Chiral-scale effective field theory for dense and thermal systems

Abstract: In this contribution, I will present some properties of nuclear matter (NM) by using the chiral-scale effective field theory that is anchored on the chiral, scale and hidden local flavor symmetries of QCD. We show that the sound velocity (SV) of the compact star matter can saturate the conformal limit, the SV exhibits a peak configuration in the intermediate density. To extend the chiral-scale effective field theory to both dense and tnermal systems, we setup a chiral-scale density counting (CSDC) rule and explore the contributions up to $\mathcal{O}(k_c^{12})$.

• The paper develops a novel chiral-scale EFT that reinstates the σ meson as a dilaton to realistically describe dense nuclear matter.
• The methodology integrates mesonic and baryonic sectors with density-dependent couplings to capture sound velocity saturation and peak behaviors.
• The approach introduces a chiral-scale density counting rule validated up to N⁴LO, offering insights into neutron star interiors and thermal QCD systems.

Chiral-Scale Effective Field Theory for Dense and Thermal Systems: An Authoritative Analysis

Introduction and Motivation

Chiral effective field theories (EFTs), rooted in the chiral symmetry of quantum chromodynamics (QCD), have achieved significant success in describing nuclear interactions. Nonetheless, conventional chiral EFTs are hampered by the absence of the $\sigma$ meson—critical for the attractive nuclear force—due to integration in nonlinear realizations of chiral symmetry. The present work leverages the QCD trace anomaly to reinstate the iso-scalar scalar $\sigma$ meson, interpreting it as the dilaton: the pseudo-Nambu-Goldstone boson associated with scale symmetry breaking [Crewther & Tunstall, (Crewther et al., 2013)]. This chiral-scale EFT framework integrates the nucleons, pions, vector mesons, and the dilaton, explicitly encoding chiral, scale, and hidden local flavor symmetries.

Formal Structure of Chiral-Scale EFT

The Lagrangian structure decomposes into mesonic $(\mathcal{L}_M)$ and baryonic $(\mathcal{L}_B)$ components. The mesonic sector encapsulates pionic, rho, and omega interactions, scalar meson kinetic and potential terms, and chiral mass terms, regulated through the dilaton compensator $\chi$. The baryonic sector features scale-dependent kinetic and interaction terms for nucleons, incorporating density- and scale-dependent coupling modifications—reflecting Brown-Rho scaling. Notably, parameters are allowed explicit density dependence, consistent with the topological transition insights from the skyrmion crystal approach [Ma et al., (Ma et al., 2013)]. This framework naturally accommodates modifications of $f_\pi$, $m_N$, $m_\rho$, etc., and their saturation behavior at high density, integral to modeling the transition from skyrmions to half-skyrmions in dense matter.

Pseudoconformal Structure and Sound Velocity Saturation

A salient prediction is that the sound velocity (SV) in compact star matter can saturate the conformal limit ($c_s^2 \to 1/3$ for the speed of sound squared) post topological transition at density $n_{1/2}$. This transition, parameterized and tuned around nuclear matter saturation density $\sigma$0, is supported by empirical constraints ($\sigma$1, as established by heavy-ion and nuclear data [Ma & Rho, (Ma et al., 2018)]). Importantly, the model achieves this SV saturation in a density regime previously considered inaccessible to conformal behavior—contradicting prior consensus that such a phenomenon could only arise above the perturbative QCD threshold [Tews et al., (Tews et al., 2018)]. In this framework, SV behavior occurs despite the energy-momentum tensor trace remaining a nonzero density-independent constant, distinguishing this pseudoconformal scenario from true scale invariance.

Peak Structure of Sound Velocity

Another distinctive result is the emergence of a pronounced peak in SV at intermediate densities. This phenomenon is typically attributed to configuration or phase transitions but had eluded unified hadronic modeling. Within the chiral-scale EFT, mean field calculations demonstrate that the peak arises inherently due to the model’s treatment of medium-modified vector meson masses, which are coupled to the dilaton compensator. The screening mass of the omega meson, $\sigma$2, is prevented from vanishing by the stabilization mechanism intrinsic to the dilaton dynamics, ensuring system stability against unbounded vector repulsion. Comparative studies confirm this mechanism is unique to chiral-scale EFT, absent in standard Walecka-type models [Zhang et al., (Zhang et al., 2024)].

Chiral-Scale Density Counting (CSDC) Rule: Extension to Dense and Thermal Systems

Extension to dense and thermal environments requires systematic expansion of the dilaton field, encapsulated by Taylor expansion in $\sigma$3. The author establishes a chiral-scale density counting (CSDC) rule, validated up to densities relevant for neutron star cores ($\sigma$4 MeV, corresponding to $\sigma$5), allowing consistent ordering of interaction terms to $\sigma$6 (N$\sigma$7LO). Mesonic fields and couplings are assigned orders respecting their derivative structure and background field nature. Numerical validation confirms that NM properties (binding energy, symmetry energy, incompressibility, etc.) converge appropriately under this scheme, aligning closely with empirical values and supporting the sufficiency of including terms up to N$\sigma$8LO [Xiong et al., (Xiong et al., 6 Nov 2025)]. The CSDC rule is argued to remain robust up to thermal systems with temperatures $\sigma$9 MeV.

Implications and Prospective Developments

The theoretical implications are substantial. The restoration of the scalar degree of freedom via the dilaton enables a realistic and tractable description of dense nuclear matter within an EFT framework more congruent with QCD symmetries. The saturation and peak structure of the SV provide quantitative predictions for neutron star interiors and inform constraints on the nuclear EoS, with repercussions for astrophysics and compact star phenomenology. The pseudoconformal scenario, realized below the perturbative QCD regime, potentially influences the modeling of hadron-quark continuity and compact star properties [Ma & Rho, (Ma et al., 2019)]. The CSDC rule facilitates controlled expansion for both dense and hot matter, enhancing the reach of chiral-scale EFTs into thermal QCD environments.

Future research trajectories include parameter space exploration, higher loop corrections to nuclear matter, and incorporation of strangeness degrees of freedom—critical for advancing towards full QCD effective descriptions of dense stellar objects.

Conclusion

The paper develops a chiral-scale effective field theory codifying chiral, scale, and local flavor symmetries, introducing the dilaton to resolve longstanding shortcomings in describing dense nuclear matter. Strong numerical results establish that sound velocity can saturate the conformal limit in neutron stars, and a peak structure at intermediate densities emerges naturally due to dilaton-modified vector meson masses. The chiral-scale density counting rule enables systematic calculations in dense and thermal regimes. The approach constitutes a theoretically rigorous and numerically validated framework for modeling nuclear matter, with broad implications for QCD-based effective theories and astrophysical applications.