Violation of the Conformal Limit at Finite Density: Insights from Effective Models and Lattice QCD

TL;DR

The paper investigates the violation of the conformal limit for the speed of sound in dense QCD matter using the Nambu–Jona-Lasinio (NJL) model equipped with the Medium Separation Scheme (MSS). It analyzes three contexts—QCD at finite isospin density, two-color QCD, and two-flavor color superconductivity—showing that the MSS properly separates medium from vacuum contributions and reproduces a robust peak in $c_s^2$ above the conformal value while guiding the system toward $c_s^2=1/3$ at high density. The work demonstrates that regulator artifacts in traditional schemes can obscure essential nonperturbative physics, and that MSS yields regulator-independent, lattice- and perturbative-consistent thermodynamics, including softer equations of state and regulated condensates. These insights provide a consistent, physically motivated framework for connecting low-density lattice results to high-density QCD and have implications for neutron-star phenomenology and the interpretation of dense-matter observables.

Abstract

In this work, we discuss recent results obtained with the application of the medium separation scheme (MSS) in different contexts where a clear violation of the conformal limit for the speed of sound at finite density has been observed in Quantum Chromodynamics (QCD). We analyze several scenarios, including QCD at finite isospin density, two-color QCD, and two-flavor color superconductivity. Whenever possible, we compare our findings with lattice QCD (LQCD) results, showing that the Nambu--Jona-Lasinio (NJL) model combined with the MSS provides a consistent description across different regimes of the QCD phase diagram. Our analysis highlights how effective models, when properly regularized, can capture essential nonperturbative features of dense QCD matter, offering complementary insights to lattice simulations.

Figures (13)

• Figure 1: Effective quark mass $M$ as a function of the normalized isospin chemical potential $\mu_I/m_\pi$. Dashed, solid, dotted, and dot-dashed lines correspond to TRS + 3D, MSS + 3D, TRS + PV, and MSS + PV, respectively. The vertical black dashed line indicates the scale $\Lambda_{\text{3D}}$, while the vertical orange dotted line represents the corresponding Pauli--Villars (PV) cutoff, $\Lambda_{\text{PV}}$.
• Figure 2: Pion condensate $\Delta_\pi$ as a function of the normalized isospin chemical potential $\mu_I/m_\pi$. The line styles follow the same convention as in Figure \ref{['fig:iso_mass']}. The vertical black dashed line indicates the scale $\Lambda_{\text{3D}}$, while the vertical orange dotted line represents the corresponding Pauli--Villars (PV) cutoff, $\Lambda_{\text{PV}}$.
• Figure 3: Equation of state $P \times \varepsilon$ of isospin dense matter. The line styles follow the same convention as in Figure \ref{['fig:iso_mass']}, and guidelines corresponding to constant values of the speed of sound, $c_s^2 = 1/3$ and $c_s^2 = 1$, are also shown.
• Figure 4: Speed of sound squared $c_s^2$ as a function of the normalized isospin chemical potential $\mu_I/m_\pi$. The line styles follow the same convention as in Figure \ref{['fig:iso_mass']}. The vertical black dashed line indicates the scale $\Lambda_{\text{3D}}$, while the vertical orange dotted line represents the corresponding Pauli--Villars (PV) cutoff, $\Lambda_{\text{PV}}$.
• Figure 5: Effective mass $M$ as a function of quark chemical potential $\mu$. The vertical black dashed line indicates the scale $\Lambda_{\text{3D}}$, while the vertical orange dotted line represents the corresponding Pauli--Villars (PV) cutoff, $\Lambda_{\text{PV}}$. The line styles follow the same convention as in Figure \ref{['fig:iso_mass']}.