Approaching Stable Quark Matter

TL;DR

This work asks whether quark matter could be the true ground state of baryon matter in QCD by constraining the bag parameter $B$ through a synthesis of perturbative QCD EOS (up to $ ext{O}( ext{α}_s^2)$) with color superconductivity and nonperturbative vacuum energy, and lattice QCD data for isospin-dense matter. A two-parameter model with $X$ (renormalization-scale proxy) and $B$ is fit to LQCD$_{ m I}$ data via $p(oldsymbol{ extmu},X,B)=p_{ m pQCD}(oldsymbol{ extmu},X)+p_{ m CS}(oldsymbol{ extmu},X)-B$, enabling inferences about stability for 2-flavor and 2+1-flavor quark matter; general thermodynamic bounds are also derived as a robust cross-check. The analysis finds an upper bound $B^{1/4}\lesssim 160$ MeV from current data, with LQCD favoring $B o 0$, and GMOR-based lower bounds pushing $B_{ m min}$ to around $(136$–$153)$ MeV$^4$, which excludes stable 2+1-flavor quark matter while leaving a narrow window for stable 2-flavor quark matter. Together, these results push toward a near-conclusive answer about quark-matter stability and highlight the synergy between high-density pQCD and isospin-dense LQCD data, complemented by model-independent thermodynamic bounds that remain informative where perturbation theory is uncertain.

Abstract

The determination of whether the ground state of baryon matter in Quantum Chromodynamics (QCD) is the ordinary nucleus or a quark matter state remains a long-standing question in physics. A critical parameter in this investigation is the bag parameter $B$, which quantifies the QCD vacuum energy and can be computed using nonperturbative methods such as Lattice QCD (LQCD). By combining the equation of state derived from perturbative QCD (pQCD) with the bag parameter to fit the LQCD-simulated data for isospin-dense matter, we address the stability of quark matter within the LQCD+pQCD framework. Our findings suggest that the current data imposes an upper bound on $B^{1/4} \lesssim 160$ MeV, approaching a conclusive statement on quark matter stability. Given the lower bound on $B$ from the quark condensate contribution to the vacuum energy, the stable 2-flavor quark matter remains possible, whereas the stable 2+1-flavor quark matter is excluded, assuming complete deconfinement and chiral-symmetry restoration and the reliability of pQCD at baryon chemical potentials around the proton mass. Additionally, we derive more general thermodynamic bounds on the quark matter energy-per-baryon and $B$, which, while weaker, provide complementary insights.

Figures (14)

• Figure 1: Left panel: A schematic plot of the $p(\mu_{\text{B}})$ curves of the hadron matter phase and the quark matter phase at zero temperature, when there is only one intersection at $\mu_{\text{B}}=\mu_{\rm t}$ (green square point) between the two phases. The blue circle point at $\mu_{\text{B}}=\mu_{\rm h}$ with $p=0$ gives the energy per baryon of the (free) hadron matter as $\mu_{\rm h}$, while the red circle point at $\mu_{\text{B}}=\mu_{\rm q}$ gives that of the quark matter. The lower endpoint of the quark matter phase at $\mu_{\text{B}}=\mu_0$ and $p = - B$ has $n_{\text{B}} = 0$. The globally stable baryon matter phase (green solid line) follows the trajectory of the maximum $p$ at any given $\mu_{\text{B}}$. Right panel: The same as the left panel but with two intersections at $\mu_{\text{B}}=\mu_{{\rm t}_1}$ and $\mu_{{\rm t}_2}$ between the two phases.
• Figure 2: Left panel: The $p(\mu_{\text{B}}, X) = p_{\rm pQCD} + p_{\rm CS}$ distributions of the benchmarks with $X=1.6$ and $2.2$ for the 2+1- and 2-flavor systems. The black dotted vertical line stands for $\mu_{\text{B}} = 930$ MeV, whose intersections with the $p$ distributions imply the upper bounds on the bag parameter $B$ that allow stable quark matter. Right panel: The parameter region (white) of stable quark matter with trustable pQCD calculations. The regions above the blue solid and orange dashed lines exclude the existence of stable 2+1- and 2-flavor quark matter, respectively. The pQCD calculation breaks down for $\mu_{\text{B}} \leq 930$ MeV in the shaded gray region of $X \leq 1.42$ (see Appendix \ref{['sec:quark']} for details) and thus cannot give conclusive remarks about the stability of the quark matter. Above (below) the black dashed line, the 2+1-flavor (2-flavor) quark matter is more energetically stable, as indicated by the blue (orange) arrows. The two asterisks correspond to the two asterisks marked in the left panel.
• Figure 3: Reproduced distributions for $c_s^2(\mu)$, $p(\mu)/p_{\rm id}(\mu)$, and $\epsilon(\mu)/\epsilon_{\rm id}(\mu)$ using the publicly available LQCD data in Ref. Abbott:2024vhj. We also show the projected distributions with an expected decrease of $1/N_{\rm scaled}=1/\sqrt{10}$ in the bin-wise uncertainties.
• Figure 4: The ratio of the gluon Debye mass $m_{\rm D}(\mu)$ over the color-superconducting gap $\Delta(\mu)$ as a function of the quark chemical potential $\mu$ for $X = 1.2, 1.5, 2.0$.
• Figure 5: The $c_s^2$, $p/p_{\rm id}$, and $\epsilon/\epsilon_{\rm id}$ distributions of the best-fit points (red and green solid lines) with $\mu_{\rm start} = 1000$ MeV and $k = 49, 53$, whose benchmark parameters are $(X, B^{1/4}) = (1.66, 0~{\rm MeV})$ and $(1.78, 0~{\rm MeV})$. Also shown are the distributions of the benchmarks $(X, B^{1/4}) = (1.66, 161~{\rm MeV})$ (red dashed line), which sits on the $90\%$ CL contour around the $1\,\sigma$ ($k=49$) best-fit point, $(X, B^{1/4}) = (1.66, 200~{\rm MeV})$ (red dotted line), and $(1.66, 250~{\rm MeV})$ (red dot-dashed line).