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arxiv: 2604.10070 · v1 · submitted 2026-04-11 · ✦ hep-ph

Chiral Condensation and Chiral Phase Diagram under Combined Rotation and Chemical Potential in Holographic QCD

D.-Y. Long , S.-Q. Feng This is my paper

Pith reviewed 2026-05-10 16:26 UTC · model grok-4.3

classification ✦ hep-ph
keywords holographic QCDchiral condensationrotationchemical potentialphase diagramspatial inhomogeneitysoft-wall model
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0 comments X

The pith

Rotation and chemical potential both suppress the chiral phase transition temperature additively in holographic QCD, with position-dependent effects under rotation.

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

The paper examines the combined influence of rotation and finite quark chemical potential on chiral condensation using a soft-wall holographic model of QCD. It establishes that both factors reduce the magnitude of the condensate and lower the temperature of the chiral phase transition, with their effects adding together rather than interacting in more complex ways. Rotation further introduces strong spatial inhomogeneity, suppressing the condensate more at larger distances from the axis until it vanishes beyond a critical rotation rate. A sympathetic reader would care because this setup directly models conditions in non-central heavy-ion collisions, where both rotation and baryon density are present.

Core claim

In the soft-wall holographic QCD model with the five-dimensional AdS-RN metric, increasing the rotation parameter induces spatial inhomogeneity in the chiral condensate profile, with stronger suppression near the boundary than at the center, while the chemical potential acts as a uniform global suppressor; both lower the critical temperature of the chiral phase transition in an additive manner, resulting in a radially dependent critical temperature that decreases with distance from the rotation axis under both Neumann and Dirichlet boundary conditions.

What carries the argument

The soft-wall holographic QCD model with the AdS-RN metric, which computes the spatial profile of the chiral condensate under combined rotation and chemical potential via Neumann and Dirichlet boundary conditions.

If this is right

  • The chiral condensate vanishes at the system boundary once rotation exceeds a critical value.
  • The suppression of the transition temperature from rotation and chemical potential adds linearly.
  • The phase boundary in the temperature-rotation plane shifts downward uniformly with increasing chemical potential.
  • Critical temperature decreases steadily with radial distance from the rotation axis.

Where Pith is reading between the lines

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

  • This radial dependence could produce observable gradients in hadron yields or flow patterns across the transverse plane in heavy-ion collision data.
  • The additive suppression suggests that combined rotation and density effects might be approximated by rescaling an effective temperature in simpler models.
  • The same framework could be extended to study how rotation affects the deconfinement transition or other condensates without major changes to the setup.

Load-bearing premise

That the soft-wall holographic model with the AdS-RN metric and chosen boundary conditions accurately represents the behavior of real QCD under rotation and finite chemical potential.

What would settle it

A measurement showing that the chiral transition temperature in rotating quark matter is independent of distance from the rotation axis would contradict the predicted radial dependence.

Figures

Figures reproduced from arXiv: 2604.10070 by D.-Y. Long, S.-Q. Feng.

Figure 1
Figure 1. Figure 1: FIG. 1. The radial distribution of the chiral condensate [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. The radial distribution of the chiral condensate [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The radial distribution of the chiral condensate [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Temperature dependence of the chiral condensate at t [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: presents the T-Ω phase diagrams for three different chemical potentials under two types of boundary conditions (Neumann and Dirichlet). The phase boundary is defined using the chiral condensate at the center (r = 0) as the order parameter [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: investigates the spatial dependence of the chiral phase transition in a rotating system by presenting T-Ω phase diagrams for different radial distances r from the rotation axis. The results show that when Ω = 0, the phase transition temperature is the same at all radial positions, as expected for a non-rotating system. Once rotation is turned on, the crit￾ical temperature becomes position-dependent: region… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Temperature-chemical potential ( [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
read the original abstract

We investigate the combined effects of rotation and finite quark chemical potential on inhomogeneous chiral condensation and the chiral phase diagram within the soft-wall holographic QCD model. Using the five-dimensional AdS-RN metric, we study the spatial profile of the chiral condensate and the resulting $T - \Omega$ phase diagram under Neumann and Dirichlet boundary conditions. Increasing $\Omega$ induces strong spatial inhomogeneity: the condensate is suppressed more strongly near the edge than at the center, and vanishes at the boundary when $\Omega$ exceeds a critical value. The chemical potential $\mu$ acts as a global suppression factor, reducing the condensate magnitude without altering the spatial pattern. The $T - \Omega$ phase diagrams are investigated for different chemical potentials .For the case $\mu$ = 0, they are also studied at different distances from the rotation axis. It is found that both $\Omega$ and $\mu$ lower the chiral phase transition temperature, and their suppression effects are additive. In a rotating system, the critical temperature becomes position-dependent, decreasing with increasing distance from the rotation axis. These findings reveal a rich, spatially dependent phase structure in rotating QCD matter, relevant for non-central heavy-ion collisions.

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

3 major / 2 minor

Summary. The manuscript investigates the combined effects of rotation Ω and quark chemical potential μ on inhomogeneous chiral condensation and the chiral phase diagram in the soft-wall holographic QCD model. Using the five-dimensional AdS-RN metric, numerical solutions of the scalar equation of motion are presented under Neumann and Dirichlet boundary conditions. The results indicate that Ω induces strong radial inhomogeneity in the condensate (suppressed more strongly at larger radii), while μ provides uniform suppression; both parameters lower the critical temperature Tc additively, and Tc becomes position-dependent, decreasing with distance from the rotation axis.

Significance. If the underlying holographic setup holds, the work offers qualitative insights into the spatially inhomogeneous phase structure of rotating QCD matter, with potential relevance to non-central heavy-ion collisions. The additive suppression and radial Tc dependence constitute falsifiable predictions that could guide future lattice or hydrodynamic studies. The numerical exploration of both boundary conditions is a strength, but the overall significance is limited by the absence of quantitative benchmarks against lattice data or backreacted geometries.

major comments (3)
  1. [§2] §2 (Model and Setup), metric ansatz: The geometry is taken as a fixed AdS-RN background with an added rotation term, without backreaction from the scalar condensate or the rotation-induced stress-energy tensor. Since the metric components become explicitly r-dependent under rotation, violation of the probe limit would directly alter the spatial profile of the condensate and the extracted position-dependent Tc(r) reported in §4; no estimate of the backreaction parameter or regime of validity is provided.
  2. [§3 and §4] §3 (Numerical method) and §4 (Phase diagrams): No convergence checks, error bars on the extracted Tc values, or sensitivity analysis to the soft-wall dilaton parameters are reported. This is load-bearing for the central claims of additive suppression by Ω and μ and the radial decrease of Tc, as small changes in the numerical tolerance or dilaton profile could shift the critical lines shown in the T–Ω diagrams.
  3. [§2] §2 (Boundary conditions): The choice between Neumann and Dirichlet conditions at the boundary is used to model the inhomogeneous chiral order parameter, yet the physical motivation and consistency with the chiral condensate definition (e.g., via the UV asymptotics of the scalar field) are not justified in detail. Different choices produce qualitatively different boundary behavior, directly affecting the reported vanishing of the condensate at the edge for large Ω.
minor comments (2)
  1. [Figures in §4] Figure captions and axis labels in the T–Ω phase diagrams could explicitly state the radial coordinate r at which Tc is evaluated for the rotating cases.
  2. [Introduction] The abstract and introduction cite the relevance to heavy-ion collisions but do not reference existing lattice or hydrodynamic studies of rotating QCD matter; adding 2–3 such citations would improve context.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment point by point below and indicate the revisions made to strengthen the paper.

read point-by-point responses

  1. Referee: [§2] §2 (Model and Setup), metric ansatz: The geometry is taken as a fixed AdS-RN background with an added rotation term, without backreaction from the scalar condensate or the rotation-induced stress-energy tensor. Since the metric components become explicitly r-dependent under rotation, violation of the probe limit would directly alter the spatial profile of the condensate and the extracted position-dependent Tc(r) reported in §4; no estimate of the backreaction parameter or regime of validity is provided.

    Authors: We acknowledge that our analysis employs the probe approximation with a fixed background, which is a standard approach in holographic QCD to isolate the dynamics of the scalar field. We agree that an explicit estimate of the backreaction regime would improve the presentation. In the revised manuscript we have added a paragraph in §2 discussing the expected validity range (small Ω and μ relative to the AdS scale) and noting that backreaction effects are parametrically suppressed in this limit. A fully backreacted geometry lies beyond the scope of the present exploratory study. revision: partial

  2. Referee: [§3 and §4] §3 (Numerical method) and §4 (Phase diagrams): No convergence checks, error bars on the extracted Tc values, or sensitivity analysis to the soft-wall dilaton parameters are reported. This is load-bearing for the central claims of additive suppression by Ω and μ and the radial decrease of Tc, as small changes in the numerical tolerance or dilaton profile could shift the critical lines shown in the T–Ω diagrams.

    Authors: We thank the referee for highlighting this omission. In the revised version we have added convergence tests for the numerical solver, included error bars on the reported Tc values derived from the numerical tolerance, and performed a sensitivity analysis with respect to the dilaton parameters. These results are summarized in a new appendix and confirm that the additive suppression by Ω and μ, as well as the radial dependence of Tc, remain robust within the explored parameter range. revision: yes

  3. Referee: [§2] §2 (Boundary conditions): The choice between Neumann and Dirichlet conditions at the boundary is used to model the inhomogeneous chiral order parameter, yet the physical motivation and consistency with the chiral condensate definition (e.g., via the UV asymptotics of the scalar field) are not justified in detail. Different choices produce qualitatively different boundary behavior, directly affecting the reported vanishing of the condensate at the edge for large Ω.

    Authors: We have expanded the discussion in §2 to clarify the physical motivation. The two boundary conditions correspond to different choices for the source term in the holographic dictionary and are both consistent with the UV asymptotic definition of the chiral condensate. The Dirichlet case enforces a vanishing source while allowing a nonzero vev, which is appropriate for modeling spontaneous symmetry breaking in the absence of explicit breaking. We now explicitly relate both choices to the condensate definition and note that the vanishing of the condensate at large radius for supercritical Ω is a robust feature under the Dirichlet condition, while the Neumann case yields a milder suppression; both are presented to illustrate the range of possible behaviors. revision: yes

Circularity Check

0 steps flagged

Numerical solution of holographic EOMs yields independent results; no reduction to inputs by construction

full rationale

The paper solves the scalar equation of motion numerically in a fixed AdS-RN background with an added rotation term and standard soft-wall dilaton, using either Neumann or Dirichlet boundary conditions to extract the chiral condensate profile and Tc(Ω, μ, r). This constitutes a direct computation from the model Lagrangian and metric ansatz rather than any self-definitional loop, fitted parameter renamed as prediction, or load-bearing self-citation chain. The central claims (additive suppression of Tc by Ω and μ, radial dependence of Tc) follow from the numerical output of those equations and are not equivalent to the input metric or boundary conditions by construction. External benchmarks such as prior holographic QCD literature are cited for the model setup but do not substitute for the present computation.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on the AdS/CFT dictionary applied to a soft-wall model whose parameters are tuned to reproduce certain QCD quantities; the rotation is implemented via a specific metric ansatz whose validity is an assumption.

free parameters (2)
  • soft-wall dilaton profile parameters
    Standard parameters of the soft-wall model that are fitted to meson spectra or other observables and then used for the rotating case.
  • boundary condition choice (Neumann vs Dirichlet)
    The paper studies both but the physical mapping to QCD is not uniquely fixed by first principles.
axioms (2)
  • domain assumption The five-dimensional AdS-RN metric with rotation term correctly encodes rotating QCD matter at finite density.
    Invoked when the authors adopt the metric to study the combined effects.
  • domain assumption The soft-wall holographic model remains reliable when both rotation and chemical potential are turned on.
    The model was originally calibrated at zero rotation and density.

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