07, 225 (2023), arxiv:2212.06591 [hep-ph]

2023 arXiv
[68]

F. R. Brown, F. P. Butler, H. Chen, N. H. Christ, Z.-h. Dong, W. Schaffer, L. I. Unger, and A. Vaccarino, Phys. Rev. Lett.65, 2491 (1990)

1990
[69]

H.-T. Ding, F. Karsch, and S. Mukherjee, Int. J. Mod. Phys. E24, 1530007 (2015), arXiv:1504.05274 [hep-lat]

2015 arXiv
[70]

Cherman, T

A. Cherman, T. D. Cohen, and E. S. Werbos, Phys. Rev. C79, 045203 (2009), arXiv:0804.1096 [hep-ph]

2009 arXiv
[71]

J. P. Boyd, Chebyshev and Fourier spectral methods (Dover Publications, 2001)

2001

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [49]

X. Cao, J. Chao, H. Liu, and D. Li, Phys. Rev. D107, 086001 (2023), arXiv:2204.11604 [hep-ph]

work page Pith review arXiv 2023

[2] [1]

T. Mori, T. N. Ikeda, E. Kaminishi, and M. Ueda, J. Phys. B51, 112001 (2018), arXiv:1712.08790 [cond- mat.stat-mech]

arXiv 2018

[3] [2]

Berges, S

J. Berges, S. Borsányi, and C. Wetterich, Phys. Rev. Lett.93, 142002 (2004)

work page 2004

[4] [3]

Gagel, P

P. Gagel, P. P. Orth, and J. Schmalian, Phys. Rev. Lett. 113, 220401 (2014)

work page 2014

[5] [4]

Chiocchetta, M

A. Chiocchetta, M. Tavora, A. Gambassi, and A. Mitra, Phys. Rev. B91, 220302 (2015)

2015

[6] [5]

Chiocchetta, A

A. Chiocchetta, A. Gambassi, S. Diehl, and J. Marino, Phys. Rev. Lett.118, 135701 (2017)

work page 2017

[7] [6]

Nicklas, M

E. Nicklas, M. Karl, M. Höfer, A. Johnson, W. Mues- sel, H. Strobel, J. Tomkovič, T. Gasenzer, and M. K. Oberthaler, Phys. Rev. Lett.115, 245301 (2015)

work page 2015

[8] [7]

Arseneet al

I. Arseneet al. (BRAHMS), Nucl. Phys. A757, 1 (2005), arXiv:nucl-ex/0410020

arXiv 2005