Cold quark matter

We perform an $\mathcal{O}({\alpha }_s^2)$ perturbative calculation of the equation of state of cold but dense QCD matter with two massless and one massive quark flavor, finding that perturbation theory converges reasonably well for quark chemical potentials above 1 GeV. Using a running coupling constant and strange quark mass, and allowing for further nonperturbative effects, our results point to a narrow range where absolutely stable strange quark matter may exist. Absent stable strange quark matter, our findings suggest that quark matter in (slowly rotating) compact star cores becomes confined to hadrons only slightly above the density of atomic nuclei. Finally, we show that equations of state including quark matter lead to hybrid star masses up to $M\sim 2M_{\odot }$, in agreement with current observations. For strange stars, we find maximal masses of $M\sim 2.75M_{\odot }$ and conclude that confirmed observations of compact stars with $M>2M_{\odot }$ would strongly favor the existence of stable strange quark matter.

Figure 1

Renormalization scale dependence of ${\alpha }_s$ and $m$ from Eqs. (9, 10), using values of $\overline{\Lambda }$ relevant for compact stars. The full lines and dark shaded regions correspond to the central value and uncertainty of ${\Lambda }_{\overline{\mathrm{MS}}}=0.378+0.034-0.032\mathrm{GeV}$, respectively. The light shaded region in $m$ corresponds to the uncertainty in $m(2\mathrm{GeV})$. The circles with error bars correspond to reference points obtained from Ref. 32.

Figure 2

The two- and three-loop 2GI diagrams contributing to the grand potential of QCD.

Figure 3

The generic form of the ring or plasmon sum.

Figure 4

The renormalization scale dependence of the massless quark number density and pressure for perturbation theory to order $\mathcal{O}({\alpha }_s^1)$ and $\mathcal{O}({\alpha }_s^2)$ (light and dark shaded regions, respectively), with ${\Lambda }_{\overline{\mathrm{MS}}}=0.378\mathrm{GeV}$ and $\overline{\Lambda }$ ranging from $\mu$ (right boundary) to $4\mu$ (left boundary). The dashed (full) lines correspond to the first (second) order results with $\overline{\Lambda }=2\mu$.

Figure 5

The total quark number density evaluated to $\mathcal{O}({\alpha }_s^2)$ for locally charge neutral systems of 2 and 3 massless quark flavors, as well as for the two light and one massive flavor case ($2+1$). All results are normalized to the density of three free massless flavors $3{\mu }^3/{\pi }^2$, and assume the values ${\Lambda }_{\overline{\mathrm{MS}}}=0.378\mathrm{GeV}$, $m(2\mathrm{GeV})=0.1\mathrm{GeV}$, while the renormalization scale takes the values $3\overline{\Lambda }/({\mu }_u+{\mu }_d+{\mu }_s)=1$, 2, 4 (for $N_f=2$, $2\overline{\Lambda }/({\mu }_u+{\mu }_d)=1$, 2, 4). As expected, the $2+1$ flavor result matches the three flavor result at large $\mu$ and approaches the two flavor result at small $\mu$.

Figure 6

Exclusion plots of stable strange quark matter, where the smallest possible value of $E/A$ (required to be lower than 0.93 GeV for stability) is plotted as a function of the parameters of the theory. Here the value of ${\Lambda }_{\overline{\mathrm{MS}}}$ is fixed since reducing it corresponds to linearly increasing $\overline{\Lambda }/({\mu }_u+{\mu }_d+{\mu }_s)$. The effect of varying △ can be seen by comparing the left and right figure. Left: Normal (unpaired) stable strange quark matter can only occur for comparatively large values of the renormalization scale. Right: The incorporation of the effects of the CSC gap significantly increases the allowed parameter space.

Figure 7

The matching of perturbative results for the quark matter EOS to hadronic EOSs at high temperature and density. Left: The pressure at $\mu =0$, obtained from resummed $\mathcal{O}({\alpha }_s^{5/2})$ perturbative QCD (pQCD) [4] [using Eq. (9) for ${\alpha }_s$ with ${\Lambda }_{\overline{\mathrm{MS}}}=0.378\mathrm{GeV}$ and $N_f=3$] and compared with the result of summing up the effects of all hadron resonances with masses smaller than 2 GeV [32, 93]. For a crossover transition, the pQCD results should match smoothly onto the hadronic EOS. The range in $\overline{\Lambda }$ for which this is possible corresponds quite well to the transition region determined by lattice QCD [2, 94] (the grey band labeled LQCD in the figure). Right: The $T=0$ quark matter pressure, taken from our $\mathcal{O}({\alpha }_s^2)$ result with ${\Lambda }_{\overline{\mathrm{MS}}}=0.378\mathrm{GeV}$, $m(2\mathrm{GeV})=0.1\mathrm{GeV}$, and $B=0$, compared to three different hadronic EOSs [69, 71]. The matching in general involves $B\ne 0$ and does not have to be smooth, as the phase transition at zero temperature could be of first order.

Figure 8

The equation of state of hybrid quark matter, obtained by matching the quark matter and nucleonic EOSs at low ($n_B\lesssim 0.36{\mathrm{fm}}^{-3}$, case I) and high ($n_B\gtrsim 0.64{\mathrm{fm}}^{-3}$, case II) densities, respectively. The shaded regions are obtained by varying $\overline{\Lambda }$, the integration constant $B$, and thus also the exact matching point, while keeping ${\Lambda }_{\overline{\mathrm{MS}}}$ and $m(2\mathrm{GeV})$ at their central values 0.378 and 0.1 GeV, respectively.

Figure 9

The functional dependence between the pressure and energy density, given for a selection of different EOSs considered in this section.

Figure 10

The mass-radius relation for compact stars, obtained using $\Delta =0$ (left) and $\Delta =100\mathrm{MeV}$ (right) in the quark matter EOS. We display the results for purely hadronic stars (containing only nucleons [69], nucleons with kaon condensation [70], or nucleons and hyperons [71]), pure quark matter stars (strange stars, cf. Sec. 5a) and hybrid stars including both hadronic and quark matter (see text for details). Also shown in the plots are compact star mass observations from Refs. 81, 82, 83, 84, 85.

Figure 11

Left: The effect of the variation of the baryon chemical potential ${\mu }_{\mathrm{pt}}$, at which the case I matching between the hadronic and quark matter EOSs is performed (with $3\overline{\Lambda }/({\mu }_u+{\mu }_d+{\mu }_s)=4$, ${\Lambda }_{\overline{\mathrm{MS}}}=0.346\mathrm{GeV}$, $m(2\mathrm{GeV})=0.07\mathrm{GeV}$). The value ${\mu }_{\mathrm{pt}}=0.99\mathrm{GeV}$ corresponds to the maximal value where the matching is possible. Lowering ${\mu }_{\mathrm{pt}}$ towards 0.93 GeV, the result smoothly approaches that of the strange stars, which are marginally stable with respect to neutron stars. Right: The effect of CSC obtained by varying the value of the gap parameter △. Relative to unpaired quark matter ($\Delta =0$), CSC generally decreases the radii of stable stars while the masses are hardly affected.

Figure 12

The topologies resulting from the scalarization procedure. Our notation for the functions ${\mathcal{D}}_i$ from Eqs. (b1) onwards is such that a lower index at position $k$ indicates the power of the propagator $k$ in the figure, while the corresponding upper index gives the mass of the line in question, with $c$ indicating that the line has been cut. For the uncut graphs, the chemical potential is present in each massive propagator, but not in the massless ones.