CFL Phase Research Articles

Detecting superfluid transition in the pulsar core

ABSTRACT It is believed that the core of a neutron star can be host to various novel phases of matter, from nucleon superfluid phase to exotic high baryon density quantum chromodynamics (QCD) phases. Different observational signals for such phase transitions have been discussed in the literature. Here, we point out a unique phenomenon associated with phase transition to a superfluid phase, which may be the nucleon superfluid phase or a phase like the colour-flavour locked phase, allowing for superfluid vortices. In any superfluid phase transition, a random network of vortices forms via the so-called Kibble–Zurek mechanism, which eventually mostly decays away, finally leaving primarily vortices arising from the initial angular momentum of the core. This transient, random vortex network can have a non-zero net angular momentum for the superfluid component, which will generally be oriented in an arbitrary direction. This is in contrast to the final vortices, which arise from initial rotation and hence have the initial angular momentum of the neutron star. The angular momentum of the random vortex network is balanced by an equal and opposite angular momentum in the normal fluid due to the conservation of angular momentum, thereby imparting an arbitrarily oriented angular momentum component to the outer shell of the neutron star. This will affect the pulse timing and pulse profile of a pulsar. These changes in the pulses will decay away in a characteristic manner that this as the random vortex network decays, obeying specific scaling laws leading to universal features for the detection of superfluid transitions occurring in a pulsar core.

Impact of energy–momentum conservation violation on the configuration of compact stars and their GW echoes

This work investigates the impacts of energy–momentum conservation violation on the configuration of strange stars constraint with gravitational wave (GW) event GW190814 as well as eight recent observations of compact objects. The GW echoes from these interesting classes of compact objects are also calculated. To describe the matter of strange stars, we have used two different equations of state (EoSs): first an ad-hoc exotic EoS, the stiffer MIT Bag model and next realistic CFL phase of quark matter EoS. We choose Rastall gravity as a simple model with energy–momentum conservation violation with a set of model parameter values. Our results show that this gravity theory permits stable solutions of strange stars and the resulting structures can foster GW echoes. We illustrate the implication of the gravity theory and found that the negative values of the Rastall parameter result in more compact stellar configurations and lower GW echo frequency. With an increase in the Rastall parameter, both the compactness of the stellar configurations and echo time decrease. It is worth mentioning here that with the chosen set of some probable strange star candidates from observational data and also in light of GW 190814, we have evaluated the radii of stellar models. Also, the GW echo frequencies associated with strange stars are found to be in the range of ≈ 9–27 kHz for both cases. From this work, it is also inferred that the assumption regarding the equivalence of Rastall’s theory to Einstein’s theory is refuted as we have noticed many deviations in the physical properties of the considered compact stars.

New mass limit of a strange star admitting a colour flavor locked equation of state

A class of strange stars is analysed in the present article in hydrostatic equilibrium, whose state is defined by a CFL phase equation of state. We have compared our results with those obtained from the MIT equation of state for strange quark matter, which is regarded as free particles. We have noted that if we consider quarks to form a cooper pair and if their description is made by the CFL equation of state, the maximum mass of strange star reaches a value as high as 3.61 M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\odot }$$\\end{document}. This value is well above the value of 2.03 M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\odot }$$\\end{document} obtained by using the MIT bag equation of state for massless free quarks. Both the maximum masses are determined by solving the TOV equation for different values of the strange quark mass ms\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$m_s$$\\end{document}. Thus, the inclusion of the possibility of quark pair formation in the theory permits us to accommodate a wider class of compact objects such as PSR J1614-2230, PSR J0740+6620, PSR J0952-0607 etc. and the mass of the companion star in the GW190814 event in our model. The consideration of such a high value of mass is hardly theoretically obtainable from normal strange star models in general relativity even with a fast rotation effect. The object PSR J0952-0607 is found to be the fastest and heaviest pulsar in the disk of Milky Way Galaxy, having a mass of 2.35 M⊙\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_{\\odot }$$\\end{document}, which may be predicted in our model, as observational evidence supports the existence of strange quark matter in its composition.

Dyson-Schwinger approach to color superconductivity at finite temperature and density

We investigate the phases of dense QCD matter at finite temperature with Dyson-Schwinger equations for the quark propagator for N_f=2+1 flavors. For the gluon propagator we take a fit to quenched lattice data and add quark-loop effects perturbatively in a hard-thermal-loop--hard-dense-loop approximation. We consider 2SC and CFL-like pairing with chiral up and down quarks and massive strange quarks and present results for the condensates and the phase diagram. We find a dominant CFL phase at chemical potentials larger than 500-600 MeV. At lower values of the chemical potential we find a 2SC phase, which also exists in a small band at higher temperatures for larger chemical potentials. With values of 20-30 MeV, the critical temperatures to the normal phase turn out to be quite small.

Maximum mass of neutron stars and strange neutron-star cores

Recent measurement of mass of PSR J1614-2230 rules out most of existing models of equation of state (EOS) of dense matter with high-density softening due to hyperonization, based on the recent hyperon-nucleon and hyperon-hyperon interactions, leading to a "hyperon puzzle". We study a specific solution of "hyperon puzzle", consisting in replacing a too soft hyperon core by a sufficiently stiff quark core. We construct an analytic approximation fitting very well modern EOSs of 2SC and CFL color superconducting phases of quark matter. This allows us for simulating continua of sequences of first-order phase transitions from hadronic matter to the 2SC, and then to the CFL state of color superconducting quark matter. We obtain constraints in the parameter space of the EOS of superconducting quark cores, resulting from M_max> 2 M_sol. We also derive constraints that would result from significantly higher measured masses. For 2.4 M_sol required stiffness of the CFL quark core should have been close to the causality limit, the density jump at the phase transition being very small. Condition M_max > 2 M_sol puts strong constraints on the EOSs of the 2SC and CFL phases of quark matter. Density jumps at the phase transitions have to be sufficiently small and sound speeds in quark matter - sufficiently large. A strict condition of thermodynamic stability of quark phase results in the maximum mass of hybrid stars similar to that of purely baryon stars. Therefore, to get M_max>2 M_sol for stable hybrid stars, both sufficiently strong additional hyperon repulsion at high density baryon matter and a sufficiently stiff EOS of quark matter would be needed. However, it is likely that the high density instability of quark matter (reconfinement) indicates actually the inadequacy of the point-particle model of baryons in dense matter at very high densities.

BCS Theory of Hadronic Matter at High Densities

The equilibrium between the so-called 2SC and CFL phases of strange quark matter at high densities is investigated in the framework of a simple schematic model of the NJL type. Equal densities are assumed for quarks u,d and s. The 2SC phase is here described by a color-flavor symmetric state, in which the quark numbers are independent of the color-flavor combination. In the CFL phase the quark numbers depend on the color-flavor combination, that is, the number of quarks associated with the color-flavor combinations ur,dg,sb is different from the number of quarks associated with the color flavor combinations ug,ub,dr,db,sr,sg. We find that the 2SC phase is stable for a chemical potential μ below μ c = 0.505 GeV, while the CFL phase is stable above, the equilibrium pressure being P c = 0.003 GeV4. We have used a 3-momentum regularizing cutoff Λ = 0.8 GeV, which is somewhat larger than is usual in NJL type models. This should be adequate if the relevant chemical potential does not exceed 0.6 GeV.

Constraining phases of quark matter with studies ofr-mode damping in compact stars

With the goal of constraining the phase of matter at high density, we study $r$-mode damping in the color-flavor-locked phase with kaon condensation (CFL-K0) and contrast it with the CFL phase. The mode frequencies in these phases are found to differ very slightly. While the bulk viscosity in either phase is only effective at damping the $r$-mode at temperatures $T\ensuremath{\gtrsim}{10}^{11}$ K, the shear viscosity in the CFL-K0 phase is the only effective damping agent all the way down to temperatures $T\ensuremath{\gtrsim}{10}^{8}$ K characteristic of cooling neutron stars. However, it cannot keep the star from becoming unstable to gravitational wave emission for rotation frequencies $\ensuremath{\nu}\ensuremath{\approx}56$--11 Hz at $T\ensuremath{\approx}{10}^{8}$--${10}^{9}$ K. Stars composed almost entirely of CFL or CFL-K0 matter are ruled out by observation of rapidly rotating neutron stars, indicating that dissipation at the quark-hadron interface or nuclear crust interface must play a key role in damping the instability.

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.

Nambu–Jona-Lasinio model of homogeneous neutral quark matter: Pseudoscalar diquark condensates revisited

We use a Nambu-Jona Lasinio type model to investigate the phase diagram of dense quark matter under neutron star conditions in mean field approximation. The model contains selfconsistently determined quark masses and allows for diquark condensation in the scalar as well as in the pseudoscalar channel. The latter gives rise to the possibility of K^0 condensation in the CFL phase. In agreement with earlier studies we find that this CFLK^0 phase covers large regions of the phase diagram and that the predominant part of this phase is fully gapped. We show, however, that there exists a region at very low temperatures where the CFLK^0 solutions become gapless, possibly indicating an instability towards anisotropic or inhomogeneous phases. The physical significance of solutions with pseudoscalar diquark condensates in the 2SC phase is discussed as well.

A diagrammatic derivation of the meson effective masses in the neutral color-flavor-locked phase of quantum chromodynamics

We offer a diagrammatic derivation of the effective masses of the axial flavor excitations in the electrical and color neutral CFL phase of QCD. In particular we concentrate on the excitations with the quantum numbers of the kaons: we show how their effective chemical potentials, responsible of their Bose-Einstein condensation and found previously on the basis of pure symmetry arguments, arise at the microscopic level by loop effects. We perform also the numerical evaluation of the relevant loops in the whole CFL regime Ms2/2μΔ ≤ 1, showing the existence of the enhancement of the kaon condensation with respect to the lowest order result. Finally we discuss the role of color neutrality in the microscopic calculation.

The color flavor locked phase in the chromodielectric model and quark stars

Recent results obtained in the Chromodielectric Model (CDM) have shown that strange quark matter at very high densities may appear in two phases, namely a chiral broken and a chiral symmetric phase, which may not be absolutely stable. In the chiral symmetric phase, the abundance of the quarks flavors u, d and s is the same and there are no electrons. In this paper we study an extended version of the Chromodielectric model (CDM) with a BCS quark pairing implemented, and analyze the superconducting color flavor locked phase. We show that the inclusion in the free energy density of a negative term of the diquark condensate guarantees the stability of quark matter. We also analyze the phase transition between matter described by different equations of state and only find a first order transition, at a very low pressure, from the CFL phase to the unpaired strange quark matter, which opens the possibility for a quark-hadron phase transition. Our study has implications in astrophysics, in particular regarding the formation and the structure of compact quark stars. We explicitly show that CFL stars can be absolutely stable and more compact than strange stars.

Crystallography of three-flavor quark matter

We analyze and compare candidate crystal structures for the crystalline color superconducting phase that may arise in cold, dense but not asymptotically dense, three-flavor quark matter. We determine the gap parameter $\ensuremath{\Delta}$ and free energy $\ensuremath{\Omega}(\ensuremath{\Delta})$ for many possible crystal structures within a Ginzburg-Landau approximation, evaluating $\ensuremath{\Omega}(\ensuremath{\Delta})$ to order ${\ensuremath{\Delta}}^{6}$. In contrast to the two-flavor case, we find a positive ${\ensuremath{\Delta}}^{6}$ term and hence an $\ensuremath{\Omega}(\ensuremath{\Delta})$ that is bounded from below for all the structures that we analyze. This means that we are able to evaluate $\ensuremath{\Delta}$ and $\ensuremath{\Omega}$ as a function of the splitting between Fermi surfaces for all the structures we consider. We find two structures with particularly robust values of $\ensuremath{\Delta}$ and the condensation energy, within a factor of 2 of those for the CFL phase which is known to characterize QCD at asymptotically large densities. The robustness of these phases results in their being favored over wide ranges of density. However, it also implies that the Ginzburg-Landau approximation is not quantitatively reliable. We develop qualitative insights into what makes a crystal structure favorable, and use these to winnow the possibilities. The two structures that we find to be most favorable are both built from condensates with face-centered cubic symmetry: in one case, the $⟨ud⟩$ and $⟨us⟩$ condensates are separately face-centered cubic; in the other case $⟨ud⟩$ and $⟨us⟩$ combined make up a face-centered cube.

The crystallography of strange quark matter

Cold three-flavour quark matter at large (but not asymptotically large) densities may exist as a crystalline colour superconductor. We explore this possibility by calculating the gap parameter Δ and free energy Ω(Δ) for possible crystal structures within a Ginzburg–Landau approximation, evaluating Ω(Δ) to order Δ6. We develop a qualitative understanding of what makes a crystal structure stable and find two structures with particularly large values of Δ and the condensation energy, within a factor of 2 of those for the CFL phase known to characterize QCD at asymptotically large densities. The robustness of these phases results in their being favoured over wide ranges of density and though it also implies that the Ginzburg–Landau approximation is not quantitatively reliable, previous work suggests that it can be trusted for qualitative comparisons between crystal structures. We close with a look ahead at the calculations that remain to be done in order to make contact with observed pulsar glitches and neutron star cooling.

Study of Strange Quark Mass in CFL Phase

In this paper we introduce bilocal fields in the globalcolor symmetry model and consider color and electricalneutrality conditions simultaneously to study the effect of strangequark mass MS for the momentum-dependent condensate ofcolor-flavor locked phase. Consequently we find thatthere will be a quantum phase transition occurring.

Fate of the inert three-flavor, spin-zero, color-superconducting phases

I investigate some of the inert phases in three-flavor, spin-zero color-superconducting quark matter: the CFL phase (the analogue of the B phase in superfluid $^3\rm He$), the A and A* phases, and the 2SC and sSC phases. I compute the pressure of these phases with and without the neutrality condition. It is shown that the 2SC phase is identical to the A* phase up to a color rotation. The CFL phase is the energetically favored phase except for a small region of intermediate densities where the 2SC/A* phase is favored.

Color superconductivity in the strong-coupling regime of Landau gauge QCD

The chirally unbroken and the superconducting 2SC and CFL phases are investigated in the chiral limit within a Dyson-Schwinger approach for the quark propagator in QCD. The hierarchy of Green's functions is truncated such that at vanishing density known results for the vacuum and at asymptotically high densities the corresponding weak-coupling expressions are recovered. The anomalous dimensions of the gap functions are analytically calculated. Based on the quark propagator the phase structure is studied, and results for the gap functions, occupation numbers, coherence lengths and pressure differences are given and compared with the corresponding expressions in the weak-coupling regime. At moderate chemical potentials the quasiparticle pairing gaps are several times larger than the extrapolated weak-coupling results.

Review Article

Investigations of deconfined quark matter within NJL-type models are reviewed, focusing on the regime of low temperatures and “moderate” densities, which is not accessible by perturbative QCD. Central issue is the interplay between chiral symmetry restoration and the formation of color superconducting phases. In order to lay a solid ground for this analysis, we begin with a rather detailed discussion of two- and three-flavor NJL models and their phase structure, neglecting the possibility of diquark pairing in a first step. An important aspect of this part is a comparison with the MIT bag model. The NJL model is also applied to investigate the possibility of absolutely stable strange quark matter. In the next step the formalism is extended to include diquark condensates. We discuss the role and mutual influence of several conventional and less conventional quark–antiquark and diquark condensates. As a particularly interesting example, we analyze a spin-1 diquark condensate as a possible pairing channel for those quarks which are left over from the standard spin-0 condensate. For three-flavor systems, we find that a self-consistent calculation of the strange quark mass, together with the diquark condensates, is crucial for a realistic description of the 2SC–CFL phase transition. We also study the effect of neutrality constraints which are of relevance for compact stars. Both, homogeneous and mixed, neutral phases are constructed. Although neutrality constraints generally tend to disfavor the 2SC phase we find that this phase is again stabilized by the large values of the dynamical strange quark mass which follow from the self-consistent treatment. Finally, we combine our solutions with existing hadronic equations of state to investigate the existence of quark matter cores in neutron stars.

Resummed one-loop gluonic contributions to the color superconducting color charge density vanish

It is shown that gluonic corrections to the tadpole diagrams vanish in the 2SC and CFL phases at the order where one might have expected NLO corrections. This implies that the gluonic part of the color charge density is negligible at the order of our computation. This statement remains true after inclusion of the gluon vertex correction and contributions from Nambu-Goldstone bosons.

Meissner masses in the gCFL phase of QCD

We calculate the Meissner masses of gluons in neutral three-flavor color superconducting matter for finite strange quark mass. In the CFL phase the Meissner masses are slowly varying function of the strange quark mass. For large strange quark mass, in the so-called gCFL phase, the Meissner masses of gluons with colors a=1,2 become imaginary, indicating an instability.

Supersymmetric Color Superconductivity

Recent interest in novel phases in high density QCD motivates the study of high density supersymmetric QCD (SQCD), where powerful exact results for supersymmetric gauge theories can be brought to bear in the strongly coupled regime. We begin by describing how a chemical potential can be incorporated into a supersymmetric theory as a spurion vector superfield. We then study supersymmetric SU(N_c) gauge theories with N_f flavors of quarks in the presence of a baryon chemical potential mu, and describe the global symmetry breaking patterns at low energy. Our analysis requires mu < Lambda and is thus complementary to the variational approach that has been successful for mu << Lambda. We find that for N_f < N_c a modified U(1)_B symmetry is preserved, analogous to the non-supersymmetric 2SC phase, whereas for N_f=N_c there is a critical chemical potential above which the U(1)_B is broken, as it is in the non-supersymmetric CFL phase. We further analyze the cases with N_c+1 <= N_f < 3/2 N_c and find that baryon number is broken dynamically for mu > mu_c. We also give a qualitative description of the phases in the `conformal window', 3/2 N_c < N_f < 3N_c, at finite density.