Speed of sound peak in two-color dense QCD: confronting effective models with lattice data

TL;DR

This work tackles the problem of reproducing the distinctive peak in the speed of sound $c_s^2$ observed in dense two-color QCD by extending the NJL framework with the Medium Separation Scheme (MSS). By separating medium contributions from UV vacuum divergences, the MSS yields finite gap equations for the diquark condensate $\\Delta$ and the effective mass $M$, and finite expressions for the baryon density. The MSS-modified NJL model reproduces the lattice QCD peak in $c_s^2$ and shows a rising diquark gap with increasing chemical potential, consistent with high-density pQCD expectations, while TRS fails to capture the peak and tends to nonphysical behavior. The results support MSS as a robust regularization tool for dense quark matter in QC$_2$D, with good agreement to lattice data and plausible extrapolations toward the high-density limit, and they motivate future finite-temperature extensions to benchmark against broader lattice results and refine the QCD equation of state.

Abstract

Lattice simulations of two-color, two-flavor Quantum Chromodynamics (QCD) at finite quark chemical potential have revealed a distinctive peak structure in the sound velocity. Although chiral perturbation theory (ChPT) and the Nambu-Jona-Lasinio (NJL) model have been employed to explain this phenomenon, neither approach has fully captured the observed behavior. To address this discrepancy, we have extended the NJL framework by incorporating the Medium Separation Scheme (MSS). This approach isolates medium contributions from divergent integrals, allowing for a more accurate treatment of finite-density effects. Our results indicate a clear increase in the diquark gap ($Δ$) with increasing chemical potential, consistent with what is also seen in perturbative QCD predictions at high densities. {}Furthermore, the MSS-modified NJL model successfully reproduces the observed peak in the sound velocity.

Figures (5)

• Figure 1: Effective quark mass $M$ (in units of GeV) results for ChPT (solid red line), TRS (black dash-dotted line) and for MSS (solid green line). The results for the diquark condensate $\Delta$ (in units of GeV) for ChPT (dashed red line), TRS (dotted black line) and for MSS (dashed green line). All quantities are expressed as functions of the baryon chemical potential divided by the pion mass, $\mu_B/m_\pi$.
• Figure 2: The pressure density $P$ as a function of energy density $\varepsilon$ (all in units of MeV/fm$^3$), for the MSS (continuous green line), TRS (dashed black line) and ChPT (dotted red line). The solid and dashed gray lines correspond to the values of constant speed of sound EoS, $c_s^2=1$ and $c_s^2=1/3$, respectively.
• Figure 3: The energy density (panel a) and pressure (panel b) scaled by the critical baryonic chemical potential to the fourth power, as a function of baryon chemical potential normalized by the pion mass, $\mu_B/m_\pi$, for the MSS (continuous green line), TRS (dashed black line), and ChPT (dotted red line). Blue dots with error bars indicate the lattice data points (taken from Ref. Itou:2023pcl).
• Figure 4: The barion number density divided by baryonic chemical potential to the third power, as a function of baryon chemical potential normalized by the pion mass, $\mu_B/m_\pi$, for ChPT (red dotted line), TRS (black dashed line), MSS (green solid line), and the massless free fermion gas (blue solid line).
• Figure 5: Speed of sound squared, $c_s^2$, as a function of baryonic chemical potential normalized by the pion mass, $\mu_B/m_\pi$, for ChPT (red dotted line), TRS (black dashed line), the analytical expression Eq. (\ref{['fit']}) (blue dot-dashed line), and MSS (green solid line). Blue dots with error bars indicate the lattice datapoints (data taken from Ref. Itou:2023pcl).