Cosmic QCD transition-from quark to strangeon and nucleon

Technical Summary: Cosmic QCD Phase Transition: From Quark to Strangeon and Nucleon?

Problem Statement
The paper addresses the nature of the Quantum Chromodynamics (QCD) phase transition in the early Universe and its potential role in generating dark matter. While the Standard Model suggests a smooth crossover transition from Quark-Gluon Plasma (QGP) to hadronic matter at a temperature of $T \sim 170$ MeV, the exact dynamics remain unclear due to the breakdown of perturbative theory. A central question is whether stable "strong matter" nuggets could survive this transition to constitute Cold Dark Matter (CDM) without invoking exotic particles beyond the Standard Model. Previous work by Witten (1984) suggested stable quark nuggets could form in a first-order transition, but lattice QCD and effective models favor a crossover. This study investigates whether a crossover transition can still produce stable, macroscopic nuggets of strange quark matter ("strangeon nuggets") that survive to the present day.

Methodology
The authors employ a multi-stage theoretical framework to model the thermodynamics of the early Universe during the QCD epoch:

1. Equations of State (EOS):

   • Quark Phase: Modeled using the 2+1 flavor Polyakov-Nambu-Jona-Lasinio (PNJL) model. This incorporates the $SU(3)$ Nambu-Jona-Lasinio interaction coupled with a temporal background gauge field (Polyakov loop) to describe the deconfined phase.
   • Hadron Phase: Described using the Relativistic Mean Field (RMF) model (specifically the GM1 parameter set), which accounts for meson exchanges ($\sigma$, $\omega$, $\rho$) to describe nucleon interactions.
   • Strangeon Nugget Phase: A non-relativistic equation of state is applied to stable strangeon nuggets (clusters of $u, d, s$ quarks with net strangeness). These are treated as classical particles following a Maxwell-Boltzmann distribution.

2. Crossover Transition Modeling:

   • The transition is modeled as a "three-window" scenario around the critical temperature $T_c \sim 170$ MeV.
   • A smooth interpolation of the Helmholtz free energy per baryon ($f$) is used to connect the QGP and Hadron-Strangeon (HS) phases, avoiding the latent heat characteristic of first-order transitions.
   • Weight functions ($\chi_{\pm}$) based on a hyperbolic tangent function define the transition region ($T_c \pm \Gamma$, where $\Gamma = 30$ MeV).

3. Strangeon Nugget Formation and Stability:

   • The authors propose that during the crossover, quarks collide and nucleate into "strangeons" (clusters with net strangeness), which then merge into nuggets.
   • A critical baryon number, $A_c$, is introduced. Nuggets with baryon number $A < A_c$ decay rapidly via weak interactions or evaporation into nucleons. Nuggets with $A > A_c$ are thermodynamically stable.
   • The distribution of nugget sizes is assumed to follow an exponential function $n(D) = n_0 e^{-D/R_c}$, where $D$ is the diameter and $R_c$ is the critical radius corresponding to $A_c$.
   • The study explores values of $A_c$ ranging from $10^5$ to $10^9$, based on constraints from weak decay scales and strong interaction scales.

Key Results

• Thermodynamic Negligibility: Calculations of the pressure ($P_S$), entropy density ($s_S$), and energy density ($\epsilon_S$) for the strangeon nugget component show that their thermodynamic contributions are negligible compared to the hadronic phase. This is because the number density of stable nuggets is extremely low due to their large mass (large $A$), even though they contain a significant fraction of the total baryon mass.
• Mass Fraction: Under the assumed exponential distribution and the condition that nuggets with $A > A_c$ are stable, the model predicts that stable strangeon nuggets constitute approximately 85% of the total baryon mass density. The remaining ~15% consists of ordinary nucleons ($A < A_c$).
• Phase Transition Dynamics: The interpolation of thermodynamic quantities (pressure, energy density, free energy) between the QGP and HS phases is smooth, consistent with a crossover transition. The transition occurs at $T_c \sim 170$ MeV.
• Dark Matter Candidate: The resultant mass density of the surviving strangeon nuggets is comparable to the observed dark matter density. The authors argue that these nuggets interact negligibly with normal matter via strong, weak, or electromagnetic forces (for sufficiently large $A$), making them viable Cold Dark Matter candidates.

Significance and Claims
The paper claims to provide a mechanism for the production of Cold Dark Matter within the regime of "old" physics, i.e., without introducing new exotic particles (such as axions or WIMPs) beyond the Standard Model. By proposing that stable strangeon nuggets can form and survive a crossover QCD phase transition, the authors offer an explanation for the dark matter abundance that relies solely on the known properties of quarks and the strong/weak interactions.

The authors explicitly note that their calculation of the mass fraction (~85%) assumes the nuggets are free from other cosmological constraints, such as those from Big Bang Nucleosynthesis (BBN). They acknowledge that a detailed BBN network study is required to provide realistic constraints on the distribution function of the nuggets, particularly regarding the interaction of smaller nuggets with light nuclei during the nucleosynthesis epoch. However, the current work establishes the thermodynamic feasibility of such nuggets surviving the early Universe and contributing significantly to the matter density.