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Recent observations of neutron stars, combined with causality, thermodynamic stability, and nuclear constraints, indicate rapid stiffening of QCD matter at density slightly above nuclear saturation density ($n_0 \simeq 0.16\,{\rm fm}^{-3}$). The evolution of the stiffening is quicker than expected from purely nucleonic models with many-body repulsion. Taking into account the quark substructure of baryons, we argue that the saturation of quarks states occur at $\sim$ 2-3$n_0$, driving the quark matter formation even before baryonic cores of the radii $\sim$0.5 fm spatially overlap. We describe the continuous transitions from hadronic to quark matter are described within a quarkyonic matter model in which gluons are assumed to be confining at density of interest. To obtain analytic insights for the transient regime, we construct an ideal model of quarkyonic matter, IdylliQ model, in which one can freely switch from baryonic to quark languages and vice versa., Comment: 7 pages, 5 figures; presentation given at "Aspects of Criticality II," Wroclaw, Poland

A crossover involving three-fermion clusters is relevant to the hadron-quark crossover, which, if occurring in a neutron star, could naturally reproduce the dense-matter equation of state recently deduced from simultaneous observations of neutron-star masses and radii. To understand the crossover mechanism, we examine the role of tripling fluctuations induced by the formation of three-fermion clusters. The phase-shift representation of fluctuations manifests an interplay of bound and scattering states, leading to non-monotonic momentum distributions of baryon-like clusters and peaked sound speed at finite densities. We demonstrate them by applying our approach to a nonrelativistic system of one-dimensional three-color fermions analogous to the hadron-quark matter., Comment: 7 pages, 5 figures

Matter composed of hyperons has been hypothesized to occur in neutron stars at densities slightly above the nuclear saturation density and in many descriptions gives rise to a significant softening in the equation of state (EoS). This softening would be at odds with the constraints from neutron star observations and ab initio nuclear matter computations at low density. This inconsistency is known as the hyperon puzzle. We show that Quarkyonic Matter models, which take into account the quark substructure of baryons, can mitigate the hyperon puzzle. We demonstrate two important consequences of the quark substructure effects. First, the hyperon threshold is shifted to a higher density as neutrons preoccupy the phase space for down quarks, preventing the emergence of hyperons at low energy. Secondly, the softening in the EoS becomes mild even above the hyperon threshold density because only little phase space is available for low-energy hyperons; increasing hyperon density quickly drives hyperons into the relativistic regime. In this work, we illustrate these two effects for a matter composed of charge-neutral baryons, using the ideal dual Quarkyonic (IdylliQ) model for three flavors. This model incorporates the Quarkyonic duality and allows us to manifestly express the quark Pauli blocking constraints in terms of the baryon occupation probability. The extension to neutron star matter is also briefly discussed., Comment: 20 pages, 6 figures

By taking into account the latest observations and theoretical constraints, we investigate the merger and post-merger of binary neutron stars (NSs) with general-relativistic numerical simulations employing hadronic and hybrid equations of state (EOSs). We name our hybrid stars neutron-quark stars (NQS), because the transition from hadrons to quarks starts at a density lower than the central density of $\sim 1 M_{\odot}$ stars. We address two viable scenarios for the transition to quark matter: a crossover or a strong first-order phase transition (1PT). We find that a crossover transition is in principle observable when both the inspiral and post-merger signals are detected because the post-merger gravitational-wave (GW) main frequency $f_2$ is generally lower than that of hadronic models with the same tidal deformability ($\Lambda$). Since it is viable according to current multi-messenger constraints, we also highlight the possibility of an EOS with a strong 1PT that takes place at 1.8 times the nuclear saturation density, with a stiff quark EOS after the transition. It is the first time that mergers of binary NQSs with a deconfined quark-matter core are studied numerically in full general relativity. In this case, although ($\Lambda$, compactness) lies significantly outside the hadronic relation, ($\Lambda$, $f_2$) is close to the relation valid for hadronic EOSs. We also point out a linear correlation, valid within the observational constraints and not sensitive to the presence of a hadron-quark transition, between the emitted energy in GWs and their frequency., Comment: 6+7 pages, 3+10 figures

QCD with the isospin chemical potential, $\mu_I$, is a useful laboratory to delineate the microphysics in dense QCD. To study the quark-hadron-continuity we use a quark-meson model that interpolates hadronic and quark matter physics at microscopic level. The equation of state is dominated by mesons at low density but taken over by quarks at high density. We extend our previous studies with two-flavors to the three-flavors case to study the impact of the strangeness which may be brought by kaons $(K_+, K_0) = (u\bar{s}, s\bar{d})$ and the U$_A$(1) anomaly. In the normal phase the excitation energies of kaons are reduced by $\mu_I$ in the same way as hyperons in nuclear matter at finite baryon chemical potential. Once pions condense, kaon excitation energies increases as $\mu_I$ does. Moreover, strange quarks become more massive through the U$_A$(1) coupling to the condensed pions. Hence at zero and low temperature the strange hadrons and quarks are highly suppressed. The previous findings in two-flavor models, sound speed peak, negative trace anomaly, gaps insensitve to $\mu_I$, persist in our three-flavor model and remain consistent with the lattice results to $\mu_I \sim 1$ GeV. We discuss the non-perturbative power corrections and quark saturation effects as important ingredients to understand the crossover equations of state measured on the lattice., Comment: 29 pages,11 figures, a contribution to the Special Issue "Studies in Neutron Stars"

Parity doublet model is an effective chiral model that includes the chiral variant and invariant masses of baryons. The chiral invariant mass has large impacts on the density dependence of models which can be constrained by neutron star observations. In the previous work, models of two-flavors have been considered up to a few times nuclear saturation density, but in such dense region it is also necessary to consider hyperons. With the chiral invariant masses baryons can stay massive in extreme environments (e.g., neutron stars) where the chiral symmetry restoration takes place. In this work, we generalize the previous $\mbox{SU(2)}_L \times \mbox{SU(2)}_R$ parity models of nucleons to $\mbox{SU(3)}_L \times \mbox{SU(3)}_R$ models of the baryon octet, within the linear realization of the chiral symmetry. The major problem in constructing such models has been too many candidates for the chiral representations of baryons. Motivated by the concepts of diquarks and the mended symmetry, we choose the $(3_L, \bar{3}_R) + (\bar{3}_L, 3_R)$, $(3_L, 6_R) + (6_L, 3_R)$ and $(1_L, 8_R) + (8_L, 1_R)$ representations and use quark diagrams to constrain the possible types of Yukawa interactions. The masses of the baryon octets for positive and negative baryons up to the first excitations are successfully reproduced. As expected from the diquark considerations, the ground state baryons are well dominated by $(3_L, \bar{3}_R) + (\bar{3}_L, 3_R)$ and $(1_L, 8_R) + (8_L, 1_R)$ representations, while the excited states require $(3_L, 6_R) + (6_L, 3_R)$ representations. Important applications of our model are the chiral restoration for strange quarks at large density and the continuity of diquarks from hadronic to quark matter. We also address the problem of large Yukawa couplings which are enhanced in three-flavor construction.

We study thermal effects on equations of state (EOS) in isospin QCD, utilizing a quark-meson model coupled to a Polyakov loop. The quark-meson model is analyzed at one-loop that is the minimal order to include quark substructure constraints on pions which condense at finite isospin density. In the previous study we showed that the quark-meson model at zero temperature produces the sound velocity peak and the negative trace anomaly in the domain between the chiral effective theory regime at low density and the perturbative QCD regime at high density, in reasonable agreement with lattice simulations. We now include thermal effects from quarks in the Polyakov loop background and examine EOS, especially the sound velocity and trace anomaly along isentropic trajectories. At large isospin density, there are three temperature windows; (i) the pion condensed region with almost vanishing Polyakov loops, (ii) the pion condensed region with finite Polyakov loops, and (iii) the quark gas without pion condensates. In the domain (i), the gap associated with the pion condensate strongly quenches thermal excitations. As the system approaches the domain (ii), thermal quarks, which behave as non-relativistic particles, add energy density but little pressure, substantially reducing the sound velocity to the value less than the conformal value while increasing the trace anomaly toward the positive value. Approaching the domain (iii), thermal quarks become more relativistic as pion condensates melt, increasing sound velocity toward the conformal limit. Corrections from thermal pions are also briefly discussed., Comment: 12 pages, 8 figures

We examine the gravitational wave frequencies of the fundamental ($f$-) and 1st pressure ($p_1$-) modes excited in the neutron star models constructed with the quark-hadron crossover (QHC) type equations of state (EOS). We find that the $f$-mode frequencies with QHC EOS basically are smaller and the $p_1$-mode frequencies with QHC EOS are larger than those with hadronic EOS, focusing on the neutron star model with a fixed mass. We also find that the universality in the $f$-mode frequencies multiplied by the stellar mass as a function of the stellar compactness or as a function of the dimensionless tidal deformability, which is derived with various hadronic EOSs, can keep even with QHC EOS. That is, using these universal relations, one cannot distinguish QHC EOS from hadronic EOSs. Instead, using the relations one can extract the stellar radii whose evolution from low to high mass neutron stars can differentiate QHC from hadronic EOSs. On the other hand, we find that the $p_1$-mode frequencies multiplied by the stellar mass with QHC EOS significantly deviate in a certain mass range from the corresponding empirical relations derived with various hadronic EOSs, with which one may distinguish QHC EOS from hadronic EOSs., Comment: Accepted for publication in RPD

We construct $ {\rm SU(3)}_{\rm L} \otimes {\rm SU(3)}_{\rm R}$ invariant parity doublet models within the linear realization of the chiral symmetry. Describing baryons as the superposition of linear representations should be useful description for transitions toward the chiral restoration. The major problem in the construction is that there are much more chiral representations for baryons than in the two-flavor cases. To reduce the number of possible baryon fields, we introduce a hierarchy between representations with good or bad diquarks (called soft and hard baryon representations, respectively). We use $(3,\bar3)+(\bar3,3)$ and $(8,1)+(1,8)$ as soft to construct a chiral invariant Lagrangian, while the $(3,6)+(6,3)$ representations are assumed to be integrated out, leaving some effective interactions. The mass splitting associated with the strange quark mass is analyzed in the first and second order in the meson fields $M$ in $(3,\bar3)+(\bar3,3)$ representations. We found that the chiral $ {\rm SU(3)}_L \otimes {\rm SU(3)}_R$ constraints are far more restrictive than the $ {\rm SU(3)}_V$ constraints used in conventional models for baryons. After extensive analyses within $(3,\bar3)+(\bar3,3)$ and $(8,1)+(1,8)$ models, we found that models in the first order of $M$ do not reproduce the mass hierarchy correctly, although the {\GO} is satisfied. In the second order, the masses of the positive parity channels are reproduced well up to the first radial excitations, while some problem in the mass ordering remains in a negative parity channel. Apparently the baryon dynamics is not well-saturated by just $(3,\bar3)+(\bar3,3)$ and $(8,1)+(1,8)$ representations, as indicated by the necessity of terms higher order in $M$.

We present a model of cold QCD matter that bridges nuclear and quark matter through the duality relation between quarks and baryons. The baryon number and energy densities are expressed as functionals of either the baryon momentum distribution, $f_{\rm B}$, or the quark distribution, $f_{\rm Q}$, which are subject to the constraints on fermions, $0 \le f_{\rm B,Q} \le 1$. The theory is ideal in the sense that the confinement of quarks into baryons is reflected in the duality relation between $f_{\rm Q}$ and $f_{\rm B}$, while other possible interactions among quarks and baryons are all neglected. The variational problem with the duality constraints is formulated and we explicitly construct analytic solutions, finding two distinct regimes: A nuclear matter regime at low density and a Quarkyonic regime at high density. In the Quarkyonic regime, baryons underoccupy states at low momenta but form a momentum shell with $f_{\rm B}=1$ on top of a quark Fermi sea. Such a theory describes a rapid transition from a soft nuclear equation of state to a stiff Quarkyonic equation of state. At this transition, there is a rapid increase in the pressure., Comment: 6 pages, 3 figures; v2: published version

We study zero temperature equations of state (EOS) in isospin QCD within a quark-meson model which is renormalizable and hence eliminates high density artifacts in models with the ultraviolet cutoff (e.g., NJL models). The model exhibits a crossover transition of pion condensations from the Bose-Einstein-Condensation regime at low density to the Bardeen-Cooper-Schrieffer regime at high density. The EOS stiffens quickly and approaches the quark matter regime at density significantly less than the density for pions to spatially overlap. The sound velocity develops a peak in the crossover region, and then gradually relaxes to the conformal value $1/3$ from above, in contrast to the perturbative QCD results which predicts the approach from below. In the context of QCD computations, this opposite trend is in part due to the lack of gluon exchanges in our model, and also due to the non-perturbative power corrections arising from the condensates. We argue that with large power corrections the trace anomaly can be negative. In quantitative level, our EOS is consistent with the lattice results in the BEC regime but begins to get stiffer at higher density. The sound velocity peak also appears at higher density. The BCS gap in our model is $\Delta \simeq 300$ MeV in the quark matter domain, and naive application of the BCS relation for the critical temperature $T_c \simeq 0.57\Delta$ yields the estimate $T_c \simeq 170$ MeV, in good agreement with the lattice data., Comment: 19 pages, 16 figures, added description of renormalization procedure, fixed n0 scaling

We review the chiral variant and invariant components of nucleon masses and its consequence on the chiral restoration in extreme conditions, neutron star matter in particular. We consider a model of linear realization of chiral symmetry with the nucleon parity doublet structure that permits the chiral invariant mass, $m_0$, for positive and negative parity nucleons. Nuclear matter is constructed with the parity doublet nucleon model coupled to scalar fields $\sigma$, vector fields $(\omega, \rho)$, and to mesons with strangeness through the U(1)$_A$ anomaly. In models with large $m_0$, the nucleon mass is insensitive to the medium, and the nuclear saturation properties can be reproduced without demanding strong couplings of nucleons to scalar fields $\sigma$ and vector fields $\omega$. We confront the resulting nuclear equations of state with nuclear constraints and neutron star observations, and delineate the chiral invariant mass and effective interactions. To further examine nuclear equations of state beyond the saturation density, we supplement quark models to set the boundary conditions from the high density side. The quark models are constrained by the two-solar mass conditions, and such constraints are transferred to nuclear models through the causality and thermodynamic stability conditions. We also calculate various condensates and matter composition from nuclear to quark matter in a unified matter, by constructing a generating functional that interpolates nuclear and quark matter with external fields. Two types of chiral restoration are discussed; the one due to the positive scalar charges of nucleons, and the other triggered by the evolution of the Dirac sea. We found the U(1)$_A$ anomaly softens equations of state from low to high density.

Smooth transitions from hadronic matter to hot and dense matter of quantum chromodynamics accompany continuous transformations in effective degrees of freedom. The microscopic descriptions should include relativistic quarks interacting inside of hadrons. In this work we construct a schematic constituent quark model with relativistic kinematics which captures the global trends of meson spectra in the light, strange, and charm quark sectors. We examine the roles of the scalar- and vector-confining potentials as well as semishort range correlations in estimating the strength of central, spin-spin, and spin-orbit interactions. The quark dynamics in low-lying mesons is very sensitive to relativistic kinematics and short range interactions, while in high-lying mesons are sensitive to the composition of scalar- and vector-confinement. After expressing mesons in terms of quark wave functions, we use them to describe the quark occupation probability in a meson resonance gas, and discuss how it can be related to its counterpart in a quark-gluon-plasma., Comment: 24 pages, 10 figures

We examine the role of the $U(1)_A$ anomaly in a parity doublet model of nucleons which include the chiral variant and invariant masses. Our model expresses the $U(1)_A$ anomaly by the Kobayashi-Maskawa-'t\,Hooft (KMT) interaction in the mesonic sector. After examining the roles of the KMT term in vacuum, we discuss its impacts on nuclear equations of state (EOS). The $U(1)_A$ anomaly increases the masses of the $\eta'$ and $\sigma$ mesons and enhances the chiral symmetry breaking. The $U(1)_A$ anomaly enlarges the energy difference between chiral symmetric and symmetry broken vacuum; in turn, the chiral restoration at high density adds a larger energy density (often referred as a bag constant) to EOSs than in the case without the anomaly, leading to softer EOSs. Including these $U(1)_A$ effects, we update the previously constructed unified equations of state that interpolate the nucleonic EOS at $n_B \le 2n_0$ ($n_{0} = 0.16\, \rm{fm^{-3}}$: nuclear saturation density) and quark EOS at $n_B \ge 5n_0$. The unified EOS is confronted with the observational constraints on the masses and radii of neutron stars. The softening of EOSs associated with the $U(1)$ anomaly reduces the overall radii, relaxing the previous constraint on the chiral invariant mass $m_0$. Including the attractive nonlinear $\rho$-$\omega$ coupling to improved estimates for the slope parameter in the symmetry energy, our new estimate is $400\,{\rm MeV} \leq m_0 \leq 700\,{\rm MeV}$, with $m_0$ smaller than our previous estimate by $\sim 200$ MeV., Comment: 14 pages

Neutron stars and supernovae provide cosmic laboratories of highly compressed matter at supra nuclear saturation density which is beyond the reach of terrestrial experiments. The properties of dense matter is extracted by combining the knowledge of nuclear experiments and astrophysical observations via theoretical frameworks. A matter in neutron stars is neutron rich, and may further accommodate non-nucleonic degrees of freedom such as hyperons and quarks. The structure and composition of neutron stars are determined by equations of state of matter, which are the primary subject in this chapter. In case of supernovae, the time evolution includes several dynamical stages whose descriptions require equations of state at finite temperature and various lepton fractions. Equations of state also play essential roles in neutron star mergers which allow us to explore new conditions of matter not achievable in static neutron stars and supernovae. Several types of hadron-to-quark transitions, from first order transitions to crossover, are reviewed, and their characteristics are summarized., Comment: 53 pages, 24 figures, Contribution to the "Handbook of Nuclear Physics", Springer, 2022, edited by I. Tanihata, H. Toki and T. Kajino

Fully general-relativistic binary-neutron-star (BNS) merger simulations with quark-hadron crossover (QHC) equations of state (EOSs) are studied for the first time. In contrast to EOSs with purely hadronic matter or with a first-order quark-hadron phase transition (1PT), in the transition region QHC EOSs show a peak in sound speed, and thus a stiffening. We study the effects of such stiffening in the merger and post-merger gravitational (GW) signals. Through simulations in the binary-mass range $2.5 < M/M_{\odot} < 2.75$, characteristic differences due to different EOSs appear in the frequency of the main peak of the post-merger GW spectrum ($f_2$), extracted through Bayesian inference. In particular, we found that (i) for lower-mass binaries, since the maximum baryon number density ($n_{\rm max}$) after the merger stays below $3\text{--}4$ times the nuclear-matter density ($n_0$), the characteristic stiffening of the QHC models in that density range results in a lower $f_2$ than that computed for the underlying hadronic EOS and thus also than that for EOSs with a 1PT, (ii) for higher-mass binaries, where $n_{\rm max}$ may exceed $4\text{--}5 n_0$ depending on the EOS model, whether $f_2$ in QHC models is higher or lower than that in the underlying hadronic model depends on the height of the sound-speed peak. Comparing the values of $f_2$ for different EOSs and BNS masses gives important clues on how to discriminate different types of quark dynamics in the high-density end of EOSs and is relevant to future kilohertz GW observations with third-generation GW detectors., Comment: 6+8 pages, 5+6 figures

In this contribution, we summarize our recent studies on the chiral invariant mass and the chiral condensates in neutron star matter. We construct a unified equations of state assuming the crossover phase transition from hadronic matter described by a parity doublet model to quark matter by an Nambu--Jona-Lasinio type quark model. We first show that the chiral invariant mass is constrained to be 600 MeV $\lesssim m_0 \lesssim$ 900 MeV from recent observations of neutron stars. We then determine the density dependence of the chiral condensate in the crossover description, and show that the chiral condensates are actually smoothly connected from the hadronic matter where the change is driven by the positive chiral scalar charge in a nucleon, to the quark matter where the change is by the modification of the quark Dirac sea, reflecting the hadron-quark crossover.

The recent NICER measurement of the radius of the neutron star PSR J0740+6620, and the inferred small variation of radii from 1.4$M_\odot$ to 2.1$M_\odot$, reveal key features of the equation of state of neutron star matter. The pressure rises rapidly in the regime of baryon density $n \sim$ 2-4 times nuclear saturation density, $n_0$ -- the region where we expect hadronic matter to be undergoing transformation into quark matter -- and the pressure in the nuclear regime is greater than predicted by microscopic many-body variational calculations of nuclear matter. To incorporate these insights into the microscopic physics from the nuclear to the quark matter regimes, we construct an equation of state, QHC21, within the framework of quark-hadron crossover (QHC). We include nuclear matter results primarily based on the state-of-the-art chiral effective field theory, but also note results of using nuclear matter variational calculations based on empirical nuclear forces. We employ explicit nuclear degrees of freedom only up to $n \sim 1.5n_0$, in order to explore the possibility of further physical degrees of freedom than nucleonic here. The resulting QHC21, which has a peak in sound velocity in $\sim 2$-$4 n_0$, is stiffer than the earlier QHC19 below 2$n_0$, predicting larger radii in substantial agreement with the NICER data., Comment: 10 pages, 11 figures; v2) 12 pages, 11 figures

We discuss stiffening of dense matter in two color QCD (QC$_2$D) where hadrons are mesons and diquark baryons. We study two models which describe a transition of matter from the Bose-Einstein-Condensation regime at low density to the Bardeen-Cooper-Schrieffer regime at high density. The first model is based on coherent states of diquarks, and the second is the Nambu-Jona-Lasinio model with diquark pairing terms. We particularly focus on how quark states are occupied as baryon density increases. We find that, due to the occupied quark levels, the ideal gas picture of diquarks breaks down at density significantly less than the density where baryon cores overlap. The saturation of quark states at low momenta stiffens equations of state. We also study the effects of interactions which depend on the quark occupation probability. We argue that equations of state become very stiff when the bulk part of the quark Fermi sea has the effective repulsion but the Fermi surface enjoys the attractive correlations. This disparity for different momentum domains is possible due to the strong channel dependence in gluon exchanges with momentum transfer of $0.2-1$ GeV. These concepts can be transferred from QC$_2$D to QCD in any numbers of colors., Comment: 14 pages, 14 figures; v2, minor changes, published in PRD

We study the chiral condensates in neutron star matter from nuclear to quark matter domain. We describe nuclear matter with a parity doublet model (PDM), quark matter with the Nambu--Jona-Lasino (NJL) model, and a matter at the intermediate density by interpolating nuclear and quark matter equations of state. The model parameters are constrained by nuclear physics and neutron star observations. Various condensates in the interpolated domain are estimated from the chemical potential dependence of the condensates at the boundaries of the interpolation. The use of the PDM with substantial chiral invariant mass ($m_0 \gtrsim 500$ MeV, which is favored by the neutron star observations) predicts the mild chiral restoration, and the significant chiral condensate remains to baryon density $n_B \sim 2-3n_0$ ($n_0\simeq 0.16\,{\rm fm}^{-3}$: nuclear saturation density), smoothly approaching the NJL predictions for the color-flavor-locked phase at $n_B \gtrsim 5n_0$. The same method is applied to estimate diquark condensates, number densities of up-, down- and strange-quarks, and the lepton fraction. In our descriptions the chiral restoration in the interpolated domain proceeds with two conceptually distinct chiral restoration effects; the first is associated with the positive scalar density in a nucleon, relevant in dilute regime, and the other primarily arises from the modification of the quark Dirac sea, which is triggered by the growth of the quark Fermi sea. We discuss several qualitative conjectures to interpolate the microphysics in nuclear and quark matter.

We discuss stiffening of matter in quark-hadron continuity. We introduce a model that relates quark wave functions in a baryon and the occupation probability of states for baryons and quarks in dense matter. In a dilute regime, the confined quarks contribute to the energy density through the masses of baryons, but do not directly contribute to the pressure; hence, the equations of state are very soft. This dilute regime continues until the low momentum states for quarks get saturated; this may happen even before baryons fully overlap, possibly at density slightly above the nuclear saturation density. After the saturation the pressure grows rapidly while changes in energy density are modest, producing a peak in the speed of sound. If we use baryonic descriptions for quark distributions near the Fermi surface, we reach a description similar to the quarkyonic matter model of McLerran and Reddy. With a simple adjustment of quark interactions to get the nucleon mass, our model becomes consistent with the constraints from 1.4-solar mass neutron stars, but the high density part is too soft to account for two-solar mass neutron stars. We delineate the relation between the saturation effects and short range interactions of quarks, suggesting interactions that leave low density equations of state unchanged but stiffen the high density part., Comment: 14 pages, 14 figures; v2) 15 pages, 16 figures; v3) published in PRD

We study the chiral separation effect (CSE) in two-color and two-flavor QCD (QC$_2$D) to delineate quasiparticle pictures in dense matter from low to high temperatures. Both massless and massive quarks are discussed. We particularly focus on the high density domain where diquarks form a color singlet condensate with the electric charge $1/3$. The condensate breaks baryon number and $U(1)_A$ axial symmetry, and induces the electromagnetic Meissner effects. Within a quark quasiparticle picture, we compute the chiral separation conductivity at one-loop. We have checked that Nambu-Goldstone modes, which should appear in the improved vertices as required by the Ward-Takahashi identities, do not contribute to the chiral separation conductivity due to their longitudinal natures. In the static limit, the destructive interferences in the particle-hole channel, as in usual Meissner effects, suppress the conductivity (in chiral limit, to $1/3$ of the normal phase's). This locally breaks the universality of the CSE coefficients, provided quasiparticle pictures are valid in the bulk matter., Comment: 12 pages, 6 figures

We analyze mesons in constant magnetic fields ($B$) within a non-relativistic constituent quark model. Our quark model contains a harmonic oscillator type confining potential, and we perturbatively treat short range correlations to account for the spin-flavor energy splittings. We study both neutral and charged mesons taking into account the internal quark dynamics. The neutral states are labelled by two-dimensional momenta for magnetic translations, while the charged states by two discrete indices related to angular momenta. For $B \ll \Lambda_{\rm QCD}^2$ ($\Lambda_{\rm QCD} \sim 200$ MeV: the QCD scale), the analyses proceed as in usual quark models, while special precautions are needed for strong fields, $B \sim \Lambda_{QCD}^2$, especially when we treat short range correlations such as the Fermi-Breit-Pauli interactions. We compute the energy spectra of mesons up to energies of $\sim 2.5$ GeV and use them to construct the meson resonance gas. Within the assumption that the constituent quark masses are insensitive to magnetic fields, the phase space enhancement for mesons significantly increases the entropy, assisting a transition from a hadron gas to a quark gluon plasma. We confront our results with the lattice data, finding reasonable agreement for the low-lying spectra and the entropy density at low temperature less than $\sim 100$ MeV, but our results at higher energy scale suffer from artifacts of our confining potential and non-relativistic treatments., Comment: 14 pages, 9 figures; A contribution to the topical issue "The QCD Phase Diagram in Strong Magnetic Fields"; v2 revised, 18 pages, 10 figures, comparisons with the lattice results were added

We study Landau gauge gluon propagators in two-color QCD at finite quark chemical potential ($\mu_q$) and temperature ($T$). We include medium polarization effects at one-loop by quarks into massive gluon propagators, and compared the analytic results with the available lattice data. We particularly focus on the high density phase of color-singlet diquark condensates whose critical temperature is $\sim 100$ MeV with weak dependence on $\mu_q$. At zero temperature the color singlet condensates protect the IR limit of electric and magnetic gluon propagators from the medium screening effects. At finite temperature, this behavior remains true for the magnetic sector, but the electric screening mass should be generated by thermal, and hence gapless, particles which are unbound from the diquark condensates. Treating thermal excitations as quasi-quarks, we found that the electric screening develops too fast compared to the lattice results. Beyond the critical temperature for diquark condensates the analytic results are consistent with the lattice results., Comment: 10 pages, 10 figures; v2 published in PRD

We delineate the quark-hadron continuity by constructing QCD equations of state for neutron star dynamics, covering the wide range of charge chemical potential ($\mu_Q$) and temperatures ($T$). Based on the nuclear-2SC continuity scenario, we match equations of state for nuclear and two-flavor color-superconducting (2SC) quark matter, where the matching baryon density is $n_B\simeq 1.5n_0$ ($n_0\simeq 0.16\, {\rm fm^{-3}}$: nuclear saturation density). The effective vector and diquark couplings in a quark matter model evolve as functions of $n_B, \mu_Q, T$, whose low density values are constrained by the nuclear matter properties and neutron star radii, with the high density behavior by the two-solar mass ($2M_\odot$) constraint. With couplings dependent on $n_B$, we examined how smooth the nuclear-2SC continuity can be, and found problems in matching nuclear and 2SC entropies at low temperatures. To proceed with the continuity scenario, we enforce smooth matching by making the couplings ($n_B, \mu_Q, T$)-dependent. In effect, this adds phenomenological contributions which we call "X". After the phenomenological matching, we take the rest as our predictions. The 2SC and color-flavor-locked (CFL) phases computed with these evolving couplings are called 2SCX and CFLX. The CFLX appears around $n_B\simeq 2$-$4n_0$ and, in contrast to the conventional CFL, has non-negligible dependence on $(\mu_Q,T)$. To examine the astrophysical consequences of our modeling, we add charged leptons and neutrinos, and study the composition of matter for lepton fractions relevant for protoneutron stars and neutron star mergers. The abundance of neutrinos and thermal effects reduce the strangeness fraction and stiffen equations of state. For a neutrino trapped neutron star at $T\simeq 30 $ MeV with a lepton fraction $Y_L\simeq 0.05$, the mass is larger than its cold static counterpart by $\sim 0.1M_\odot$., Comment: 23 pages, 16 figures; v2, 24 pages, 16 figures, published in PRD

We construct an equation of state (EOS) for neutron stars by interpolating hadronic EOS at low density and quark EOS at high density. A hadronic model based on the parity doublet structure is used for hadronic matter and a quark model of Nambu--Jona-Lasinio type is for quark matter. We assume crossover between hadronic matter and quark matter in the the color-flavor locked phase. The nucleon mass of the parity doublet model has a mass associated with the chiral symmetry breaking, and a chiral invariant mass $m_0$ which is insensitive to the chiral condensate. The value of $m_0$ affects the nuclear EOSs at low density, and has strong correlations with the radii of neutron stars. Using the constraint to the radius obtained by LIGO-Virgo and NICER, we find that $m_0$ is restricted as $600\,\mathrm{MeV}\lesssim m_0 \lesssim 900\,\mathrm{MeV}$.

Neutron stars are cosmic laboratories to study dense matter in Quantum Chromodynamics (QCD). The observable mass-radius relations of neutron stars are determined by QCD equations of state, and can reflect the properties of QCD phase transitions. In the last decade there have been historical discoveries in neutron stars, the discoveries of two-solar mass neutron stars and neutron star merger events, which have imposed tight constraints on equations of state. While a number of equations of state are constructed to satisfy these constraints, a theoretical challenge is how to reconcile those constructions with the microphysics expected from the hadron physics and in-medium calculations. In this short article we briefly go over recent observations and discuss their implications for dense QCD matter, referring to QCD constraints in the low and high density limits, QCD-like theories, and lattice QCD results for baryon-baryon interactions., Comment: 9 pages, 2 figures, a review contributed to the AAPPS Bulletin; v2 published version

We propose a conceptual distinction between hard and soft realizations of deconfinement from nuclear to quark matter. In the high density region of Hard Deconfinement the repulsive hard cores of baryons overlap each other and bulk thermodynamics is dominated by the core properties that can be experimentally accessed in high-energy scattering experiments. We find that the equation of state estimated from a single baryon core is fairly consistent with those empirically known from neutron star phenomenology. We next discuss a novel concept of Soft Deconfinement, characterized by quantum percolation of quark wave-functions, at densities lower than the threshold for Hard Deconfinement. We make a brief review of quantum percolation in the context of nuclear and quark matter and illustrate a possible scenario of quark deconfinement at high baryon densities., Comment: 14 pages, 12 figures; v2 published in PRD

The properties of dense QCD matter are delineated through the construction of equations of state which should be consistent with the low and high density limits of QCD, nuclear laboratory experiments, and the neutron star observations. These constraints, together with the causality condition of the sound velocity, are used to develop the picture of hadron-quark continuity in which hadronic matter continuously transforms into quark matter (modulo small 1st order phase transitions). The resultant unified equation of state at zero temperature and $\beta$-equilibrium, which we call Quark-Hadron-Crossover (QHC19), is consistent with the measured properties of neutron stars as well as the microphysics known for the hadron phenomenology. In particular to $\sim 10n_0$ ($n_0$: saturation density) the gluons remain as non-perturbative as in vacuum and the strangeness can be as abundant as up- and down-quarks at the core of two-solar mass neutron stars. Within our modeling the maximum mass is found less than $\simeq 2.35$ times solar mass and the baryon density at the core ranges in $\sim 5$-8$n_0$., Comment: 14 pages, 10 figures, talk given at 37th International Symposium on Lattice Field Theory - Lattice2019, 16-22 June 2019, Wuhan, China. arXiv admin note: text overlap with arXiv:1904.05080

We study the string tension $\sigma$ of quantum chromodynamics (QCD) in external magnetic fields $B$ by combining the quasi-particle treatments for quarks with the collective treatments for gluons. We utilize the 1/Nc-expansion (Nc: number of colors) and the framework of Wilson's strong coupling expansion at long distance, and compare the resulting string tensions with the lattice data. The effects of magnetic fields are taken into account through the dielectric functions, which are evaluated from the one-loop polarization due to light flavor quarks with $B$-dependent effective masses $M_{\rm f}(B)\ ({\rm f}=u,d,s)$. All Landau levels are adopted in the calculations so that we can explore the string tensions from weak to strong magnetic fields. For the effective quark masses, we use a simple parameterization: $M_{\rm f}(B) = M_{\rm f}(0) + C_B |q_{\rm f}B|^{1/2}$ with $q_{\rm f}$ the electric charge of a given flavor and $C_B$ an adaptive parameter. We find that the parameter $C_B$ needs to be small, $C_B \lesssim 0.05$, to qualitatively reproduce the trends of lattice QCD results for $0 \leq |eB| \lesssim 1.5\, {\rm GeV}^2$. The quantitative agreement is found for $C_B \simeq 0$ and the strong coupling $\alpha_S \simeq 5$ in the infrared., Comment: 5 pages, 3 figures

We study the Landau gauge gluon propagators in dense two-color QCD at quark chemical potential, $\mu_q$, in the range from 0.5 to 1.0 GeV not reachable by the perturbative method at weak coupling. In order to take into account the non-perturbative effects, at tree level we use a massive Yang-Mills model for the Yang-Mills theory (or the Curci-Ferrari model) which has successfully described the lattice results of the gluon and ghost propagators in the Landau gauge. We couple quarks to this theory and compute the one-loop polarization effects in medium. The presence of the gluon mass significantly tempers the medium effects and uncertainties associated with the strong coupling constant $\alpha_s$. The diquark condensate in two-color QCD is color-singlet, for which neither electric nor magnetic screening masses should appear at the scale less than the diquark gap. The presence of the gap helps to explain the lattice results which are not very sensitive to the quark density. Meanwhile we also found the limitation of the one-loop estimate as well as the lack of some physics in perturbative medium corrections., Comment: 16 pages, 13 figures

A moderately strong vector repulsion between quarks in dense quark matter is needed to explain how a quark core can support neutron stars heavier than two solar masses. We study this repulsion, parametrized by a four-fermion interaction with coupling g_V, in terms of non-perturbative gluon exchange in QCD in the Landau gauge. Matching the energy of quark matter, g_V n_q^2 (where n_q is the number density of quarks) with the quark exchange energy calculated in QCD with a gluon propagator parametrized by a finite gluon mass m_g and a frozen coupling alpha_s, at moderate quark densities, we find that gluon masses m_g in the range 200 - 600 MeV and alpha_s = 2 - 4 lead to a g_V consistent with neutron star phenomenology. Estimating the effects of quark masses and a color-flavor-locked (CFL) pairing gap, we find that g_V can be well approximated by a flavor-symmetric, decreasing function of density. We briefly discuss similar matchings for the isovector repulsion and for the pairing attraction., Comment: 9 pages, 6 figures

The properties of dense QCD matter are delineated through the construction of equations of state which should be consistent with QCD calculations in the low and high density limits, nuclear laboratory experiments, and the neutron star observations. These constraints, together with the causality condition of the sound velocity, are used to develop the picture of hadron-quark continuity in which hadronic matter continuously transforms into quark matter (modulo small 1st order phase transitions). For hadronic matter (at baryon density nB > ~2n0 with n0 ~ 0.16 fm^(-3) being the nuclear saturation density) we use equations of state by Togashi et al. based on microscopic variational many-body calculations, and for quark matter (nB > ~5n0) we construct equations of state using a schematic quark model (with strangeness) whose interactions are motivated by the hadron phenomenology. The region between hadronic and quark matters (~2n0 < nB < ~5n0), which is most difficult to calculate, is treated by highly constrained interpolation between nuclear and quark matter equations of state. The resultant unified equation of state at zero temperature and beta-equilibrium, which we call Quark-Hadron-Crossover (QHC18 and QHC19), is consistent with the measured properties of neutron stars and in addition gives us microscopic insights into the properties of dense QCD matter. In particular to ~10n0 the gluons can remain as non-perturbative as in vacuum and the strangeness can be as abundant as up- and down-quarks at the core of two-solar mass neutron stars. Within our modeling the maximum mass is found less than ~2.35 times solar mass and the baryon density at the core ranges in ~5-8n0., Comment: 18 pages 11 figures, AIP Proceedings of the Xiamen-CUSTIPEN Workshop on the EOS of Dense Neutron-Rich Matter in the Era of Gravitational Wave Astronomy, Jan. 3-7, Xiamen, China; v2 references are added

We present a much improved equation of state for neutron star matter, QHC19, with a smooth crossover from the hadronic regime at lower densities to the quark regime at higher densities. We now use the Togashi et al.~equation of state (Togashi:2017), a generalization of the Akmal-Pandharipande-Ravenhall equation of state of uniform nuclear matter, in the entire hadronic regime; the Togashi equation of state consistently describes non-uniform as well as uniform matter, and matter at beta equilibrium without the need for an interpolation between pure neutron and symmetric nuclear matter. We describe the quark matter regime at higher densities with the Nambu--Jona--Lasinio model, now identifying tight constraints on the phenomenological universal vector repulsion between quarks and the pairing interaction between quarks arising from the requirements of thermodynamic stability and causal propagation of sound. The resultant neutron star properties agree very well with the inferences of the LIGO/Virgo collaboration, from GW170817, of the pressure vs. baryon density, neutron star radii, and tidal deformabilities. The maximum neutron star mass allowed by QHC19 is 2.35 $M_\odot$, consistent with all neutron star mass determinations., Comment: 8 pages, 11 figures; v3) published version, appendix was added

We analyze the zero point energy of composite particles (or resonances) which are dynamically created from relativistic fermions. We compare the zero point energies in medium to the vacuum one, taking into account the medium modification of the constituent particles. Treating composite particles as if quasi-particles, their zero point energies contain the quadratic and logarithmic UV divergences even after the vacuum subtraction. The coefficients of these divergences come from the difference between the vacuum and in-medium fermion propagators. We argue that such apparent divergences can be cancelled by consistently using fermion propagators to compute the quasi-particle contributions as well as their interplay, provided that the self-energies of the constituents at large momenta approach to the vacuum ones sufficiently fast. In the case of quantum chromodynamics, mesons and baryons, which may be induced or destroyed by medium effects, yield the in-medium divergences in the zero point energies, but the divergences are assembled to cancel with those from the quark zero-point energy. This is particularly important for unified descriptions of hadronic and quark matter which may be smoothly connected by the quark-hadron continuity., Comment: v3) 32 pages, 13 figures; published in PRD

We analyze the zero point energy in a dense matter of quarks or hadrons with particular attention on the renormalization of the UV divergences. Besides divergences removable by the vacuum subtraction and counter terms, there are also UV divergences associated with non-perturbative modifications of quark bases appearing in the in-medium propagators. The latter would remain after the self-energies and vertices are renormalized, unless a proper set of medium contributions is included at a given truncation. We use the formalism of the two particle irreducible action to clarify how the UV divergences are assembled to cancel. An example is given for the thermodynamic potentials with mesons as composite particles whose zero point energies apparently diverge but can be cancelled by the quark self-energy contributions. Important applications of this work are quark matter with hadronic correlations which may be realized at the core of neutron stars., Comment: v2: 12 pages, 4 figures, discussions of renormalizability were revised

We delineate the properties of dense QCD matter through equations of state constrained by the neutron star observations. The two solar mass constraint, the radius constraint of 11-13 km, and the causality constraint on the speed of sound, are used to develop the picture of hadron-quark continuity in which hadronic matter continuously transforms into quark matter. A unified equation of state at zero temperature and beta-equilibrium is constructed by a phenomenological interpolation between nuclear and quark matter equations of state. For applications to supernovae and neutron star mergers, the unified equation of state is perturbed by temperature corrections., Comment: 13 pages, 9 figures, a talk presented at Critical Point and Onset of Deconfinement, 7-11 August, 2017, The Wang Center, Stony Brook University, Stony Brook, NY

We study the phase diagram of two-flavor massless QCD at finite baryon density by applying the functional renormalization group (FRG) for a quark-meson model with $\sigma, \pi$, and $\omega$ mesons. The dynamical fluctuations of quarks, $\sigma$, and $\pi$ are included into the flow equations, while the amplitudes of $\omega$-fields are also allowed to fluctuate. At high temperature the effects of the $\omega$-field on the phase boundary are qualitatively similar to the mean-field calculations; the phase boundary is shifted to the higher chemical potential region. As the temperature is lowered, however, the transition line bends back to the lower chemical potential region, irrespective to the strength of the vector coupling. In our FRG calculations, the driving force of the low temperature first order line is the fluctuations rather the quark density, and the effects of $\omega$-fields have little impact. At low temperature, the effective potential at small $\sigma$ field is very sensitive to the infrared cutoff scale, and this significantly affects our determination of the phase boundaries. The critical chemical potential at the tricritical point is affected by the $\omega$-field effects but its critical temperature stays around the similar value. Some caveats are given in interpreting our model results., Comment: 10 pages, 11 figures

We review the equation of state of matter in neutron stars from the solid crust through the liquid nuclear matter interior to the quark regime at higher densities. We focus in detail on the question of how quark matter appears in neutron stars, and how it affects the equation of state. After discussing the crust and liquid nuclear matter in the core we briefly review aspects of microscopic quark physics relevant to neutron stars, and quark models of dense matter based on the Nambu--Jona-Lasinio framework, in which gluonic processes are replaced by effective quark interactions. We turn then to describing equations of state useful for interpretation of both electromagnetic and gravitational observations, reviewing the emerging picture of hadron-quark continuity in which hadronic matter turns relatively smoothly, with at most only a weak first order transition, into quark matter with increasing density. We review construction of unified equations of state that interpolate between the reasonably well understood nuclear matter regime at low densities and the quark matter regime at higher densities. The utility of such interpolations is driven by the present inability to calculate the dense matter equation of state in QCD from first principles. As we review, the parameters of effective quark models -- which have direct relevance to the more general structure of the QCD phase diagram of dense and hot matter -- are constrained by neutron star mass and radii measurements, in particular favoring large repulsive density-density and attractive diquark pairing interactions. We describe the structure of neutron stars constructed from the unified equations of states with crossover. Lastly we present the current equations of state -- called "QHC18" for quark-hadron crossover -- in a parametrized form practical for neutron star modeling., Comment: v2, 42 pages, 36 figures, 3 tables; to be published in Reports on Progress in Physics; new sections for cooling, X-ray analyses, and gravitational waves are added; the results for tidal deformability are included; equations of state and the numerical tables are updated; v3, typos corrected in eq.8

Thermal QCD equations of state at high baryon density are sensitive to the phase structure and the resulting excitation modes. The leading contribution at low temperature can be either ~p_F^2 T^2 (pF: Fermi momentum, T: temperature) for phases with gapless quarks, or ~T^4 for phases with gapped quarks. In the latter the thermal pressure is dominated by collective modes. Starting with a schematic quark model developed for neutron star structure, we estimate the thermal contributions and zero point energy from the Nambu-Goldstone modes by building them upon the mean field background for the color-flavor-locked quark matter. Applying the phase shift representation for thermodynamic potentials, we include not only the bound state pairs but also resonating pairs. According to the Levinson's theorem, the high energy contributions tend to cancel the pole contributions to the thermodynamics, tempering the UV behaviors in the zero point energy. Our primary target in this talk is the domain with baryon density nB as large as ~ 5-10n_0 (n_0 = 0.16 fm^{-3}: nuclear saturation density), and the temperature T of the order ~30-100 MeV. The insights into this domain may be obtained through the future detection of gravitational waves from neutron star merging events., Comment: 5 pages, 2 figures; prepared for quark matter 2017, Chicago, Illinois, USA

We discuss cold dense QCD by examining constraints from neutron stars, nuclear experiments, and QCD calculations at low and high baryon density. The two solar mass constraint and suggestive small radii (10~13 km) of neutron stars constrain the strength of hadron-quark matter phase transitions. Assuming the adiabatic continuity from hadronic to quark matter, we use a schematic quark model for hadron physics and examine the size of medium coupling constants. We find that to baryon density nB ~ 10n0 (n0: nuclear saturation density), the model coupling constants should be as large as in the vacuum, indicating that gluons remain non- perturbative even after the quark matter formation., Comment: 6 pages, 2 figures. A talk prepared for "CPOD2016" in Wroclaw, Poland May 30th - June 4th, 2016

If quarks and gluons are either gapped or confined in neutron stars (NSs), the most relevant light modes are Nambu-Goldstone (NG) modes. We study NG modes within a schematic quark model whose parameters at high density are constrained by the two-solar mass constraint. Our model has the color-flavor-locked phase at high density, with the effective couplings as strong as in hadron physics. We find that strong coupling effects make NG modes more massive than in weak coupling predictions, and would erase several phenomena caused by the stressed pairings in mismatched Fermi surfaces. For instance, we found that charged kaons, which are dominated by diquark and anti-diquark components, are not light enough to condense at strong coupling. Implications for gravitational wave signals for NS-NS mergers are also briefly discussed., Comment: 15 pages, 5 figures; v2 published in PLB

We delineate the properties of QCD matter at baryon density n_B=1-10n_0 (n_0: nuclear saturation density), through the construction of neutron star equations of state that satisfy the neutron star mass-radius constraints as well as physical conditions on the speed of sound. The QCD matter is described in the 3-window modeling: at n_B < 2n_0 purely nuclear matter; at n_B > 5n_0 percolated quark matter; and at 2n_0 < n_B < 5n_0 matter intermediate between these two which are constructed by interpolation. Using a schematic quark model with effective interactions inspired from hadron and nuclear physics, we analyze the strength of interactions necessary to describe observed neutron star properties. Our finding is that the interactions should remain as strong as in the QCD vacuum, indicating that gluons at n_B =1-10 n_0 remain non-perturbative even after quark matter formation., Comment: 5 pages, 3 figures, proceedings of the XXV Quark Matter conference, Sep. 27 to Oct. 3 in 2015, Kobe (Japan)

We study properties of neutral and charged mesons in strong magnetic fields |eB|>> Lambda_QCD^2 with Lambda_QCD being the QCD renormalization scale. Assuming long-range interactions, we examine magnetic-field dependences of various quantities such as the constituent quark mass, chiral condensate, meson spectra, and meson wavefunctions by analyzing the Schwinger-Dyson and Bethe-Salpeter equations. Based on the density of states obtained from these analyses, we extend the hadron resonance gas (HRG) model to investigate thermodynamics at large B. As B increases the meson energy behaves as a slowly growing function of the meson's transverse momenta, and thus a large number of meson states is accommodated in the low energy domain; the density of states at low temperature is proportional to B^2. This extended transverse phase space in the infrared regime significantly enhances the HRG pressure at finite temperature, so that the system reaches the percolation or chiral restoration regime at lower temperature compared to the case without a magnetic field; this simple picture would offer a gauge invariant and intuitive explanation of the inverse magnetic catalysis., Comment: 36 pages, 6 figures; v2, minor corrections, published in Nuclear Physics A

We discuss theoretical scenarios on crossover between nuclear matter (NM) and quark matter (QM). We classify various possibilities into three major scenarios according to the onset of diquark degrees of freedom that characterizes color-superconducting (CSC) states. In the conventional scenario NM occurs at the liquid-gas (or liquid-vacuum at zero temperature) phase transition and QM occurs next, after which CSC eventually appears. With the effect of strong correlation, the BEC-BCS scenario implies that CSC occurs next to NM and QM comes last in the BCS regime. We adopt the quarkyonic scenario in which NM, QM, and CSC are theoretically indistinguishable and thus these names refer to not distinct states but relevant descriptions of the same physical system. Based on this idea we propose a natural scheme to interpolate NM near normal nuclear density and CSC with vector coupling at high baryon density. We finally discuss the mass-radius relation of the neutron star and constraints on parameters in the proposed scheme., Comment: 15 pages, 6 figures. v2

We discuss the 3-window modeling of cold, dense QCD matter equations of state at density relevant to neutron star properties. At low baryon density, n_B < ~ 2n_s (n_s: nuclear saturation density), we utilize purely hadronic equations of state that are constrained by empirical observations at density n_B ~ n_s and neutron star radii. At high density, n_B > ~ 5n_s, we use the percolated quark matter equations of state which must be very stiff to pass the two-solar mass constraints. The intermediate domain at 2 < n_B/n_s < 5 is described as neither purely hadronic nor percolated quark matter, and the equations of state are inferred by interpolating hadronic and percolated quark matter equations of state. Possible forms of the interpolation are severely restricted by the condition on the (square of) speed of sound, 0 < c_s^2 < 1. The characteristics of the 3-window equation of state are compared with those of conventional hybrid and self-bound quark matters. Using a schematic quark model for the percolated domain, it is argued that the two-solar mass constraint requires the model parameters to be as large as their vacuum values, indicating that the gluon dynamics remains strongly non-perturbative to n_B ~ 10n_s. The hyperon puzzle is also briefly discussed in light of quark descriptions., Comment: 18 pages, 22 figures, prepared for the 2015 EPJA Topical Issue on "Exotic Matter in Neutron Stars"; v2 published version, discussions are extended