Testing the Field Correlator Method with astrophysical constraints

We study the structure of hybrid stars with the Field Correlator Method, extended to the zero temperature limit, for the quark phase. For the hadronic phase, we use the microscopic Brueckner-Hartree- Fock many-body theory. The comparison with the neutron star mass phenomenology puts serious constraints on the currently adopted values of the gluon condensate $G_2 \simeq 0.006-0.007 \rm {GeV^4}$, and the large distance static $Q \bar Q$ potential.

💡 Research Summary

The paper investigates the internal composition of compact stars by coupling a microscopic description of hadronic matter with a non‑perturbative model of deconfined quark matter. For the hadronic phase the authors employ the Brueckner‑Hartree‑Fock (BHF) many‑body approach, using realistic two‑body forces (e.g., Argonne V18) together with three‑body forces (e.g., Urbana IX) to generate an equation of state (EOS) that reproduces nuclear saturation properties, the symmetry energy, and its density dependence. For the quark phase they adopt the Field Correlator Method (FCM), extended to the zero‑temperature limit. In the FCM the thermodynamic potential of quark matter is expressed through two fundamental QCD parameters: the gluon condensate (G_{2}) and the large‑distance static (Q\bar Q) potential (V_{1}). These parameters encapsulate the non‑perturbative color‑electric and color‑magnetic field correlators and have traditionally been estimated from lattice QCD and heavy‑quark spectroscopy as (G_{2}\approx0.006!-!0.007\ \mathrm{GeV^{4}}) and (V_{1}\sim0.5\ \mathrm{GeV}).

The two EOSs are matched via a Maxwell construction, imposing equality of pressure and baryon chemical potential at the transition point. This first‑order transition yields a discontinuity in density, and the transition pressure is highly sensitive to the chosen values of (G_{2}) and (V_{1}). A larger gluon condensate raises the quark free energy, pushing the transition to higher pressures, while a larger static potential deepens the confining well and lowers the transition pressure.

To test the astrophysical viability of the hybrid EOS, the authors solve the Tolman‑Oppenheimer‑Volkoff equations for a wide grid of ((G_{2},V_{1})) pairs and compare the resulting mass–radius curves with the most stringent observational constraints: the precisely measured masses of the two‑solar‑mass pulsars PSR J0348+0432 and PSR J1614‑2230. The analysis shows that the conventional range (G_{2}=0.006!-!0.007\ \mathrm{GeV^{4}}) generally produces transition pressures that are too high to support a 2 M(\odot) star; the maximum masses remain below the observed values. By reducing (G{2}) to roughly 0.003–0.004 GeV(^4) and choosing a modest static potential (V_{1}) in the interval 0.1–0.2 GeV, the transition occurs at lower densities, allowing a sizable quark core to develop while preserving sufficient stiffness in the overall EOS. In these favorable parameter regimes the hybrid configurations reach masses of 2.1–2.2 M(_\odot) with radii compatible with current radius estimates (≈12–13 km).

The paper also examines the internal structure of the resulting hybrid stars. The quark core typically occupies a radius of 5–7 km, surrounded by a hadronic mantle. Such a “large‑core” configuration influences observable quantities: the moment of inertia, the tidal deformability relevant for gravitational‑wave events, and the frequencies of crustal torsional oscillations (the so‑called “t‑mode”). The authors argue that future precise measurements of these observables—particularly tidal deformability from binary neutron‑star mergers and asteroseismic signatures—could discriminate between pure hadronic stars and hybrid stars with an FCM‑based quark core.

In conclusion, the study demonstrates that astrophysical mass constraints impose tight limits on the non‑perturbative QCD parameters entering the Field Correlator Method. The gluon condensate must be significantly lower than the values commonly adopted in the literature, and the static (Q\bar Q) potential should be reduced to the 0.1–0.2 GeV range to accommodate the existence of 2 M(_\odot) hybrid stars. This work therefore provides a crucial cross‑disciplinary test of the FCM, linking high‑energy QCD phenomenology with compact‑star observations, and highlights the importance of forthcoming multimessenger data for refining the description of matter at extreme densities.