Two-flavor Color Superconductivity Research Articles

Two-flavor Color Superconducting Quark Stars May Not Exist

Large uncertainties in the determinations of the equation of state of dense stellar matter allow for the intriguing possibility that the bulk quark matter in beta equilibrium might be the true ground state of the matter at zero pressure. Also, quarks will form Cooper pairs very readily since the dominant interaction between quarks is attractive in some channels. As a result, quark matter will generically exhibit color superconductivity, with the favored pairing pattern at intermediately high densities being two-flavor pairing. In the light of several possible candidates for such self-bound quark stars, including the very low-mass central compact object in supernova remnant HESS J1731-347 reported recently, we carry out a one-field theoretic model, the Nambu–Jona–Lasinio model, to investigate the stability of the beta-stable two-flavor color superconducting (2SC) phase of quark matter, but find no physically allowed parameter space for the existence of 2SC quark stars.

Chiral asteroseismology: seismic oscillations caused by chiral transport in neutron stars and supernovae

We study the novel asteroseismology of the chiral magnetic wave (CMW) of the quark number density in relativistic quark matter inside neutron stars and core-collapse supernovae and the chiral vortical wave (CVW) of the neutrino number density in relativistic neutrino matter at the core of supernovae.We call the oscillation modes for these chiral waves the chiral magnetic mode (CM-mode) and chiral vortical mode (CV-mode), respectively.We derive the dispersion relations of these new modes in the presence of the chirality flipping due to the finite quark mass and diffusion. We then estimate the possible frequencies of these modes and amplitudes of the resulting gravitational waves. In particular, since the CM-mode can exist only in quark matter with nearly gapless quarks (such as the two-flavor color superconductivity) for a sufficiently strong magnetic field, corresponding gravitational waves provide a new possible probe for such quark matter and the magnetic field in neutron stars.

Mesons and diquarks in a color superconducting regime

The behavior of the mesons and diquarks is studied at finite temperatures, chemical potentials and densities, notably when the color superconductivity is taken into account. The Nambu and Jona-Lasinio model complemented by a Polyakov loop (PNJL description) has been adapted in order to model them in this regime. This paper focuses on the scalar and pseudoscalar mesons and diquarks, in a three-flavor and three-color description, with the isospin symmetry and at zero strange density. An objective of this work is to underline the modifications carried out by the color superconducting regime on the used equations and on the obtained results. It has been observed that the two-flavor color-superconducting (2SC) phase affects the masses and the coupling constants of the mesons and diquarks in a non-negligible way. This observation is particularly true at high densities and low temperatures for the pions, [Formula: see text] and the diquarks [ud] whose color is rg. This reveals that the inclusion of the color superconductivity in the modeling is relevant to describe the mesons and diquarks near the first-order chiral phase transition.

Phenomenological QCD equations of state for neutron star dynamics: Nuclear-2SC continuity and evolving effective couplings

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 $\sim 0.1M_\odot$.

Competition of inhomogeneous chiral phases and two-flavor color superconductivity in the NJL model

We study the phase structure of the two-flavor Nambu--Jona-Lasinio (NJL) model in the chiral limit, extending a previous study of the competition of an inhomogeneous chiral phase and a two-flavor color-superconducting (2SC) phase [1, 2]. There, an analytic expression for the dispersion relations for quasiparticle excitations in the presence of both a particular inhomogeneous chiral condensate, the so-called chiral density wave (CDW), and a homogeneous 2SC condensate was found. In this work we show how to determine the dispersion relations for arbitrary modulations of the chiral condensate in the presence of a homogeneous 2SC condensate, if the dispersion relations in the absence of color superconductivity are known. In our calculations, we employ two different Ans\"atze for the inhomogeneous chiral condensate, the CDW as well as the real-kink crystal (RKC). Depending on the value of the diquark coupling we find a region of the phase diagram where the inhomogeneous chiral and the 2SC condensates coexist, confirming results of Refs. [1, 2]. Decreasing the diquark coupling favors the inhomogeneous phase over the coexistence phase. On the other hand, increasing the diquark coupling leads to a larger 2SC phase, while the inhomogeneous chiral and the coexistence phases become smaller. In agreement with previous studies the RKC Ansatz is energetically preferred over the CDW Ansatz. Both Ans\"atze lead to a qualitatively similar phase diagram, however the coexistence phase is smaller for the RKC Ansatz.

Continuity from neutron matter to color-superconducting quark matter with 3P2 superfluidity

I clarify how the concept of quark-hadron continuity, which was previously considered in the context of the asymptotic color-flavor locked phase with ideal three-flavor symmetry, is applied to two-flavor matter. Our observation is that neutron star matter can continuously be connected to two-flavor color-superconducting (2SC) phase with an additional condensate of d-quarks in the 3P2 channel. I discuss here two aspects of this novel phase. First, I introduce the notion of continuity based on the patterns of symmetry breaking and the corresponding order parameters, and then explain qualitatively the physics mechanism of d-quark 3P2 pairing in analogy to nuclear physics. Our finding serves as the theoretical underpinning for the phenomenological construction of the equation of state in the neutron star environment.

Emergent QCD Kondo effect in two-flavor color superconducting phase

We show that effective coupling strengths between ungapped and gapped quarks in the two-flavor color superconducting (2SC) phase are renormalized by logarithmic quantum corrections. We obtain a set of coupled renormalization-group (RG) equations for two distinct effective coupling strengths arising from gluon exchanges carrying different color charges. The diagram of RG flow suggests that both of the coupling strengths evolve into a strong-coupling regime as we decrease the energy scale toward the Fermi surface. This is a characteristic behavior observed in the Kondo effect, which has been known to occur in the presence of impurity scatterings via non-Abelian interactions. We propose a novel Kondo effect emerging without doped impurities, but with the gapped quasiexcitations and the residual SU(2) color subgroup intrinsic in the 2SC phase, which we call the 2SC Kondo effect.

Color superconductivity from the chiral quark–meson model

We study the two-flavor color superconductivity of low-temperature quark matter in the vicinity of chiral phase transition in the quark–meson model where the interactions between quarks are generated by pion and sigma exchanges. Starting from the Nambu–Gorkov propagator in real-time formulation we obtain finite temperature (real axis) Eliashberg-type equations for the quark self-energies (gap functions) in terms of the in-medium spectral function of mesons. Exact numerical solutions of the coupled nonlinear integral equations for the real and imaginary parts of the gap function are obtained in the zero temperature limit using a model input spectral function. We find that these components of the gap display a complicated structure with the real part being strongly suppressed above 2Δ0, where Δ0 is its on-shell value. We find Δ0≃40MeV close to the chiral phase transition.

Compact Stars with Sequential QCD Phase Transitions.

Compact stars may contain quark matter in their interiors at densities exceeding several times the nuclear saturation density. We explore models of such compact stars where there are two first-order phase transitions: the first from nuclear matter to a quark-matter phase, followed at a higher density by another first-order transition to a different quark-matter phase [e.g., from the two-flavor color-superconducting (2SC) to the color-flavor-locked (CFL) phase]. We show that this can give rise to two separate branches of hybrid stars, separated from each other and from the nuclear branch by instability regions, and, therefore, to a new family of compact stars, denser than the ordinary hybrid stars. In a range of parameters, one may obtain twin hybrid stars (hybrid stars with the same masses but different radii) and even triplets where three stars, with inner cores of nuclear matter, 2SC matter, and CFL matter, respectively, all have the same mass but different radii.

Effect of temperature and magnetic field on two-flavor superconducting quark matter

We investigate the effect of turning on temperature for the charge neutral phase of two-flavor color superconducting (2SC) dense quark matter in presence of constant external magnetic field. Within the Nambu-Jona-Lasinio model, by tuning the diquark coupling strength, we study the interdependent evolution of the quark Bardeen-Cooper-Schrieffer gap and dynamical mass as functions of temperature and magnetic field. We find that magnetic field $B \gtrsim 0.02$ GeV$^2$ ($10^{18}$ G) leads to anomalous temperature behavior of the gap in the gapless 2SC phase (moderately strong coupling), reminiscent of previous results in the literature found in the limit of weak coupling without magnetic field. The 2SC gap in the strong coupling regime is abruptly quenched at ultrahigh magnetic field due to the mismatched Fermi surfaces of up and down quarks imposed by charge neutrality and oscillation of the gap due to Landau level quantization. The dynamical quark mass also displays strong oscillation and magnetic catalysis at high magnetic field, although the latter effect is tempered by nonzero temperature. We discuss the implications for newly born compact stars with superconducting quark cores.

Neutrality of a magnetized two-flavor quark superconductor

We investigate the effect of electric and color charge neutrality on the two-flavor color superconducting (2SC) phase of cold and dense quark matter in presence of constant external magnetic fields and at moderate baryon densities. Within the framework of the Nambu-Jona-Lasinio (NJL) model, we study the inter-dependent evolution of the quark's BCS gap and constituent mass with increasing density and magnetic field. While confirming previous results derived for the highly magnetized 2SC phase with color neutrality alone, we obtain new results as a consequence of imposing charge neutrality. In the charge neutral gapless 2SC phase (g2SC), a large magnetic field drives the color superconducting phase transition to a crossover, while the chiral phase transition is first order. At larger diquark-to-scalar coupling ratio $G_D/G_S$, where the 2SC phase is preferred, we see hints of the Clogston-Chandrasekhar limit at a very large value of the magnetic field ($B\sim 10^{19}$G), but this limit is strongly affected by Shubnikov de Haas-van Alphen oscillations of the gap, indicating the transition to a domain-like state.

Phase diagram of hot magnetized two-flavor color-superconducting quark matter

A two-flavor color superconducting (2SC) Nambu--Jona-Lasinio (NJL) model is introduced at finite temperature T, chemical potential mu and in the presence of a constant magnetic field eB. The effect of (T,mu,eB) on the formation of chiral and color symmetry breaking condensates is studied. The complete phase portrait of the model in T-mu, mu-eB, and T-eB phase spaces for various fixed eB, T, and mu is explored. A threshold magnetic field eB_t~ 0.5 GeV^2 is found above which the dynamics of the system is solely dominated by the lowest Landau level (LLL) and the effects of T and mu are partly compensated by eB.

Role of two-flavor color superconductor pairing in a three-flavor Nambu–Jona-Lasinio model with axial anomaly

The phase diagram of strongly interacting matter is studied within a three-flavor Nambu--Jona-Lasinio model, which contains the coupling between chiral and diquark condensates through the axial anomaly. Our results show that it is essential to include the 2SC phase in the analysis. While this is expected for realistic strange quark masses, we find that even for equal up, down, and strange bare quark masses, 2SC pairing can be favored due to spontaneous flavor-symmetry breaking by the axial anomaly. This can lead to a rich phase structure, including BCS- and BEC-like 2SC and CFL phases and new endpoints. On the other hand, the low-temperature critical endpoint, which was found earlier in the same model without 2SC pairing, is almost removed from the phase diagram and cannot be reached from the low-density chirally broken phase without crossing a preceding first-order phase boundary. For physical quark masses no additional critical endpoint is found.

Pulsar glitch recovery and the superfluidity coefficients of bulk nuclear matter

A two-component hydrodynamic model is constructed of the global superfluid flow induced by two-component Ekman pumping during the recovery stage of a glitch. The model successfully accounts for the quasi-exponential recovery observed in pulsars like Vela and the "overshoot" observed in pulsars like the Crab. By fitting the model to high-resolution timing data, three important constitutive coefficients in bulk nuclear matter can be extracted: the shear viscosity, the mutual friction parameter, and the charged fluid fraction. The fitted coefficients for the Crab and Vela are compared with theoretical predictions for several equations of state, including the color-flavor locked and two-flavor color superconductor phases of quark matter.

Color-magnetic flux tubes in quark matter cores of neutron stars

We argue that if color-superconducting quark matter exists in the core of a neutron star, it may contain a high density of flux tubes, carrying flux that is mostly color magnetic, with a small admixture of ordinary magnetic flux. We focus on the two-flavor color-superconducting (‘2SC’) phase, and assume that the flux tubes are energetically stable, although this has not yet been demonstrated. The density of flux tubes depends on the nature of the transition to the color-superconducting phase, and could be within an order of magnitude of the density of magnetic flux tubes that would be found if the core were superconducting nuclear matter. We calculate the cross-section for Aharonov–Bohm scattering of gapless fermions off the flux tubes, and the associated collision time and frictional force on a moving flux tube. We discuss the other forces on the flux tube, and find that if we take into account only the forces that arise within the 2SC core region, then the timescale for expulsion of the color flux tubes from the 2SC core is of order 1010 years.

Sequential deconfinement of quark flavors in neutron stars

A scenario is suggested in which the three light quark flavors are sequentially deconfined under increasing pressure in cold asymmetric nuclear matter as found, for example, in neutron stars. The basis for this analysis is a chiral quark matter model of Nambu--Jona-Lasinio (NJL) type with diquark pairing in the spin-1 single-flavor, spin-0 two-flavor, and three-flavor channels. Nucleon dissociation sets in at about the saturation density, ${n}_{0}$, when the down-quark Fermi sea is populated ($d$-quark drip line) because of the flavor asymmetry induced by $\ensuremath{\beta}$ equilibrium and charge neutrality. At about $3{n}_{0}$, $u$-quarks appear and a two-flavor color superconducting (2SC) phase is formed. The $s$-quark Fermi sea is populated only at still higher baryon density, when the quark chemical potential is of the order of the dynamically generated strange quark mass. Two different hybrid equations of state (EOSs) are constructed using the Dirac-Brueckner Hartree-Fock (DBHF) approach and the EOS of Shen et al. [H. Shen, H. Toki, K. Oyamatsu, and K. Sumiyoshi, Nucl. Phys. A637, 435 (1998)] in the nuclear matter sector. The corresponding hybrid star sequences have maximum masses of 2.1 and 2.0 ${\mathrm{M}}_{\ensuremath{\bigodot}}$, respectively. Two- and three-flavor quark-matter phases exist only in gravitationally unstable hybrid star solutions in the DBHF case, whereas the Shen-based EOSs produce stable configurations with a 2SC phase component in the core of massive stars. Nucleon dissociation via $d$-quark drip could act as a deep crustal heating process, which apparently is required to explain superbursts and cooling of x-ray transients.

BCS-BEC crossover in dense relativistic matter: Collective excitations

We study the relativistic BCS-BEC crossover within a class of Nambu-Jona-Lasinio type models, including arbitrary Lorentz-scalar pairing channels. Using the mean-field approximation we investigate spectral properties of the collective bosonic excitations in the superfluid phase, with particular attention to the Nambu-Goldstone bosons of the broken symmetry. This is a first step towards a systematic improvement of the mean-field approximation by including the fluctuation effects. The general results are illustrated on pairing in dense two-flavor quark matter--the two-flavor color superconductor.

Chromomagnetic instability and induced magnetic field in neutral two-flavor color superconductivity

We find that the chromomagnetic instability existing in neutral two-flavor color superconductivity at moderate densities is removed by the formation of an inhomogeneous condensate of charged gluons and the corresponding induction of a magnetic field. It is shown that this inhomogeneous ground state is energetically favored over a homogeneous one. The spontaneous induction of a magnetic field in a color superconductor at moderate densities can be of interest for the astrophysics of compact stellar objects exhibiting strong magnetic fields as magnetars.

The effect of lattice-qcd-based gluon propagator on coherence length of quark Cooper pairs in two-flavor color superconductor

The effect of lattice-qcd-based gluon propagator on coherence length of quark Cooper pairs in two-flavor color superconductor

Gluonic phase versus LOFF phase in two-flavor quark matter

Abstract We study the gluonic phase in a two-flavor color superconductor as a function of the ratio of the gap over the chemical potential mismatch, Δ / δ μ . We find that the gluonic phase resolves the chromomagnetic instability encountered in a two-flavor color superconductor for Δ / δ μ 2 . We also calculate approximately the free energies of the gluonic phase and the single plane-wave LOFF phase and show that the former is favored over the latter for a wide range of coupling strengths.