Excluded volume effects in the quark-mass density-dependent model: Implications for the equation of state and compact star structure

We present a significant extension of the quark-mass density-dependent model (QMDDM), initially revised in our prior study [Phys. Rev. D 107, 043025 (2023)], where thermodynamic inconsistencies were addressed. Our current work enriches the QMDDM by incorporating excluded volume effects, as a step toward a more realistic representation of the quark matter equation of state (EOS) at zero temperature. We introduce the concept of “available volume” in the Helmholtz free energy formulation, accounting for the space excluded by each quasiparticle due to its finite size or repulsive interactions. We present a methodology to modify the EOS for pointlike particles, allowing for a simple and direct incorporation of excluded volume effects. This is first addressed in a simple one-flavor model and then extended to a more realistic three-flavor system, incorporating both mass and volume dependencies on the baryon number density. We examine various Ansätze for the excluded volume, ultimately adopting one that aligns with the asymptotic freedom behavior of quantum chromodynamics. The EOS for electrically neutral systems in chemical equilibrium is computed, focusing on self-bound and hybrid matter scenarios. We show that the incorporation of excluded volume effects renders the EOS stiffer and that excluded volume effects are essential to align the mass-radius relation of self-bound and hybrid stars with modern astrophysical constraints.

Figure 1

Constant excluded volume Ansatz: Effective quasiparticle mass $M$ depicted as a function of (a) particle number density $n/n_0$, (b) chemical potential, and (c) pressure. Parameters used are $m=5\mathrm{MeV}$, $a=2$, and $C=100$, with units assigned such that $n$ is in ${\mathrm{fm}}^{-3}$ and $M$ is in MeV.

Figure 2

EOS for a one-component gas employing the constant excluded volume Ansatz. (a) Total pressure $p$ plotted against particle number density $n$ in units of the nuclear saturation density $n_0$. (b) Energy per particle $\epsilon /n$ as a function of $p$. (c) Pressure versus energy density. (d) Speed of sound as a function of $n/n_0$.

Figure 3

Effective quasiparticle mass $M$ in the density-dependent excluded volume Ansatz. (a)–(c) Organization is the same as in Fig. 1, with identical choices for the parameters $m$, $a$, and $C$.

Figure 4

EOS for a one-component gas with the density-dependent excluded volume Ansatz. (a)–(d) Organization is the same as in Fig. 2, with identical choices for the parameters $m$, $a$, and $C$.

Figure 5

Gibbs free energy per baryon of two- and three-flavor quark matter in bulk for different choices of the EOS parameters $a$, $C$, and $\kappa$. In (a), results are shown for a parameter choice leading to strange matter, while in (b) they are for a parameter choice yielding hybrid matter (see the main text for further details).

Figure 6

(a),(b) Total pressure $p$ and bag constant $B$ as a function of the energy density for the same parameter choices of Fig. 5.

Figure 7

(a),(b) Speed of sound as a function of the baryon number density for the same parameter choices of Fig. 5.

Figure 8

Mass-radius relationship for strange quark stars with parameters $a=3$ and $C=58$. Curves represent different values of the parameter $\kappa$. Colored bands correspond to the 95% confidence intervals for the mass and radius of the millisecond pulsars PSR $\mathrm{J}0030+0451$ [33, 34] and PSR $\mathrm{J}0740+6620$ [20, 21] measured recently by the Neutron Star Interior Composition Explorer (NICER) and the 90% confidence intervals for the merging event GW170817 [35] (LIGO/Virgo).

Figure 9

(a) The EOS for hybrid matter with parameters $a=3$ and $C=75$. The red curve represents the hadron matter EOS according to the MPA1 model. The black curves depict the quark matter EOS for pointlike quasiparticles ($\kappa =0$) and with excluded volume effects with $\kappa =0.3$ and $\kappa =0.5$. (b) The corresponding mass-radius relationship for these EOS. The colored bands in the lower diagram denote the observational constraints introduced in Fig. 8.