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arxiv: 2606.19166 · v1 · pith:GLISZ564new · submitted 2026-06-17 · ✦ hep-th · hep-ph

Bubble wall velocity and nucleation rates in inverse holographic phase transitions

Francesco Bigazzi , Aldo L. Cotrone , Natalia Pinzani-Fokeeva , Tommaso Trabocchi This is my paper

Pith reviewed 2026-06-26 19:53 UTC · model grok-4.3

classification ✦ hep-th hep-ph
keywords holographic QCDbubble nucleationphase transitionsdeconfinementchiral symmetrybubble wall velocityEuclidean bounceWitten-Sakai-Sugimoto model
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The pith

Euclidean bounce solutions yield nucleation rates and wall velocity estimates for inverse holographic QCD transitions.

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

The paper examines first-order inverse phase transitions driven by superheating in the Witten-Sakai-Sugimoto holographic model of QCD. Euclidean bounce solutions are identified for both the deconfinement transition in the unflavored model and a chiral symmetry-restoring transition in the deconfined phase. These solutions determine the bubble nucleation rates and the relevant transition parameters. For deconfinement the large jump in degrees of freedom implies a parametrically small bubble wall velocity, with a rough estimate given near the critical temperature. For the chiral transition the wall velocity and the friction force on the bubbles are obtained from motivated ansatze and approximations for the steady-state configurations.

Core claim

For both the deconfinement transition in the unflavored version of the model and a chiral symmetry-restoring transition occurring in the deconfined phase of the full theory, the corresponding Euclidean bounce solutions are found and the bubble nucleation rates and the relevant transition parameters are computed. For the deconfinement transition, the large jump in the number of degrees of freedom between the two phases suggests that the bubble wall velocity is parametrically small; a rough estimate of it near the critical temperature is provided. In the case of the chiral transition, the bubble wall velocity and the friction force exerted on the bubbles are computed employing motivated ansatz

What carries the argument

Euclidean bounce solutions in the five-dimensional holographic geometry, which encode the tunneling rate from the metastable superheated phase to the stable phase.

If this is right

  • Nucleation rates follow directly from the Euclidean bounce action for both transitions.
  • The deconfinement bubble wall velocity is parametrically small near the critical temperature.
  • The friction force on the bubble is obtained for the chiral transition.
  • Transition parameters such as the strength parameter can be extracted from the computed quantities.

Where Pith is reading between the lines

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

  • Small wall velocities may imply extended metastable lifetimes in cosmological applications of these models.
  • The chiral-transition ansatze could be checked by direct numerical evolution of the bubble profile.
  • The same bounce and velocity methods could be applied to other top-down holographic models of strong-coupling transitions.

Load-bearing premise

Motivated ansatze and approximations for the steady-state configurations are adequate to compute the bubble wall velocity and friction force for the chiral transition.

What would settle it

A full numerical solution of the steady-state bubble equations without the ansatze that produces a wall velocity differing by more than order one from the reported value.

Figures

Figures reproduced from arXiv: 2606.19166 by Aldo L. Cotrone, Francesco Bigazzi, Natalia Pinzani-Fokeeva, Tommaso Trabocchi.

Figure 1
Figure 1. Figure 1: The potential for T¯ = 1.4. The potential has two minima VD = −T¯6/(36π 4 ) for ΦD = −T¯2/(4π 2 ) and VC = −1/(36π 4 ) for ΦC = 1/(4π 2 ). These correspond, respectively, to the deconfined and the confined so￾lutions reviewed above. We will focus on the case T > ¯ 1, where the true vacuum is the deconfined one at Φ = ΦD. See figure 1 for an explicit example. The bubble-like bounce solution ΦB of the Euler-… view at source ↗
Figure 2
Figure 2. Figure 2: Solutions for the bubble profile with T¯ = 1.5 (Blue line), T¯ = 3 (Green line), T¯ = 5 (Red line). The horizontal dashed line corresponds to the (constant) value of the field in the confining (false) vacuum. 2 4 6 8 10 T 2 4 6 8 10 12 14 ρw(T ) [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Bubble wall radius as a function of T¯. wall radius increases as we approach the critical temperature, consistently with expectations from the thin-wall approximation; it then decreases as the temperature increases. This is shown in figure 3. The wall radius is always greater than the radius ρT = (2πT) −1 of the thermal circle, confirming the validity of the O(3) symmetric configuration. For large enough t… view at source ↗
Figure 4
Figure 4. Figure 4: Bounce action (divided by the coupling g) as a function of T¯. 3.1 Bubble nucleation rate The nucleation rate of the bubbles can be estimated by first computing the so-called bounce action [24, 25] S3,B T = S3(ΦB) − S3(ΦC) T , (3.7) which is the difference between the on-shell action of the bounce solution and that of the false vacuum. Our results show that (3.7) is enhanced near the critical temperature, … view at source ↗
Figure 5
Figure 5. Figure 5: Bubble nucleation rates for representative values of [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Energy density as a function of T¯ in the holographic unflavored model. Solid (resp. dashed) lines correspond to stable (resp. metastable) phases. In orange (resp. blue) the deconfined (resp. confined) branch. The dotted purple line is just sketched and corresponds to the unstable branch; its actual slope can be computed following [32]. where f/t stands for false/true vacuum, θ = ρ − 3p 4 , (3.14) is the t… view at source ↗
Figure 7
Figure 7. Figure 7: The solution for α(σ) and its derivative at L˜ = 0.685. The corresponding bounce solution xB(y, σ) that can be obtained by inserting α(σ) into (4.24) and (4.23) for the entire profile is shown in figure 8 [PITH_FULL_IMAGE:figures/full_fig_p021_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The bubble solution for L˜ = 0.685. The values of σ0 as a function of T¯ ≡ T T χ c , (4.34) are reported in figure 9. These approximate very well the dimensionless radius of the bubble 20 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison plot between σ0 trend (blue thick line) and bubble radius R trend (red dashed line), both as functions of T¯. R, 11 defined here as the value of σ for which α is halfway between the true and false vacuum α(R) = 1 2 , (4.35) and reported in the same figure 9. Thus, we can equivalently take σ0 to be the definition of the (dimensionless) bubble radius. Notice that the farther we are from the critic… view at source ↗
Figure 10
Figure 10. Figure 10: The on-shell bounce action S˜ B[xB] as a function of T¯. supercooled case studied in [4], the bounce action diverges near criticality and approaches zero at higher temperatures, following a behavior similar to that of the bubble radius. We believe that this solution provides a first reasonable approximation to the actual bounce solution. The reason for treating this as an approximation is that the solutio… view at source ↗
Figure 11
Figure 11. Figure 11: On the left, a typical profile of α(σ) as a function of σ (L˜ = 0.655). The black dashed line corresponds to αi = y −1 J,f = 0.755. On the right, the profile of α(σ) for various values of L˜ (from left to right: L˜ = 0.695, L˜ = 0.66, L˜ = 0.65, L˜ = 0.647). Black dots represent the transition points from disconnected to connected phase. We have scanned several values of L˜ over the range (4.2) as shown i… view at source ↗
Figure 12
Figure 12. Figure 12: Values of α0 and αi when varying T¯ [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The bounce solution for L˜ = 0.655 is shown on the left, and that for L˜ = 0.695 on the right. The horizontal axis denotes the radial coordinate σ, the vertical axis is the holographic direction y, and the remaining axis corresponds to x. For smaller L˜, closer to the critical value, most of the interior of the bubble is occupied by an almost vertical configuration. For larger L˜, instead, an almost verti… view at source ↗
Figure 14
Figure 14. Figure 14: Bubble radius R (blue line) and σi (orange line) as functions of T¯. action (4.49) can be computed using the solution for α(σ) in (4.23) for any given L˜. We plot its value in figure 15. 1.02 1.04 1.06 1.08T 50 100 150 200 250 300 350 S  B(T ) [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: The on-shell bounce action S˜ B[xB] as a function of T¯. To conclude this section, let us compare the fully variational solution with the approx￾imated one found in the previous section. Examples of profiles α(σ) in the two cases are provided in figure 16. It is also possible to compare the on-shell bounce actions. The full 27 [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Comparison plot between the approximate bounce solution (orange) and the [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: The on-shell bounce action for the approximated (dashed) and fully variational [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: The dimensionless bubble nucleation rate as a function of [PITH_FULL_IMAGE:figures/full_fig_p031_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The energy density of the flavor sector as a function of the temperature. [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Cartoon of a possible steady-state bubble in the rectangular approximation. Note [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Plot of the trailing wall along the holographic radial direction [PITH_FULL_IMAGE:figures/full_fig_p039_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Plot of the terminal velocity of the bubble as a function of the order parameter [PITH_FULL_IMAGE:figures/full_fig_p041_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Cartoon of an alternative configuration of the steady-state bubble in the rectan [PITH_FULL_IMAGE:figures/full_fig_p042_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Plot of the alternative brane profile along the holographic radial direction [PITH_FULL_IMAGE:figures/full_fig_p045_24.png] view at source ↗
read the original abstract

We study the dynamics of first-order inverse phase transitions (driven by superheating) at strong coupling, focusing on the top-down Witten-Sakai-Sugimoto model for holographic QCD. Two cases are considered: the deconfinement transition in the unflavored version of the model and a chiral symmetry-restoring transition occurring in the deconfined phase of the full theory. In both cases, we imagine driving the system into a metastable phase at high temperature, inducing the nucleation of bubbles of the stable phase. For both classes of transitions, we find the corresponding Euclidean bounce solutions and compute the bubble nucleation rates and the relevant transition parameters. For the deconfinement transition, the large jump in the number of degrees of freedom between the two phases suggests that the bubble wall velocity is parametrically small; we provide a rough estimate of it near the critical temperature. In the case of the chiral transition, instead, we compute the bubble wall velocity and the friction force exerted on the bubbles employing motivated ansatze and approximations for the steady-state configurations.

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 / 1 minor

Summary. The paper examines inverse (superheating-driven) first-order phase transitions in the Witten-Sakai-Sugimoto holographic QCD model. It constructs Euclidean bounce solutions and computes nucleation rates and transition parameters for both the deconfinement transition (unflavored model) and the chiral symmetry-restoring transition (flavored model in the deconfined phase). For deconfinement it argues the wall velocity is parametrically small due to the large jump in degrees of freedom and supplies a rough estimate near T_c; for the chiral case it reports wall velocity and friction using motivated ansatze and approximations for steady-state bubble profiles.

Significance. If the numerical solutions and ansatze are reliable, the work supplies concrete, top-down holographic results for nucleation rates and wall dynamics in strongly coupled inverse transitions, which are relevant for cosmological and heavy-ion applications. The explicit construction of bounces in an established model is a positive feature; however, the absence of error control on the chiral results limits the immediate utility of the velocity and friction numbers.

major comments (2)
  1. [Abstract] Abstract and the chiral-transition section: the computation of bubble wall velocity and friction force rests on 'motivated ansatze and approximations for the steady-state configurations' with no reported error estimates, convergence tests, comparison to exact solutions, or sensitivity analysis. Because these quantities are the central output for the chiral case, the lack of validation makes the reported numbers load-bearing and unquantified.
  2. [Abstract] Abstract and nucleation-rate section: the statement that 'Euclidean bounce solutions were found and rates computed' supplies no error estimates, convergence checks, or validation against known limits (e.g., thin-wall or high-T regimes), weakening in the numerical values for both transitions.
minor comments (1)
  1. Notation for the steady-state profiles and friction force should be defined explicitly with reference to the underlying holographic equations.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major point below, indicating where revisions will be made.

read point-by-point responses

  1. Referee: [Abstract] Abstract and the chiral-transition section: the computation of bubble wall velocity and friction force rests on 'motivated ansatze and approximations for the steady-state configurations' with no reported error estimates, convergence tests, comparison to exact solutions, or sensitivity analysis. Because these quantities are the central output for the chiral case, the lack of validation makes the reported numbers load-bearing and unquantified.

    Authors: We agree that the chiral-transition results rely on motivated ansatze for the steady-state bubble profiles, as obtaining fully numerical solutions to the coupled equations for the wall is numerically demanding in this top-down model. The ansatze are constructed to respect the asymptotic behavior and the dominant degrees of freedom in the Witten-Sakai-Sugimoto background, following standard practice in holographic bubble dynamics. Nevertheless, the absence of quantitative error bars or sensitivity scans does limit the precision that can be claimed. In the revised manuscript we will add an expanded discussion of the ansatz limitations, include a parameter-sensitivity study, and qualify the abstract to indicate that the velocity and friction values are approximate. revision: partial

  2. Referee: [Abstract] Abstract and nucleation-rate section: the statement that 'Euclidean bounce solutions were found and rates computed' supplies no error estimates, convergence checks, or validation against known limits (e.g., thin-wall or high-T regimes), weakening in the numerical values for both transitions.

    Authors: The Euclidean bounce solutions were obtained by numerically minimizing the on-shell action subject to the appropriate boundary conditions in the holographic geometry. We will strengthen the revised version by adding explicit convergence tests with respect to the radial discretization and by comparing the computed nucleation rates to the thin-wall limit near T_c and to the high-temperature regime where analytic approximations are available. These additions will be included for both the deconfinement and chiral cases. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivations rely on explicit solutions and model ansatze without reduction to inputs by construction

full rationale

The paper computes Euclidean bounce solutions for nucleation rates and transition parameters directly from the Witten-Sakai-Sugimoto holographic model. For the deconfinement case, the velocity estimate follows from the degrees-of-freedom jump, a model property. For the chiral case, velocity and friction are obtained via motivated ansatze for steady-state profiles. No quoted equations or steps show a fitted parameter renamed as prediction, self-definitional loop, or load-bearing self-citation chain; the central results remain independent computations within the established framework.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed on abstract only; the Witten-Sakai-Sugimoto model is treated as given, with no new free parameters, axioms, or entities introduced in the provided text.

axioms (1)
  • domain assumption The Witten-Sakai-Sugimoto model provides a valid holographic dual for the relevant QCD-like phase transitions at strong coupling.
    Invoked throughout the abstract as the framework for all calculations.

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discussion (0)

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