Stability and Structural Properties of Hot Quark Stars within Perturbative QCD

Author(s)

Raza, Tousif, Gorda, Tyler

Abstract

The hypothesis of strange quark matter (SQM) and the possible existence of strange stars have been extensively investigated within the thermodynamic bag model, typically employing the free Fermi gas approximation with or without perturbative QCD corrections at zero temperature, where quark confinement is modeled by the bag pressure. In this work, based on the perturbative inclusion of quark mass effects, we develop a thermodynamically consistent equation of state (EOS) for SQM at finite temperature, incorporating perturbative corrections up to $\mathcal{O}(α_s)$ in the strong coupling constant $α_s$ while ensuring full adherence to the Maxwell relations. These corrections are essential for accurately describing hot quark stars potentially formed during core-collapse events. We present results for both fixed and running couplings, using a phenomenological model to include the breakdown of the perturbative running of $α_s$ at low momenta. Our results demonstrate that incorporating finite-temperature perturbative QCD corrections leads to SQM configurations that fall within the absolute stability window, with equilibrium energies per baryon lying below that of iron, the most tightly bound nucleus. Our EOS supports compact stars with masses exceeding $1.4\,M_\odot$ (and above $2\,M_\odot$ for some values of the bag constant), in agreement with current astrophysical constraints from pulsar and gravitational-wave observations.

Figures

Pressure of strange quark matter as a function of the baryon chemical potential, computed using a fixed QCD coupling, $\alpha_s \approx 0.3$ (left panel), and a running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Caption Pressure of strange quark matter as a function of the baryon chemical potential, computed using a fixed QCD coupling, $\alpha_s \approx 0.3$ (left panel), and a running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Pressure of strange quark matter as a function of the baryon chemical potential, computed using a fixed QCD coupling, $\alpha_s \approx 0.3$ (left panel), and a running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Caption Pressure of strange quark matter as a function of the baryon chemical potential, computed using a fixed QCD coupling, $\alpha_s \approx 0.3$ (left panel), and a running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Energy per baryon of strange quark matter as a function of the baryon number density for a fixed QCD coupling (left panel) and a modeled running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Caption Energy per baryon of strange quark matter as a function of the baryon number density for a fixed QCD coupling (left panel) and a modeled running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Energy per baryon of strange quark matter as a function of the baryon number density for a fixed QCD coupling (left panel) and a modeled running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Caption Energy per baryon of strange quark matter as a function of the baryon number density for a fixed QCD coupling (left panel) and a modeled running QCD coupling (right panel). In both panels, we set the bag constant to $B^{1/4}=145~\mathrm{MeV}$ and use the quark masses $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$.
Mass--radius relations for strange stars obtained using a fixed coupling, $\alpha_s \simeq 0.3$ (left panel), and a running coupling with $X=1$ (right panel), at a temperature of $T=5~\mathrm{MeV}$. Solid curves correspond to $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$, whereas dashed curves represent the massless-quark case, $m_u=m_d=m_s=0$. Different colors indicate bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The cyan and green bands show observational constraints on the radius of a $1.4\,M_\odot$ compact star from GW170817~\cite{LIGOScientific:2017vwq} and on the mass of the secondary compact object in GW190814~\cite{LIGOScientific:2020zkf}. The light-pink region represents the mass--radius constraints for PSR J0030+0451~\cite{Vinciguerra:2023qxq}, whereas the dark-yellow region represents those of PSR J0740+6620~\cite{Miller:2021qha}, based on observations from NICER and XMM-Newton. The horizontal black line represents the mass measurement of PSR J0348+0432~\cite{Antoniadis:2013pzd}. The yellow region indicates constraints associated with the central compact object in the supernova remnant HESS J1731--347~\cite{2022NatAs...6.1444D,Horvath:2023uwl}, and the dark-pink region represents the constraint on the low-mass X-ray binary 4U 1702$-$429 inferred from Rossi X-ray Timing Explorer observations~\cite{Nattila:2017wtj}.
Caption Mass--radius relations for strange stars obtained using a fixed coupling, $\alpha_s \simeq 0.3$ (left panel), and a running coupling with $X=1$ (right panel), at a temperature of $T=5~\mathrm{MeV}$. Solid curves correspond to $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$, whereas dashed curves represent the massless-quark case, $m_u=m_d=m_s=0$. Different colors indicate bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The cyan and green bands show observational constraints on the radius of a $1.4\,M_\odot$ compact star from GW170817~\cite{LIGOScientific:2017vwq} and on the mass of the secondary compact object in GW190814~\cite{LIGOScientific:2020zkf}. The light-pink region represents the mass--radius constraints for PSR J0030+0451~\cite{Vinciguerra:2023qxq}, whereas the dark-yellow region represents those of PSR J0740+6620~\cite{Miller:2021qha}, based on observations from NICER and XMM-Newton. The horizontal black line represents the mass measurement of PSR J0348+0432~\cite{Antoniadis:2013pzd}. The yellow region indicates constraints associated with the central compact object in the supernova remnant HESS J1731--347~\cite{2022NatAs...6.1444D,Horvath:2023uwl}, and the dark-pink region represents the constraint on the low-mass X-ray binary 4U 1702$-$429 inferred from Rossi X-ray Timing Explorer observations~\cite{Nattila:2017wtj}.
Mass--radius relations for strange stars obtained using a fixed coupling, $\alpha_s \simeq 0.3$ (left panel), and a running coupling with $X=1$ (right panel), at a temperature of $T=5~\mathrm{MeV}$. Solid curves correspond to $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$, whereas dashed curves represent the massless-quark case, $m_u=m_d=m_s=0$. Different colors indicate bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The cyan and green bands show observational constraints on the radius of a $1.4\,M_\odot$ compact star from GW170817~\cite{LIGOScientific:2017vwq} and on the mass of the secondary compact object in GW190814~\cite{LIGOScientific:2020zkf}. The light-pink region represents the mass--radius constraints for PSR J0030+0451~\cite{Vinciguerra:2023qxq}, whereas the dark-yellow region represents those of PSR J0740+6620~\cite{Miller:2021qha}, based on observations from NICER and XMM-Newton. The horizontal black line represents the mass measurement of PSR J0348+0432~\cite{Antoniadis:2013pzd}. The yellow region indicates constraints associated with the central compact object in the supernova remnant HESS J1731--347~\cite{2022NatAs...6.1444D,Horvath:2023uwl}, and the dark-pink region represents the constraint on the low-mass X-ray binary 4U 1702$-$429 inferred from Rossi X-ray Timing Explorer observations~\cite{Nattila:2017wtj}.
Caption Mass--radius relations for strange stars obtained using a fixed coupling, $\alpha_s \simeq 0.3$ (left panel), and a running coupling with $X=1$ (right panel), at a temperature of $T=5~\mathrm{MeV}$. Solid curves correspond to $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$, whereas dashed curves represent the massless-quark case, $m_u=m_d=m_s=0$. Different colors indicate bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The cyan and green bands show observational constraints on the radius of a $1.4\,M_\odot$ compact star from GW170817~\cite{LIGOScientific:2017vwq} and on the mass of the secondary compact object in GW190814~\cite{LIGOScientific:2020zkf}. The light-pink region represents the mass--radius constraints for PSR J0030+0451~\cite{Vinciguerra:2023qxq}, whereas the dark-yellow region represents those of PSR J0740+6620~\cite{Miller:2021qha}, based on observations from NICER and XMM-Newton. The horizontal black line represents the mass measurement of PSR J0348+0432~\cite{Antoniadis:2013pzd}. The yellow region indicates constraints associated with the central compact object in the supernova remnant HESS J1731--347~\cite{2022NatAs...6.1444D,Horvath:2023uwl}, and the dark-pink region represents the constraint on the low-mass X-ray binary 4U 1702$-$429 inferred from Rossi X-ray Timing Explorer observations~\cite{Nattila:2017wtj}.
Mass--tidal-deformability relations for strange stars at $T=5~\mathrm{MeV}$, with $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$. The colored curves correspond to bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The vertical error bar indicates the tidal-deformability constraint from GW170817~\cite{LIGOScientific:2017vwq}, obtained assuming a low-spin prior. The upper and lower panels show the results for a fixed coupling, $\alpha_s\simeq0.3$, and a running QCD coupling with $X=1$, respectively.
Caption Mass--tidal-deformability relations for strange stars at $T=5~\mathrm{MeV}$, with $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$. The colored curves correspond to bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The vertical error bar indicates the tidal-deformability constraint from GW170817~\cite{LIGOScientific:2017vwq}, obtained assuming a low-spin prior. The upper and lower panels show the results for a fixed coupling, $\alpha_s\simeq0.3$, and a running QCD coupling with $X=1$, respectively.
Mass--tidal-deformability relations for strange stars at $T=5~\mathrm{MeV}$, with $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$. The colored curves correspond to bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The vertical error bar indicates the tidal-deformability constraint from GW170817~\cite{LIGOScientific:2017vwq}, obtained assuming a low-spin prior. The upper and lower panels show the results for a fixed coupling, $\alpha_s\simeq0.3$, and a running QCD coupling with $X=1$, respectively.
Caption Mass--tidal-deformability relations for strange stars at $T=5~\mathrm{MeV}$, with $m_u=m_d=0$ and $m_s=96~\mathrm{MeV}$. The colored curves correspond to bag constants in the range $B^{1/4}=135$--$155~\mathrm{MeV}$. The vertical error bar indicates the tidal-deformability constraint from GW170817~\cite{LIGOScientific:2017vwq}, obtained assuming a low-spin prior. The upper and lower panels show the results for a fixed coupling, $\alpha_s\simeq0.3$, and a running QCD coupling with $X=1$, respectively.
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