Microscopic determination of the in-medium quark couplings of cold dense quark matter — vector repulsion, flavor-non-singlet vector couplings, and diquark coupling — from nonperturbative gluon exchange

The equation of state of cold dense quark matter — and with it the stiffness of neutron-star cores, the location of quark–hadron phase boundaries, and the viability of quark–hadron continuity — is governed in effective models by a handful of in-medium four-quark couplings: the vector-isoscalar coupling $g_V$ (the effective repulsion between quarks), the flavor-non-singlet vector-isovector couplings $g_\tau^{(3)}$ and $g_\tau^{(8)}$ , and the diquark pairing coupling $H$ . Their values at the baryon densities $n_B \sim 5$ – $10\,n_0$ ( $n_0$ = nuclear saturation density), where quark matter is expected to form in compact stars, are presently fitted to neutron-star observations rather than derived from QCD: lattice QCD is blocked by the fermion sign problem at real baryon chemical potential, and perturbative QCD is controlled only at densities far above $\sim 40\,n_0$ . The single-gluon-exchange estimate of Song, Baym, Hatsuda, and Kojo (2019), using a massive gluon propagator and a frozen infrared coupling $\alpha_s$ , reproduces the phenomenologically required repulsion but leaves nonperturbative corrections beyond that single diagram uncontrolled. The problem is to determine these in-medium couplings and their density dependence microscopically from nonperturbative gluon exchange in QCD — including multi-gluon exchange, in-medium gluon polarization, and the infrared behavior of $\alpha_s$ and the quark–gluon vertex — with controlled uncertainties, and to establish how they map onto the vector and pairing couplings of hadronic descriptions.

Background

The matter in neutron-star cores is the densest strongly interacting matter directly accessible to observation. Its energy per baryon as a function of baryon density $n_B$ — the equation of state (EoS) — determines the stellar mass–radius relation. Nuclear physics is anchored at the saturation density $n_0 \simeq 0.16\ \mathrm{fm}^{-3}$ of atomic nuclei, where neutron-rich matter is described with controlled errors by chiral effective field theory (an expansion in powers of momenta over the chiral symmetry-breaking scale, with low-energy constants fitted to few-nucleon data). Shapiro-delay timing — measuring the general-relativistic delay of radio pulses as they pass near the pulsar's white-dwarf companion, which weighs the system — of the millisecond pulsar PSR J0740+6620 demonstrated a $\simeq 2.14\,M_{\odot}$ neutron star (Cromartie et al. 2020), and multimessenger observations now demand a stiff EoS — rapidly rising pressure — at supra-nuclear density.

At $n_B \gtrsim 5$ – $10\,n_0$ the relevant degrees of freedom are expected to become deconfined quarks. At asymptotically high density ( $n_B \gtrsim 40\,n_0$ ) the EoS is computable in perturbative QCD (pQCD, the expansion in the running strong coupling $\alpha_s$ ; Kurkela, Romatschke, Vuorinen 2009). In the neutron-star window between a few $n_0$ and tens of $n_0$ , however, no controlled first-principles method exists: lattice QCD, the standard nonperturbative tool, suffers the fermion sign problem (a complex Monte Carlo measure) at real baryon chemical potential, and at momentum transfers of order the quark Fermi momentum ( $\lesssim 500$ MeV) the coupling $\alpha_s$ is large and its infrared behavior poorly known.

In this window, dense quark matter is modeled by Nambu–Jona-Lasinio (NJL)-type theories: local four-fermion interactions among quarks in scalar, vector, and diquark (quark–quark) channels, meant to represent the low-momentum content of gluon exchange. The vector-isoscalar coupling $g_V$ multiplies a repulsive energy $g_V n_q^2$ ( $n_q$ = quark number density) and stiffens the EoS; the flavor-non-singlet vector-isovector couplings $g_\tau^{(3)}$ and $g_\tau^{(8)}$ act on flavor-density differences such as $n_u - n_d$ (the quark-matter analogue of the nuclear symmetry energy); the diquark coupling $H$ sets the strength of attractive quark–quark correlations that drive color superconductivity — BCS-like Cooper pairing of quarks near the Fermi surface in a two-flavor (2SC) or three-flavor color-flavor-locked (CFL) pattern with gaps of order 10–100 MeV. In the quark–hadron crossover (QHC) constructions of the EoS, which interpolate between chiral-EFT nuclear matter and pQCD, supporting stars of two solar masses requires $g_V/G_0 \simeq 0.6$ – $1.3$ and $H/G_0 \simeq 1.35$ – $1.65$ , where $G_0$ is the scalar coupling fixed in the vacuum pion sector (Kojo, Hou, Okafor, Togashi 2021; Kojo, Baym, Hatsuda 2021). These bands are phenomenological fits constrained by neutron-star observations, not QCD predictions.

Song, Baym, Hatsuda, and Kojo (2019) made the first microscopic estimate of $g_V$ : they matched the interaction energy $g_V n_q^2$ to the quark exchange energy computed from explicit single-gluon exchange in Landau gauge (a standard fixed-gauge convention in which the nonperturbative gluon propagator is commonly quoted), with the gluon propagator parametrized by a nonperturbative dynamical gluon mass $m_g = 200$ – $600$ MeV (as found in zero-density lattice simulations and Dyson–Schwinger studies (systems of integral equations for the full quark and gluon propagators)) and an infrared-frozen coupling $\alpha_s \sim 2$ – $4$ . This reproduces the required $g_V \sim 0.6$ – $1.3\,G_0$ at $5n_0$ and suggests the density dependence

$g_V(p_F) \simeq \frac{4\pi\alpha_s}{3\left(9m_g^2 + 8p_F^2\right)},$

where $p_F$ is the quark Fermi momentum. The same framework gives Fierz-related isovector couplings $g_\tau^{(3)} = g_\tau^{(8)} = \tfrac{3}{2}g_V$ at single-gluon exchange (a Fierz transformation reorganizes a Dirac–color interaction into its spin–flavor–color channels) and, from the vacuum $N$ – $\Delta$ baryon mass splitting, a rough low-density anchor for $H$ . But the authors stress that at $n_B \gtrsim 5n_0$ the nonperturbative contributions beyond the single-gluon-exchange diagram — multi-gluon exchange, in-medium gluon polarization (the density-dependent modification of the gluon propagator: electric Debye and magnetic Meissner screening masses, and Landau damping — the absorption of gluons by excitations at the quark Fermi surface — all of order $\sqrt{\alpha_s}\,\mu$ , with $\mu$ the quark chemical potential), the momentum-dependent gluon mass function, and the running and possible infrared freezing or divergence of $\alpha_s$ , which could range anywhere from 0 to 10 — are not under control. A microscopic determination also bears on quark–hadron continuity, the conjecture (reviewed in Baym, Hatsuda, Kojo, Powell, Song, Takatsuka 2018) that dense hadronic and quark matter join without a sharp phase transition, so that quark-level and nucleon-meson-level vector and pairing couplings should map onto one another.

Problem Statement

Determine microscopically, from QCD, the in-medium effective quark couplings of cold (zero-temperature) dense quark matter and their density dependence at the baryon densities $n_B \sim 5$ – $10\,n_0$ (extending to $\sim 20\,n_0$ ) where quark matter is expected to form in neutron-star cores — specifically: (i) the vector-isoscalar coupling $g_V$ that encodes the effective repulsion between quarks and stiffens the equation of state; (ii) the flavor-non-singlet (vector-isovector) couplings $g_\tau^{(3)}$ and $g_\tau^{(8)}$ ; and (iii) the diquark pairing coupling $H$ in the scalar and axial-vector quark–quark channels. The determination comprises the three couplings (i)–(iii) and their density dependence; a treatment of nonperturbative gluon exchange beyond the single-gluon-exchange diagram, whose scope includes multi-gluon exchange, in-medium gluon polarization functions (Debye/Meissner screening and Landau damping), the momentum-dependent gluon mass function, and the infrared behavior of the strong coupling $\alpha_s$ and the quark–gluon vertex; and the mapping of these quark-level couplings onto the vector and pairing couplings of hadronic (nucleon-meson) effective descriptions, as a test of quark–hadron continuity, each with controlled uncertainties. These three couplings are physically linked: they are the vector and diquark vertices of one and the same in-medium quark–quark scattering amplitude, and the common objective is to derive that amplitude's low-momentum content from QCD rather than fit it to neutron-star observations.

References

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