Impact of Localization in Early-Universe QCD Phase Transition

TL;DR

This work extends the MIT bag model by incorporating gluon-induced disorder that localizes quarks, effectively suppressing quark degrees of freedom via a suppression factor $F_{ext}(W,T)=1-F_{loc}$ with $F_{loc}=F_{max} G(W) H(T)$. The disorder and localization are modeled with $G(W)=1- ext{e}^{-(W/W_×)^q}$ and $H(T)=1/[1+(T/T_*)^p]$, with $W∝T$, leading to a temperature-dependent effective quark content $g_q^{eff}=F_{ext}(T) g_q$. Applying this to the early-universe QCD transition and solving the Friedmann equation shows a $\sim 7\%$ increase in the critical temperature to $T_c^{loc} \approx 181$ MeV, a shorter mixed-phase duration by $\sim 24\%$ (from $\Delta t_B\approx 10.80\,\mu$s to $\Delta t_{loc}\approx 8.22\,\mu$s), and earlier onset of hadronization, with the resulting thermodynamics closer to HotQCD lattice data. The localization model also pushes the energy density and pressure toward the reduced Stefan–Boltzmann limit at high temperatures, signifying improved agreement with lattice results and indicating a significant role for quark localization in early-universe cooling dynamics.

Abstract

We introduce a phenomenological modification of the MIT bag model equation of state that incorporates quark localization arising from gluon-induced disorder in the quark-gluon plasma. This model effectively reduces the quark degrees of freedom encoded in the product of the disorder activation function $G(W)$ and the localization efficiency factor $H(T)$. As a result, the critical temperature is increased roughly by 7%. Employing the Friedmann equation, we find that the onset of the phase transition occurs earlier. Consequently, the mixed phase duration is only $8.22\, μs$, which is 24% shorter than the bag model, and the hadronic phase cools faster. The Stephan-Boltzmann constant is reached at much higher temperatures, causing the energy density and pressure curves of the bag model to shift downward and yielding better agreement with the lattice QCD data from the HotQCD collaboration. Our results show that the localization of quarks plays a significant role in the cooling dynamics of the early universe.

Figures (7)

• Figure 1: Plots of the disorder activation function $G(T)$ for varying characteristic temperature scales $T_\times=$ 200 MeV, 250 MeV, 300 MeV, 350 MeV, and 400 MeV at different exponent values $q=0.5, 1.0, 1.5$, and 2.0 .
• Figure 2: The plots of the localization efficiency factor $H(T)$ as a function of temperature for varying characteristic temperature scales $T_*=$ 200 MeV, 250 MeV, 300 MeV, 350 MeV, and 400 MeV at different exponent values $p=0.5, 1.0, 1.5$, and $2.0$ .
• Figure 3: The plots of the localization and extended fractions $F_{loc}(T)$ and $F_{ext}(T)$ as functions of temperature for varying characteristic temperature scales $(T_\times,T_*)= (200\,\rm{MeV}, 400\,\rm{MeV}), (250\,\rm{MeV}, 350\,\rm{MeV}),(350\,\rm{MeV}, 250\,\rm{MeV}),\:\:\rm{and}\:\:(400\,\rm{MeV}, 200\,\rm{MeV})$. The corresponding exponent pairs are $(q,p)=(1.5,2.0), (2.0, 1.5), (1.5, 0.5)$, and $(0.5, 1.5)$ with a cap constant $F_{max}=0.8$ .
• Figure 4: The figure shows the contour plots of the localization fraction $F_{loc}(T)=0.1,0.3, 0.5, 0.6$, and $0.7$ for different temperatures $T=190$ MeV, 230 MeV, 270 MeV, and 310 MeV with exponent values $q=1.5$ and $p=2.0$ .
• Figure 5: The temperature dependence of the MIT bag and localization model for two quark flavors. The red solid (black dashed) line shows the energy density of the localization (MIT bag) model. The benchmark parameters used are $B=235\,(\textrm{MeV})^4$, $T_\times=200\,\textrm{MeV}$, $T_*=400\,\textrm{MeV}$, $p=2.0$, $q=1.5$, and $F_{max}=0.8$ .