Color-flavor Locked Phase Research Articles

Astrophysical Equation-of-State Constraints on the Color-Superconducting Gap.

We demonstrate that astrophysical constraints on the dense-matter equation of state place an upper bound on the color-superconducting gap in dense matter above the transition from nuclear matter to quark matter. Pairing effects in the color-flavor locked quark matter phase increase the pressure at high density, and if this effect is sufficiently large then the requirements of causality and mechanical stability make it impossible to reach such a pressure in a way that is consistent with what is known at lower densities. The intermediate-density equation of state is inferred by considering extensions of chiral effective field theory to neutron star densities, and conditioning these using current astrophysical observations of neutron star radius, maximum mass, and tidal deformability (PSR J0348+0432, PSR J1624-2230, PSR J0740+6620, GW170817). At baryon number chemical potential μ=2.6 GeV we find a 95% upper limit on the color-flavor locked pairing gap Δ of 457MeV using overly conservative assumptions and 216MeV with more reasonable assumptions. This constraint may be strengthened by future astrophysical measurements as well as by future advances in high-density QCD calculations.

Quark star matter in the color-flavor-locked state with a density-dependent quark mass model

The properties of strange quark matter (SQM) and color-flavor-locked (CFL) quark matter are investigated in quark stars (QSs) at zero temperature case within confined-isospin-density-dependent-mass (CIDDM) model. The mass–radius relation of QSs are also studied by considering newly proposed mass–radius constraints in CFL phase. Our results indicate that we can obtain more stable and stiffer equation of state (EOS) by considering CFL phase within CIDDM model at zero temperature. While the GW190814's secondary component with a mass around 2.6 M ⊙ cannot be QSs within CIDDM model in SQM case, it can be well described as QSs by considering CFL phase within CIDDM model in this work. In particular, we further construct a density-dependent pairing energy gap to connect the EOS of SQM and CFL quark matter with constant pairing energy gap Δ, and the results indicate that by extending the paring energy gap to include density dependence, the mass–radius lines within CIDDM model can satisfy most of the mass–radius region constraints in recent pulsar observations.

Phase transition patterns for coupled complex scalar fields at finite temperature and density

The phase transition patterns displayed by a model of two coupled complex scalar fields are studied at finite temperature and chemical potential. Possible phenomena like symmetry persistence and inverse symmetry breaking at high temperatures are analyzed. The effect of finite density is also considered and studied in combination with the thermal effects. The nonperturbative optimized perturbation theory method is considered and the results contrasted with perturbation theory. Applications of the results obtained are considered in the context of an effective model for condensation of kaons at high densities, which is of importance in the understanding of the color-flavor locked phase of quantum chromodynamics.

Gravitational wave echoes from compact stars in f(R,T) gravity

We have calculated the static and spherically symmetric solutions for compact stars in the $f(\mathcal{R},T)$ gravity metric formalism. To describe the matter of compact stars, we have used the MIT Bag model equation of state (EoS) and the color-flavor-locked (CFL) EoS. Solving the hydrostatic equilibrium equations i.e., the modified TOV equations in $f(\mathcal{R},T)$ gravity, we have obtained different stellar models. The mass-radius profiles for such stars are eventually discussed. The stability of these configurations are then analysed using different parameters. From the obtained solutions of TOV equations for mass and radius, we have checked the compactness of such objects. It is found that similar to the unrealistic EoS, like the stiffer form of the MIT Bag model, under some considerations the realistic interacting quark matter CFL EoS can give stellar structures which are compact enough to possess a photon sphere outside the stellar boundary and hence can echo GWs. The obtained echo frequencies are found to lie in the range of 39-55 kHz. Also we have shown that for different parametrizations of the gravity theory, the structure of stars and also the echo frequencies differ significantly. Moreover, we have constrained the pairing constant value $\beta$ from the perspective of emission of echo frequencies. For the stiffer MIT Bag model $\beta\geq-2.474$ and for the CFL phase with massless quark condition $\beta\geq-0.873$, whereas for the massive case $\beta\geq-0.813$.

Compact stars in the Einstein dark energy model

We investigate the properties of high density compact objects in a vector type theory, inspired by Einstein's 1919 theory of elementary particles, in which Einstein assumed that elementary particles are held together by gravitational as well as electromagnetic type forces. From a modern perspective, Einstein's theory can be interpreted as a vector type model, with the gravitational action constructed as a linear combination of the Ricci scalar, of the trace of the matter energy-momentum tensor, and of a massive self-interacting vector type field. To obtain the properties of stellar models we consider the field equations for a static, spherically symmetric system, and we investigate numerically their solutions for different equations of state of quark and neutron matter, by assuming that the self-interaction potential of the vector field either vanishes or is quadratic in the vector field potential. We consider quark stars described by the MIT bag model equation of state and in the Color Flavor Locked phase, as well as compact stars consisting of a Bose-Einstein Condensate of neutron matter, with neutrons forming Cooper pairs. Constant density stars, representing a generalization of the Interior Schwarzschild solution of general relativity, are also analyzed. Also, we consider the Douchin-Haensel (SLy) equation of state. The numerical solutions are explicitly obtained in both standard general relativity, and the Einstein dark energy model and an in depth comparison between the astrophysical predictions of these two theories are performed. As a general conclusion of our study, we find that for all the considered equations of state a much larger variety of stellar structures can be obtained in the Einstein dark energy model, including classes of stars that are more massive than their general relativistic counterparts.

Chiral non-Abelian vortices and their confinement in three flavor dense QCD

We find chiral non-Abelian vortices having windings only in one of the diquark condensations of left-handed and right-handed quarks in the color-flavor locked phase of dense QCD. They are the minimum vortices carrying half color magnetic fluxes of those of non-Abelian semi-superfluid vortices (color magnetic flux tubes) and 1/6 quantized superfluid circulations of Abelian superfluid vortices. These vortices carry ${\mathbb C}P^2$ orientational moduli in the internal space corresponding to their fluxes. The ${\mathbb C}P^2$ moduli of two chiral non-Abelian vortices with chiralities opposite to each other are energetically favored to be aligned while those of a vortex and anti-vortex to be orthogonal, and then these vortices attract each other. They are attached by chiral domain walls in the presence of the mass and axial anomaly terms explicitly breaking axial and chiral symmetries. We numerically show that two chiral non-Abelian vortices with chiralities opposite to each other are connected by a chiral domain wall, consisting a mesonic bound state which is nothing but a non-Abelian semi-superfluid vortex. We also show that Abelian and non-Abelian axial vortices attached by chiral domain walls are all unstable to decay into a set of chiral non-Abelian vortices. Furthermore, we find that chiral non-Abelian vortices exhibit unique features: one is the so-called topological obstruction implying that unbroken symmetry generators in the bulk are not defined globally around the vortices, and the other is color non-singlet Aharonov-Bohm (AB) phases implying that quarks encircling these vortices can detect the colors of magnetic fluxes of them at infinite distances.

Structural properties of charged compact stars with color-flavor-locked quarks matter

The observations of pulsars with masses close to [Formula: see text] have put strong constraints on the equation-of-state (EoS) of neutron-rich matter at supranuclear densities. Moreover, the exact internal composition of those objects is largely unknown to us. Aiming to reach the [Formula: see text] limit, here we investigate the impact of electric charge on properties of compact stars assuming that the charge distribution is proportional to the mass density. The study is carried out by solving the Tolman–Oppenheimer–Volkoff (TOV) equation for a well-motivated exotic quark matter in the color-flavor-locked (CFL) phase of color superconductivity. The existence of the CFL phase may be the true ground state of hadronic matter with the possibility of the existence of a pure stable quark star (QS). Concerning the equation-of-state, we obtain structural properties of quark stars and compute the mass, the radius as well as the total electric charge of the star. We analyze the dependence of the physical properties of these QSs depending on the free parameters with special attention on mass–radius relation. We also briefly discuss the mass versus central mass density [Formula: see text] relation for stability, the effect of electric charge and compactness. Finally, our results are compared with the recent observations data on mass–radius relationship.

Color-flavor locked compact stars: An exact solution approach

We present an exact solution that could describe compact star composed of color-flavor locked (CFL) phase. Einstein’s field equations were solved through CFL equation of state (EoS) along with a specific form of [Formula: see text] metric potential. Further, to explore a generalized solution we have also included pressure anisotropy. The solution is then analyzed by varying the color superconducting gap [Formula: see text] and its effects on the physical parameters. The stability of the solution through various criteria is also analyzed. To show the physical validity of the obtained solution we have generated the [Formula: see text] curve and fitted three well-known compact stars. This work shows that the anisotropy of the pressure at the interior increases with the color superconducting gap leading to decrease in adiabatic index closer to the critical limit. Further, the fluctuating range of mass due to the density perturbation is larger for lower color superconducting gap leading to more stable configuration.

Tidal deformability of strange stars and the GW170817 event

In this work we consider strange stars formed by quark matter in the color-flavor-locked (CFL) phase of color superconductivity. The CFL phase is described by a Nambu-Jona-Lasinio model with four-fermion vector and diquark interaction channels. The effect of the color superconducting medium on the gluons are incorporated into the model by including the gluon self-energy in the thermodynamic potential. We construct parametrizations of the model by varying the vector coupling $G_V$ and comparing the results to the data on tidal deformability from the GW170817 event, the observational data on maximum masses from massive pulsars such as the MSP J0740+6620, and the mass/radius fits to NICER data for PSR J003+0451. Our results points out to windows for the $G_V$ parameter space of the model, with and without gluon effects included, that are compatible with all these astrophysical constraints, namely, $0.21<G_V/G_S<0.4$, and $0.02<G_V/G_S<0.1$, respectively. We also observe a strong correlation between the tidal deformabilites of the GW170817 event and $G_V$. Our results indicate that strange stars cannot be ruled out in collisions of compact binaries from the structural point of view.

QCD color superconductivity in compact stars: Color-flavor locked quark star candidate for the gravitational-wave signal GW190814

At sufficiently high densities and low temperatures matter is expected to behave as a degenerate Fermi gas of quarks forming Cooper pairs, namely a color superconductor, as was originally suggested by Alford, Rajagopal and Wilczek [Nuclear Physics B 537, 443 (1999)]. The ground state is a superfluid, an electromagnetic insulator that breaks chiral symmetry, called the color-flavor locked phase. If such a phase occurs in the cores of compact stars, the maximum mass may exceed that of hadronic matter. The gravitational-wave signal GW190814 involves a compact object with mass $2.6{\rm M}_\odot$, within the so-called low mass gap. Since it is too heavy to be a neutron star and too light to be a black hole, its nature has not been identified with certainty yet. Here, we show not only that a color-flavor locked quark star with this mass is viable, but also we calculate the range of the model-parameters, namely the superconducting gap $\Delta$ and the bag constant $B$, that satisfies the strict LIGO constraints on the equation of state. We find that a color-flavor locked quark star with mass $2.6{\rm M}_\odot$ satisfies the observational constraints on the equation of state if $\Delta \geq 200{\rm MeV}$ and $B\geq 83{ \rm MeV}/{\rm fm^3}$ for a strange quark mass $m_s=95~{\rm MeV}/c^2$, and attains a radius $(12.7-13.6) {\rm km}$ and central density $(7.5-9.8) 10^{14}{\rm g}/{\rm cm}^3$.

Quark-hadron crossover equations of state for neutron stars: Constraining the chiral invariant mass in a parity doublet model

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 $600\,\mathrm{MeV}\lesssim m_0 \lesssim 900\,\mathrm{MeV}$.

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.

Chiral random matrix theory for colorful quark-antiquark condensates

In QCD at high density, the color-octet quark-antiquark condensate $\langle\overline\psi\gamma_0(\lambda^A)_C (\lambda^A)_F\psi\rangle$ is generally nonzero and dynamically breaks the $\mathrm{SU}(3)_C\times \mathrm{SU}(3)_L\times\mathrm{SU}(3)_R$ symmetry down to the diagonal $\mathrm{SU}(3)_V$. We evaluate this condensate in the mean-field approximation and find that it is of order $\mu\Delta^2\log(\mu/\Delta)$ where $\Delta$ is the BCS gap of quarks. Next we propose a novel non-Hermitian chiral random matrix theory that describes the formation of colorful quark-antiquark condensates. We take the microscopic large-$N$ limit and find that three phases appear depending on the parameter of the model. They are the color-flavor locked phase, the polar phase, and the normal phase. We rigorously derive the effective theory of Nambu-Goldstone modes and determine the quark-mass dependence of the partition function.

Topological order in the color-flavor locked phase of a ( 3+1 )-dimensional U(N) gauge-Higgs system

We study a $(3+1)$-dimensional $U(N)$ gauge theory with $N$-flavor fundamental scalar fields, whose color-flavor locked (CFL) phase has topologically stable non-Abelian vortices. The $U(1)$ charge of the scalar fields must be $Nk+1$ for some integer $k$ in order for them to be in the representation of $U(N)$ gauge group. This theory has a $\mathbb{Z}_{Nk+1}$ one-form symmetry, and it is spontaneously broken in the CFL phase, i.e., the CFL phase is topologically ordered if $k\not=0$. We also find that the world sheet of topologically stable vortices in CFL phase can generate this one-form symmetry.

Effective gauge theories of superfluidity with topological order

We discuss the low-energy dynamics of superfluidity with topological order in (3 + 1) spacetime dimensions. We generalize a topological BF theory by introducing a non-square K matrix, and this generalized BF theory can describe massless Nambu-Goldstone bosons and anyonic statistics between vortices and quasiparticles. We discuss the general structure of discrete and continuous higher-form symmetries in this theory, which can be used to classify quantum phases. We describe how to identify the appearance of topological order in such systems and discuss its relation to a mixed ’t Hooft anomaly between discrete higher-form symmetries. We apply this framework to the color-flavor locked phase of dense QCD, which shows anyonic particle-vortex statistics while no topological order appears. An explicit example of superfluidity with topological order is discussed.

Quark-Hadron Continuity beyond the Ginzburg-Landau Paradigm.

Quark-hadron continuity is a scenario in which hadronic matter is continuously connected to a color superconductor without phase transitions as the baryon chemical potential increases. This scenario is based on Landau's classification of phases, since they have the same symmetry breaking pattern. We address the question of whether this continuity is true as quantum phases of matter, which requires treatment beyond the Ginzburg-Landau description. To examine the topological nature of a color superconductor, we derive a dual effective theory for U(1) Nambu-Goldstone (NG) bosons and vortices of the color-flavor locked phase and discuss the fate of emergent higher-form symmetries. The theory has the form of a topological BF theory coupled to NG bosons, and fractional statistics of test quarks and vortices arise as a result of an emergent Z_{3} two-form symmetry. We find that this symmetry cannot be spontaneously broken, indicating that quark-hadron continuity is still a consistent scenario.

Quark-hadron continuity under rotation: Vortex continuity or boojum?

Quark-hadron continuity was proposed as crossover between hadronic matter and quark matter without a phase transition, based on the matching of the symmetry and excitations in both phases. In the limit of a light strange-quark mass, it connects hyperon matter and the color-flavor-locked (CFL) phase exhibiting color superconductivity. Recently, it was proposed that this conjecture could be generalized in the presence of superfluid vortices penetrating both phases (arXiv:1803.05115 [hep-ph]), and it was suggested that one hadronic superfluid vortex in hyperon matter could be connected to one non-Abelian vortex (color magnetic flux tube) in the CFL phase. Here, we argue that their proposal is consistent only at large distances; instead, we show that three hadronic superfluid vortices must combine with three non-Abelian vortices with different colors with the total color magnetic fluxes canceled out, where the junction is called a colorful boojum. We rigorously prove this in both a macroscopic theory based on the Ginzburg-Landau description in which symmetry and excitations match (including vortex cores), and a microscopic theory in which the Aharonov-Bohm phases of quarks around vortices match.

Continuity of vortices from the hadronic to the color-flavor locked phase in dense matter

We study how vortices in dense superfluid hadronic matter can connect to vortices in superfluid quark matter, as in rotating neutron stars, focusing on the extent to which quark-hadron continuity can be maintained. As we show, a singly quantized vortex in three-flavor symmetric hadronic matter can connect smoothly to a singly quantized non-Abelian vortex in three-flavor symmetric quark matter in the color-flavor locked (CFL) phase, without the necessity for boojums appearing at the transition.

Non-Abelian vortex in lattice gauge theory

We perform the Monte Carlo study of the SU(3) non-Abelian Higgs model. We discuss phase structure and non-Abelian vortices by gauge invariant operators. External magnetic fields induce non-Abelian vortices in the color-flavor locked phase. The spatial distribution of non-Abelian vortices suggests the repulsive vortex-vortex interaction.

Momentum space topology of QCD

We discuss the possibility to consider quark matter as the topological material. We consider hadronic phase (HP), the quark–gluon plasma phase (QGP), and the hypothetical color–flavor locking (CFL) phase. In those phases we identify the relevant topological invariants in momentum space. The formalism is developed, which relates those invariants and massless fermions that reside on vortices and at the interphases. This formalism is illustrated by the example of vortices in the CFL phase.