Speed of sound peak in isospin QCD: a natural prediction of the Medium Separation Scheme

Abstract

We present predictions for the zero-temperature equation of state at finite isospin density using the Nambu-Jona-Lasinio (NJL) model within the medium separation scheme (MSS) -- a scheme that explicitly disentangles medium effects from the ultraviolet divergent vacuum terms. Recent lattice QCD results reveal a nonmonotonic speed of sound ($c_s^2$) as a function of isospin chemical potential ($μ_I$), exhibiting explicit violation of the conformal bound $c_s^2 = 1/3$. These findings have attracted significant theoretical interest, as established models -- including the NJL model -- failed to anticipate this behavior prior to lattice simulations. Conventional NJL implementations yield unphysical artifacts, often attributed to regularization scale sensitivity stemming from nonrenormalizability. However, in this work, we demonstrate that the standard NJL framework combined with MSS quantitatively reproduces state-of-the-art lattice data for isospin QCD.

Figures (4)

• Figure 1: Speed of sound $c_s^2$ as a function of the scaled isospin chemical potential $\mu_I/m_\pi$, comparing the LQCD ensembles from Refs. Abbott:2023cojAbbott:2024vhj to NJL model results.
• Figure 2: Pion condensate $\Delta$ as a function of the scaled isospin chemical potential $\mu_I/m_\pi$, within both NJL regularization prescriptions.
• Figure 3: Speed of sound $c_s^2$ as a function of the scaled isospin chemical potential $\mu_I/m_\pi$, comparing the GP-model from Ref. Abbott:2024vhj and the QMM from Ref. Andersen:2025ezj (A) and from Ref. Brandt:2025tkg (B) to the NJL model results.
• Figure 4: The pressure ratio $P/P_{\rm SB}$ (panel a) and the energy density ratio $\epsilon/\epsilon_{\rm SB}$ (panel b), as a function of the scaled isospin chemical potential $\mu_I/m_\pi$. The results are compared with the ones from the GP-model from Ref. Abbott:2024vhj.