Pseudo-Conformal Sound Speed in Dense Matter

• Pseudo-conformal sound speed is a nonconformal behavior where the derivative dP/dε approaches 1/3 without the full requirements of true conformal symmetry.
• It emerges in varied regimes such as compact-star effective theory, dense QCD-like matter, and holographic models, displaying overshoot and universal features.
• Analyses of pseudo-conformality inform neutron-star structure by reconciling heavy star masses, realistic radii, and tidal deformability constraints.

Pseudo-conformal sound speed denotes a family of nonconformal equation-of-state behaviors in which the sound speed is organized around the conformal benchmark $c_s^2=\partial P/\partial \epsilon=1/3$ in four-dimensional relativistic matter, but without exact conformal symmetry. In one usage, prominent in compact-star effective theory, the medium develops $v_s^2/c^2\approx 1/3$ because the trace of the energy-momentum tensor becomes approximately density-independent while remaining nonzero. In another usage, common in dense QCD-like matter and isospin matter, the sound speed rises above $1/3$ at intermediate density and only later returns toward the asymptotic conformal limit. Holographic work introduces a third usage, in which nonconformal systems exhibit universal sound-speed limits that may coincide with or differ from the conformal value depending on the thermodynamic regime (Rho, 2022, Itou et al., 2023, Yang et al., 2017).

1. Definitions and thermodynamic criteria

The thermodynamic definition common to the cited literature is

$c_s^2=\frac{\partial P}{\partial \epsilon},$

or, equivalently at zero temperature, $dP/d\epsilon$. In lattice implementations at finite chemical potential one also encounters finite-difference estimators such as

$c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$

For a conformal system in four spacetime dimensions, scale invariance implies the conformal value $c_s^2=1/3$ (Shukla et al., 9 Jul 2025, Itou et al., 2023).

The compact-star literature distinguishes sharply between true conformality and pseudo-conformality. True conformality would require a vanishing trace of the energy-momentum tensor,

$\theta^\mu_{\ \mu}=\epsilon-3P=0,$

whereas the pseudo-conformal construction instead assumes that $\epsilon-3P$ is approximately density-independent but nonzero. Under the relation

$$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$

one obtains $v_s^2/c^2\approx 1/3$0 whenever $v_s^2/c^2\approx 1/3$1 and $v_s^2/c^2\approx 1/3$2 (Ma et al., 2018, Rho, 2022).

Dense QCD-like matter motivates a different but related usage. In dense two-color QCD and in isospin matter, the system is not conformal because the trace anomaly is nonzero, yet the sound speed develops a pronounced peak and can exceed $v_s^2/c^2\approx 1/3$3 before asymptotically returning toward the free-gas value. In that setting, “pseudo-conformal” refers not to a constant $v_s^2/c^2\approx 1/3$4 plateau but to a nonmonotonic approach to the conformal regime (Itou et al., 2023, Lopes et al., 18 Jul 2025).

Holographic studies broaden the terminology further. Some analyses prove or conjecture that $v_s^2/c^2\approx 1/3$5 is an upper bound in stable high-temperature branches of single-scalar models, whereas others identify universal nonconformal constants in low-temperature or large-chemical-potential limits. The shared feature is not exact scale invariance, but universal or nearly universal sound-speed structure in nonconformal systems (0905.0903, Yang et al., 2017).

2. Pseudo-conformal sound speed in compact-star effective theory

In the compact-star program based on scale-chiral effective theory, $v_s^2/c^2\approx 1/3$6HLS or the closely related GnEFT framework contains pions, nucleons, hidden local symmetry vector mesons $v_s^2/c^2\approx 1/3$7, and a light scalar dilaton $v_s^2/c^2\approx 1/3$8. Its central mechanism is a topology change from skyrmions to half-skyrmions at a density $v_s^2/c^2\approx 1/3$9 in the range $1/3$0, interpreted as a change of effective degrees of freedom without a low-order phase transition (Ma et al., 2018, Lee et al., 2021).

Above $1/3$1, the space-averaged quark condensate vanishes while remaining locally nonzero, $1/3$2 stays nonzero, and parity doubling drives the nucleon mass toward a density-independent constant $1/3$3. The dilaton sector yields

$1/3$4

so an approximately density-independent $1/3$5 makes the trace nearly density-independent as well. This is the direct origin of the pseudo-conformal value $1/3$6 in the model (Ma et al., 2021, Rho, 2022).

A convenient high-density parameterization used in this setting is

$1/3$7

with $1/3$8 and $1/3$9 fixed by matching pressure and chemical potential at $c_s^2=\frac{\partial P}{\partial \epsilon},$0. For $c_s^2=\frac{\partial P}{\partial \epsilon},$1, this form reproduces the pseudo-conformal regime and yields $c_s^2=\frac{\partial P}{\partial \epsilon},$2 without requiring $c_s^2=\frac{\partial P}{\partial \epsilon},$3 (Ma et al., 2018).

The phenomenological significance is that this construction is explicitly designed to evade the claim that the conformal sound-speed bound is incompatible with massive neutron stars. The model was reported to accommodate $c_s^2=\frac{\partial P}{\partial \epsilon},$4, with representative $c_s^2=\frac{\partial P}{\partial \epsilon},$5 radii around $c_s^2=\frac{\partial P}{\partial \epsilon},$6–$c_s^2=\frac{\partial P}{\partial \epsilon},$7 km and $c_s^2=\frac{\partial P}{\partial \epsilon},$8 for $c_s^2=\frac{\partial P}{\partial \epsilon},$9 (Ma et al., 2018). The same line of work argues that the star core is populated by baryon-charge-fractionalized quasi-fermions or quasi-baryons, neither conventional baryons nor explicit deconfined quarks, and interprets this as hadron-quark continuity realized in hadronic variables (Lee et al., 2021, Ma et al., 2021).

3. Intermediate-density overshoot in dense QCD-like matter

Lattice studies of dense two-color QCD provide a first-principles realization of super-conformal sound speed. Because $dP/d\epsilon$0 with even flavors has no sign problem, simulations at nonzero quark chemical potential reveal a low-temperature phase structure containing a hadronic phase, a hadronic-matter regime, a Bose–Einstein condensed superfluid phase, and a BCS-like regime at higher density. The onset of superfluidity occurs at $dP/d\epsilon$1, the diquark condensate $dP/d\epsilon$2 becomes nonzero, and the sound speed follows the ChPT form

$dP/d\epsilon$3

near onset before rising above $dP/d\epsilon$4 in the denser regime (Itou et al., 2023, Itou et al., 28 Nov 2025).

The interpretation given in that literature is explicitly nonconformal. The low-density side is controlled by superfluidity and ChPT; the high-density side should eventually approach perturbative QCD, where $dP/d\epsilon$5 from below; but the intermediate regime remains confining and strongly interacting, so the equation of state stiffens and $dP/d\epsilon$6 develops a peak. This is the sense in which dense $dP/d\epsilon$7 exhibits a pseudo-conformal pattern: overshoot above the conformal value followed by eventual return toward it (Itou et al., 2023).

Cold isospin QCD yields an analogous structure. In NJL analyses using the Medium Separation Scheme, pion condensation sets in when $dP/d\epsilon$8, and the model reproduces the lattice-observed nonmonotonic peak in $dP/d\epsilon$9. For one lattice ensemble the peak is at approximately $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$0 with $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$1; for later NPLQCD results it shifts to roughly $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$2 with a similar maximum height. MSS also yields the return toward $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$3 at large $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$4 (Lopes et al., 18 Jul 2025).

Magnetized hybrid-star models provide a more anisotropic version of the same pattern. In the magnetic dual chiral density wave phase, the sound speed is below $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$5 in hadronic matter, rises above $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$6 in the intermediate-density magnetized quark phase, and returns toward $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$7 in a high-density MIT-bag description. Because the magnetic field splits the pressures, the relevant quantities are

$c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$8

In the strong-field lowest-Landau-level regime the model gives $c_s^2(\mu)=\frac{\Delta p(\mu)}{\Delta e(\mu)}.$9 and $c_s^2=1/3$0, making the super-conformal enhancement explicitly directional (Ferrer et al., 2022).

A broader statistical inference from neutron-star EOS ensembles points in the same direction: among more than $c_s^2=1/3$1 EOSs and more than $c_s^2=1/3$2 stellar models consistent with nuclear theory, perturbative QCD, and astronomical observations, models with $c_s^2=1/3$3 throughout the stellar interior amount to only $c_s^2=1/3$4 of the final constrained sample. This does not prove that super-conformal sound speed is mandatory, but it makes a peak above $c_s^2=1/3$5 the natural expectation in realistic neutron-star matter (Altiparmak et al., 2022).

4. Asymptotic recovery of the conformal limit and model reliability

The asymptotic behavior of the sound speed has been proposed as a stringent test of whether an effective quark model is physically reliable at high baryon density. In particular, dense QCD matter at very large chemical potential is expected to approach a free massless quark gas, so one should recover

$c_s^2=1/3$6

This asymptotic criterion sharply distinguishes local NJL-like models from momentum-dependent dynamical quark models (Shukla et al., 9 Jul 2025).

In a standard local NJL model with vector interactions,

$c_s^2=1/3$7

the scalar density vanishes asymptotically but the vector mean field does not weaken. In the chiral limit this enforces

$c_s^2=1/3$8

which yields anomalous scaling $c_s^2=1/3$9, $\theta^\mu_{\ \mu}=\epsilon-3P=0,$0, and finally

$\theta^\mu_{\ \mu}=\epsilon-3P=0,$1

The model therefore approaches the causal upper value rather than the QCD conformal limit, making the failure structural rather than merely numerical (Shukla et al., 9 Jul 2025).

The proposed resolution is a dynamical quark model with momentum-dependent dressing functions,

$\theta^\mu_{\ \mu}=\epsilon-3P=0,$2

$\theta^\mu_{\ \mu}=\epsilon-3P=0,$3

Defining the Fermi momentum by $\theta^\mu_{\ \mu}=\epsilon-3P=0,$4, asymptotic freedom ensures that the interaction correction dies away at large external momentum, so the correct limits $\theta^\mu_{\ \mu}=\epsilon-3P=0,$5, $\theta^\mu_{\ \mu}=\epsilon-3P=0,$6, and $\theta^\mu_{\ \mu}=\epsilon-3P=0,$7 are recovered without ad hoc modifications (Shukla et al., 9 Jul 2025).

A separate corrective strategy appears in isospin matter. There the Medium Separation Scheme disentangles medium-dependent pieces from ultraviolet-divergent vacuum terms, applies the cutoff only to vacuum contributions, and leaves the $\theta^\mu_{\ \mu}=\epsilon-3P=0,$8-dependent finite parts uncut. In that context, MSS preserves the medium response needed for a nonmonotonic peak and the correct high-density conformal approach, whereas traditional cutoff regularization loses those features (Lopes et al., 18 Jul 2025). Taken together, these results suggest that pseudo-conformal behavior is highly sensitive to whether the model implements the correct momentum or medium dependence in the ultraviolet and asymptotic regimes.

5. Holographic bounds, universal limits, and violations

Holography supplies both restrictive bounds and explicit counterexamples. In a class of strongly coupled four-dimensional field theories at zero chemical potential with gravity duals described by five-dimensional Einstein gravity plus a single scalar field, high-temperature expansion around the $\theta^\mu_{\ \mu}=\epsilon-3P=0,$9-Schwarzschild background gives

$\epsilon-3P$0

with $\epsilon-3P$1 for $\epsilon-3P$2. Hence $\epsilon-3P$3 on energetically favored stable branches at high temperature, and $\epsilon-3P$4 acts as an upper bound within that class (0905.0903).

A more general Einstein–Maxwell–scalar analysis in $\epsilon-3P$5-dimensional gravity identifies three universal limits: $\epsilon-3P$6

$\epsilon-3P$7

$\epsilon-3P$8

For $\epsilon-3P$9 these become $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$0, $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$1, and $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$2. Here pseudo-conformality means that a nonconformal theory nevertheless approaches simple universal constants in extreme thermodynamic limits (Yang et al., 2017).

The bound is not universal across all holographic constructions. For planar hairy black holes with a scalar in the Breitenlohner–Freedman window and mixed boundary conditions, the sound speed satisfies a universal formula of the form

$$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$3

where the second term is not sign definite. In an $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$4-invariant truncation of type IIB supergravity, the resulting deformed theory exhibits $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$5 for an intermediate range of the deformation parameter. This establishes that holographic super-conformal sound speed can arise from scalar-sector deformation and mixed boundary conditions rather than from finite charge density alone (Anabalon et al., 2017).

The holographic literature therefore does not support a single doctrine. It instead separates sharply by assumptions: zero chemical potential versus finite density, single-scalar versus broader matter content, AdS-invariant versus mixed boundary conditions, and stable high-temperature branches versus other thermodynamic regimes.

6. Phenomenological implications and conceptual status

Pseudo-conformal sound speed directly controls the stiffness of the equation of state and therefore affects neutron-star maximum masses, radii, tidal deformabilities, and oscillation spectra. In compact-star applications, the pseudo-conformal construction is used to reconcile $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$6 in the core with heavy stars near $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$7 and with gravitational-wave constraints. In overshoot scenarios, the peak above $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$8 supplies the extra stiffness needed at intermediate density before the EOS softens again toward the asymptotic QCD limit (Ma et al., 2018, Altiparmak et al., 2022).

A common misconception is that $$\frac{\partial}{\partial n}\langle \theta_\mu^\mu\rangle = \frac{\partial \epsilon(n)}{\partial n}\left(1-3\frac{v_s^2}{c^2}\right),$$9 or $v_s^2/c^2\approx 1/3$00 automatically means conformal matter. The compact-star literature rejects that identification explicitly: the defining feature of pseudo-conformality is precisely that the sound speed takes the conformal value while $v_s^2/c^2\approx 1/3$01 (Rho, 2022). A second misconception is that the conformal value must be a universal upper bound. High-temperature single-scalar holography supports such a bound within a narrow class, but dense QCD-like lattice systems, mixed-boundary-condition holography, and several neutron-star analyses all provide controlled settings in which $v_s^2/c^2\approx 1/3$02 occurs (0905.0903, Itou et al., 2023, Anabalon et al., 2017).

Not every model with nonmonotonic $v_s^2/c^2\approx 1/3$03 realizes a clean pseudo-conformal regime. In hyperonic and deconfinement models based on continuous Gibbs constructions, the sound speed can display threshold-induced peaks, pronounced drops, and finite discontinuities at mixed-phase boundaries rather than a stable conformal-like plateau. In that setting, hyperon onset and deconfinement generate nonconformal structure more than pseudo-conformal saturation (Aguirre, 2022).

The broader concept also extends beyond dense QCD proper. Landau-hydrodynamic analyses of heavy-ion rapidity spectra treat $v_s^2/c^2\approx 1/3$04 with constant but nonconformal $v_s^2/c^2\approx 1/3$05 as an effective average over the evolution, while a five-dimensional Kaluza–Klein Fermi gas exhibits a pseudo-conformal-like thermodynamic regime in which $v_s^2/c^2\approx 1/3$06 approaches $v_s^2/c^2\approx 1/3$07, $v_s^2/c^2\approx 1/3$08, or $v_s^2/c^2\approx 1/3$09 depending on the KK spectrum and repulsive interaction. These examples do not invoke the same microscopic mechanism as dense QCD, but they reinforce the point that conformal-like sound-speed behavior is a broader structural phenomenon rather than a single theory-specific signature (Biswas et al., 2019, Horváth et al., 7 Feb 2025).

Taken across the cited literature, pseudo-conformal sound speed is best understood as a nonunique but technically precise label for equation-of-state behavior organized around conformal benchmarks in systems that remain nonconformal. Its meaning depends on context: saturation at $v_s^2/c^2\approx 1/3$10 with nonzero trace in compact-star EFT, overshoot above $v_s^2/c^2\approx 1/3$11 followed by asymptotic return in dense QCD-like matter, or universal nonconformal limits in holography. What unifies these usages is the role of sound speed as a diagnostic of hidden scales, changing degrees of freedom, and the asymptotic consistency of dense-matter models.