Thermal properties of zero sound in asymmetric nuclear matter

Jing Ye; Wei-Zhou Jiang

REVIEW 5 major objections 5 minor 86 references

Thermal properties of zero sound in asymmetric nuclear matter

T0 review · 5 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that at finite temperature zero-sound modes in asymmetric nuclear matter with stiff equations of state undergo a thermal bifurcation into first sound, providing an EOS-sensitive probe of high-density matter.

desk verdict A genuinely new finite-T zero-sound map with an overreaching 'first sound' interpretation; worth a serious referee but needs major revision. read the letter →

arxiv 2505.03437 v1 pith:RCQLU6ZZ submitted 2025-05-06 nucl-th physics.atom-ph

Jing Ye , Wei-Zhou Jiang This is my paper · ORCID

classification nucl-thphysics.atom-ph PACS 21.65.Mn24.30.Cz26.60.-c

keywords zerosoundasymmetricnuclearmatterequationofstatesymmetryenergyrelativisticrandomphaseapproximationmean-fieldmodelfinitetemperaturethermalbifurcation

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

The paper aims to show that zero sound—a collisionless density wave originally predicted by Landau—survives in hot asymmetric nuclear matter and that its thermal behavior can serve as a probe of the nuclear equation of state (EOS). Using the relativistic random phase approximation (RRPA) on four relativistic mean-field (RMF) models, the authors find that warming the system gradually destroys zero sound in soft EOS models while leaving it present over a wider density-temperature region in stiff models. The central discovery is a thermal bifurcation: in stiff models the zero-sound dispersion relation develops a second branch whose velocity approaches the first-sound speed $c_1 = \sqrt{\partial P/\partial \varepsilon}$ for momenta $Q > T$, indicating a transition from zero sound to first sound. Both the bifurcated branch (in stiff models) and the surviving zero-sound branch (in soft models) are highly sensitive to the slope of the symmetry energy $L$, so these collective modes are proposed as observables for the high-density symmetry energy.

What carries the argument

The load-bearing object is the longitudinal dielectric function $\varepsilon_L(q,q_0)=\det(1-D_L\Pi_L)$, whose zeros are the collective modes; zero sound is the root with $q_0 \to 0$ as $q\to 0$. The polarization tensor $\Pi_L$ is computed at finite temperature with Fermi–Dirac distributions instead of zero-temperature step functions, which lifts Pauli blocking and brings in intermediate-state contributions. The interaction matrix $D_L$ encodes the balance between scalar attraction and vector repulsion, including an isoscalar–isovector $\omega$–$\rho$ coupling $\Lambda_V$ that tunes the symmetry energy slope $L$; this balance is what makes the zero-sound properties EOS-sensitive.

What would settle it

A transport calculation or experiment that shows the relaxation time in dense nuclear matter is shorter than the estimate used here (e.g., $\tau \approx 4 \times 10^{-23}$ s at $T = 50$ MeV), so that zero sound is already overdamped below 50 MeV, or an observation that the bifurcated branch's velocity does not approach the first-sound speed at momenta $Q > T$, would settle the claim.

Watch

Extended reading notes

Core claim

The core claim is that at finite temperature the zero-sound mode in RMF models with a stiff EOS undergoes a thermal bifurcation: the dispersion relation, obtained from the zeros of the dielectric function $\varepsilon_L(q,q_0) = \det(1 - D_L \Pi_L)$, acquires a second branch that departs from the zero-sound dispersion and, at momenta $Q > T$, has a sound velocity consistent with the first-sound speed $c_1 = \sqrt{\partial P/\partial \varepsilon}$. This thermal bifurcation converts zero sound into first sound in stiff models (NL3w03, GM1) but not in soft models (TM1w02, FSUGarnet), where the bifurcated branch is spurious and the zero-sound region instead shrinks with temperature. The paper further establishes that the presence or absence of zero sound at high density and temperature is a qualitative marker of EOS stiffness, and that both the bifurcated branch and the soft-model zero-sound branch are strongly dependent on the symmetry energy slope $L$: in stiff models, reducing $L$ extends the bifurcated branch to lower densities; in soft models, reducing $L$ shrinks the zero-sound range. It also finds a nonlinear dispersion relation at low densities, linked to static-matter instability, that still respects the zero-sound limit $q_0 \to 0$ as $q \to 0$.

Load-bearing premise

The paper assumes the hot matter stays collisionless—that quasiparticles collide rarely enough for zero sound to exist—and estimates this with a rough kinetic theory formula for only one of the four models, valid up to about 50 MeV; if the true relaxation time is shorter, the modes would be damped away.

Editorial extensions

If this is right

At high density and temperature, the presence of zero sound becomes a qualitative indicator distinguishing stiff from soft equations of state, since stiff models retain zero sound where soft models lose it.
The density range of the zero-sound and the momentum $Q$ at which the bifurcated branch merges with first sound are sensitive to the symmetry energy slope $L$, so measurements of these modes can constrain the high-density symmetry energy.
In astrophysical settings such as hot proto-neutron stars and heavy-ion collision fireballs, the appearance of zero-sound-like density oscillations would signal a stiff EOS and a relatively long quasiparticle mean free path.
The low-density nonlinear dispersion relation connects zero sound to static-matter instability, tying the collective-mode analysis to the liquid-gas phase transition boundary.

Reading between the lines

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

Because the thermal bifurcation is found in the 1p1h RRPA truncation, including higher-order particle-hole and particle-particle correlations might shift the bifurcation momentum $Q$ or damp the branch; a beyond-RPA calculation would test how robust the effect is.
The sensitivity of both the bifurcated branch and the soft-model zero-sound branch to the symmetry energy slope $L$ suggests that the ratio of zero-sound to first-sound velocity could serve as an isovector observable, complementing neutron-skin and neutron-star radius constraints.
The relaxation-time argument could be upgraded with a full in-medium collision integral; if the collisionless regime actually extends beyond 50 MeV, zero sound would become a viable probe of the EOS at temperatures reached in supernova nucleosynthesis.

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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

First, the part that is real: this is a clean finite-T RRPA extension of the authors' earlier zero-sound work. The finding that zero-sound modes in stiff RMF models bifurcate at temperature, with a branch that exists only for Q > T and that tracks c1 at large momentum, is not in the earlier literature. The stiff/soft contrast and the sensitivity of the mode boundaries to the symmetry-energy slope L are systematic and parameter-free once the RMF sets are chosen. No data fitting, no circularity beyond using their own earlier models, which is fine. They also flag their own caveats: they call the velocity match a 'tendency', say a quantitative interpretation is absent, and concede 1p1h RRPA is likely deficient at high T.

Now the soft spot, and it is load-bearing. The abstract says zero sound 'transforms into first sound.' The calculation is collisionless — no collision integral, no quasiparticle width. Landau theory says the zero/first-sound crossover is driven by ωτ << 1, which a collisionless RPA cannot produce. What they see is a lower branch whose group velocity approaches c1 at large q. That is a kinematic hint, not a demonstration. The authors essentially admit this. Add to that the arbitrary q = 1.5T choice — the existence contours in Fig. 3 depend on it — and a relaxation-time estimate that is only shown for GM1 and only justifies T up to ~50 MeV, while Fig. 3 extends beyond that. So the 'promising signal' language in the abstract is too strong.

The fix is straightforward. Either introduce collisions in some approximation and show the first-sound conversion explicitly, or rescale the claim to 'a bifurcated branch whose velocity approaches first sound over a limited momentum window.' And show the q/T dependence or justify the choice.

Who is it for? People working on collective modes in hot asymmetric matter, heavy-ion EOS probes, or symmetry energy at high density. It deserves a serious referee. I'd send it out, but with instructions that the central claim must be tempered or reconstituted. As is, I'd not cite it in my own work because the headline conclusion is not reliable, but the maps themselves would be worth citing once the interpretation is fixed.

Referee Report

5 major / 5 minor

Summary. The paper computes finite-temperature zero-sound modes in asymmetric nuclear matter within the relativistic random phase approximation (RRPA) built on four relativistic mean-field models (NL3w03, GM1, TM1w02, FSUGarnet). It maps zero-sound existence domains in the temperature-density plane, identifies a distinction between stiff and soft equations of state, and reports that in stiff models the finite-temperature dispersion relation develops a second, lower branch that the authors call a thermal bifurcation. The paper claims that this bifurcated branch transforms into first sound for momenta Q>T, and that both the bifurcated branch in stiff models and the zero-sound branch in soft models are highly sensitive to the slope of the symmetry energy. It also reports a nonlinear low-density dispersion relation that still passes through the origin. The central quantitative output is a set of falsifiable predictions about where zero sound exists and how its properties correlate with EOS stiffness and symmetry-energy slope.

Significance. The formalism and parameter sets are standard, and the numerical scan is a genuine parameter-free prediction once the RMF Lagrangians are fixed; the paper does not fit the zero-sound modes to data. The strongest contribution is the falsifiable statement that the presence or absence of zero sound at high density and temperature is a marker of EOS stiffness, together with the reported strong sensitivity to the symmetry-energy slope L. The authors are also candid about the pragmatic exclusion of superluminal branches and about the absence of a quantitative interpretation of the velocity matching. If the first-sound identification can be put on firmer ground, the results would be interesting for neutron-star cooling and heavy-ion phenomenology. The reliability of the central claim, however, is currently limited by the collisionless nature of the calculation and by the qualitative character of the velocity comparison.

major comments (5)

[Sec. III.C, Fig. 9] The central claim that the bifurcated branch 'transforms into the first sound' is not established by the calculation. The RRPA polarization functions (Eqs. 9-10) and the dielectric function (Eq. 14) contain no collision integral and no quasiparticle width; the relaxation-time estimate in Sec. III.B is used only to argue that the system is collisionless up to about T=50 MeV, not to introduce dissipative terms. In Landau Fermi-liquid theory the zero-sound-to-first-sound crossover is controlled by the collision rate, and first sound is a hydrodynamic mode. The evidence offered is that the group velocity of the lower branch 'shows a tendency' to match c1, which the paper itself describes as lacking a quantitative interpretation (Sec. III.C). A damped RPA pole whose group velocity approaches c1 at large q is not the same as a transformation into first sound. This claim should be either reframed as a conjecture or supported by a calculation that includes collision terms.
[Sec. III.B, Figs. 8-9] The thermal bifurcation and the Q>T threshold depend on a pragmatic exclusion of superluminal branches, and the authors explicitly state that a quantitative interpretation is absent for these modes. Since the unphysical domains are excluded before the physical branches are displayed, the procedure itself shapes the reported existence contours and the claimed transformation to first sound. A quantitative criterion for identifying unphysical modes (for example, a causality condition on the group velocity, or an analysis of the mode width) should be provided, and the sensitivity of the results to that criterion should be checked.
[Fig. 3, Sec. III.B] The existence contours in Fig. 3 are computed at a single momentum-temperature ratio q=1.5T, and at zero temperature at q=1 MeV. Because the zeros of the dielectric function depend on the chosen momentum transfer, the claimed stiff-versus-soft distinction could partly reflect this choice of scale rather than an intrinsic property of the models. A sensitivity scan over q/T is needed to establish that the EOS marker and the reported boundaries in the T-rho_B plane are robust.
[Sec. III.B, after Fig. 7] The paper states that at finite temperature both branches have Im Pi != 0, so the zeros of the complex dielectric function are generally complex and the modes have finite width. The plotted loci (q0,q) need a clear definition: are they zeros of the real part of the dielectric function, minima of |epsilon|, or something else? The damping width relative to q0 also needs to be reported, because without it the physical status of the bifurcated branch is ambiguous. This is particularly important for the claim that the branch becomes first sound, since a broad overdamped pole would not be a propagating sound mode.
[Table II, Sec. III.B] In the symmetry-energy scan, both Lambda_V and g_rho are varied simultaneously, which changes Esym(rho0) as well as the slope L. The text attributes the observed changes in the sound branches specifically to the slope of the symmetry energy, but the calculation does not isolate L at fixed symmetry-energy magnitude. A separation of the L-dependence from the Esym(rho0)-dependence, or a discussion of their covariance, is needed before the claim of high sensitivity to L can be accepted as stated.

minor comments (5)

[Sec. III.B] The relaxation-time estimate is made only for the GM1 model; similar estimates or at least a rough range for NL3w03, TM1w02, and FSUGarnet would make the claim that the system is collisionless up to T=50 MeV more convincing.
[Abstract] The phrase 'resulting in the transform of zero sound into the first sound' contains a grammatical error; it should read 'resulting in the transformation of zero sound into first sound.'
[Sec. II, Eqs. (7)-(10)] The paper refers to Ref. [43] for the explicit forms of Pi_s and Pi_m at zero temperature, but only Pi_L is displayed at finite temperature. Providing the finite-temperature forms of the other polarization components, or making clear that they are obtained by the same thermal replacement, would improve readability and reproducibility.
[Sec. III.B, Fig. 8] The inset of Fig. 8 is described as showing the dispersion relation of the high-q0 branch at lower densities, but the relation between the inset and the main panel is not fully explained in the text; a sentence connecting them would help the reader.
[Sec. III.C] The comparison between c0, c1, and v_F in Fig. 9 would be easier to interpret if the curves were labeled directly in the figure or if a table of numerical values were given for a few representative momenta and temperatures.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the zero-sound results are computed from a parameter-free RRPA response once the published RMF parameter sets are fixed.

full rationale

The derivation chain is self-contained as a model study. The polarization functions (Eqs. 9-10) and the dielectric function (Eq. 14) are built from thermal mean-field Green functions with no adjustable response parameters, and the zero-sound mode is obtained by locating zeros of det(1 - D_L Π_L) (Eq. 13). The EOS-stiffness classification and the symmetry-energy sensitivity are outputs of scanning published RMF parameter sets, not inputs fitted to the zero-sound results. The comparison of the bifurcated branch velocity with the thermodynamic first-sound velocity c1 = sqrt(∂P/∂ε) in Sec. III.C is an independent benchmark; no equation in the paper forces the RPA pole velocity to equal c1, so the observed tendency is a computed coincidence rather than a construction. The self-citations to Ref. [43] (modified parameter sets and methodology) and Ref. [73] (parameter readjustment) are ordinary references to independently published model inputs; they do not carry an unverified uniqueness claim and do not smuggle the central conclusion. The paper's own caveats that the relaxation-time estimate places the system in the collisionless regime only up to about T = 50 MeV and that the 1p1h RRPA may be deficient at high temperature (Secs. III.B and III.C) are physical limitations of the calculation, not circular steps. No derivation step reduces to its own inputs, and no fitted quantity is renamed as a prediction.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The derivation is self-contained given the RMF parameter sets, which are taken from earlier literature or the authors' previous work. No new free parameters are fitted to reproduce zero sound. The only hand-chosen scale is the q=1.5T probe momentum. The load-bearing physical assumption is the collisionless regime at finite temperature.

free parameters (1)

q/T = 1.5
The momentum transfer q is set to 1.5 T following the equipartition theorem, and the zero-sound existence contours in the T-rho_B plane are computed at this fixed ratio. This choice is not fitted to data, but it is a hand-chosen scale that shapes the claimed extent of the zero-sound region.

assumptions (5)

domain assumption The RMF Lagrangian with nonlinear sigma, omega self-interactions and omega-rho coupling (Eq. 1) is an adequate basis for hot nuclear matter EOS and collective modes.
All results inherit the model space and parameter values of the chosen RMF sets.
standard math Zeros of the RRPA dielectric function (Eq. 13) identify the collective zero-sound modes.
Standard random-phase approximation criterion for collective modes, as in the cited literature.
domain assumption Finite-temperature nucleon and antinucleon distribution functions (Eq. 4) describe the thermal state.
Thermal occupation of quasiparticle states is assumed, including antiparticles in the Dirac sea.
domain assumption The system is in the collisionless regime for zero sound up to about T = 50 MeV.
Based on a relaxation-time estimate in Sec. III.B that is applied to all models though computed only for GM1.
ad hoc to paper Unphysical superluminal branches of the dispersion relation are excluded pragmatically.
Branches whose velocity exceeds the speed of light are classified as unphysical since a quantitative interpretation is absent (Sec. III.B).

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Cite this review

Pith. "Pith review of Thermal properties of zero sound in asymmetric nuclear matter." pith.science (2026). https://pith.science/paper/RCQLU6ZZ

@misc{pith2026250503437,
  author       = {Pith},
  title        = {Pith review of: Thermal properties of zero sound in asymmetric nuclear matter},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RCQLU6ZZ}},
  note         = {Machine review of arXiv:2505.03437}
}

abstract

The zero-sound modes at finite temperature are investigated with the relativistic random phase approximation to signal the uncertainty of the equation of state (EOS) of asymmetric nuclear matter. It is observed that in typically selected stiff and soft relativistic mean-field (RMF) models, zero-sound modes arise at low temperature, whereas increasing the temperature gradually breaks the zero sound in soft models, with a smaller density range compared to stiff models. At high density, the presence or absence of zero sound turns out to be correspondingly the character of the stiff or soft RMF EOS. More strikingly, we find by analyzing the dispersion relation and sound velocity that at finite temperature the zero-sound modes in RMF models with the stiff EOS undergo a thermal bifurcation, resulting in the transform of zero sound into the first sound at some momentum $Q>T$. The thermally bifurcated sound branch in the stiff models and the zero-sound branch in the soft models are both highly sensitive to the slope of the symmetry energy, providing promising signals for the pending high-density symmetry energies. In addition, it is found that there exists a nonlinear dispersion relation for both the stiff and soft models that supports the zero sound in the relatively lower density region.

Figures

Figures reproduced from arXiv: 2505.03437 by the authors.

**Figure 3.** FIG. 3: (Color online) Contours of zero sounds with various [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

**Figure 4.** FIG. 4: (Color online) Contours of zero sounds with various [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗

Figures from the paper (3 more)

**Figure 6.** Figure 6: FIG. 6: (Color online) Density profile of (zero-)sound modes [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 8.** Figure 8: FIG. 8: (Color online) Sound dispersion relation in symmet [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]

**Figure 9.** Figure 9: FIG. 9: (Color online) The sound velocities as a function of [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

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The momentum transfer ( q) is set to 1.5 times the temperature ( T )

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Reviewed August 15, 2026 · model on record in the stance chip above.

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The momentum transfer ( q) is set to 1.5 times the temperature ( T )

3) in the T − ρB space. The momentum transfer ( q) is set to 1.5 times the temperature ( T ). At zero temperature, q is set to 1 MeV. B. Zero-sound occurrence and bifurcation at ﬁnite temperature In Fig. 3, we depict the contours of the zero sounds for symmetric and asymmetric nuclear matter in the 5 TABLE I: Parameters and saturation properties for vario...

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One may think that the temperature is not guaranteed to be a Lorentz scalar [75], and the thermal ﬂuctua- tion may perhaps aﬀect the causality limit especially for low-energy modes

at ﬁ- nite temperature, which does not exclude the possibil- ity of the superluminal velocity of massless zero sound. One may think that the temperature is not guaranteed to be a Lorentz scalar [75], and the thermal ﬂuctua- tion may perhaps aﬀect the causality limit especially for low-energy modes. Alternatively, we can further analyze this phenomenon acc...

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