The generalized non-extensive statistics proposed by Tsallis have been successfully utilized in many systems where long range interactions are present. For high density quark matter an attractive long range interaction arising from single gluon exchange suggests the formation of a diquark condensate. We study the effects on this color superconducting phase for two quark flavors due to a change to Tsallis statistics. By numerically solving the gap equation we obtain a generalization of the universality condition, $\frac{2ϕ_{0}}{T_{C}}\approx 3.52$ and determine the temperature dependence of the gap. For the Tsallis parameter $q\approx 1$ the specific heat is exponential becoming more linear as q increases. This suggests that for larger values of q s-wave color superconductors behave like high $T_c$ superconductors rather than weak superconductors.
We present the dispersion relations for quasiparticle excitations about the color-flavor locked ground state of QCD at high baryon density. In the presence of condensates which pair light and strange quarks there need not be an energy gap in the quasiparticle spectrum. This raises the possibility of gapless color superconductivity, with a Meissner effect but no minimum excitation energy. Analysis within a toy model suggests that gapless color superconductivity may occur only as a metastable phase.
At sufficiently high baryon density, a quark matter is expected to become a color superconductor because of the pairing forces mediated by gluons. The theoretical aspect of this novel phase of the strong interaction is reviewed with the emphasis on the perturbation theory of QCD at high chemical potential and low temperature. The derivation of the scaling formula of the superconducting transition temperature at weak coupling is explained in detail. The Ginzburg-Landau theory of the color superconductivity is also discussed.
Solving the Schwinger-Dyson equations, we analyze the pairing of quarks in asymmetric quark matter where quarks have different chemical potentials. We show that in the asymmetric quark matter a crystalline color-superconducting gap opens when the quark coupling is stronger than a critical value. The critical coupling is nonzero, since the infrared divergence is lessened when the momenta of pairing quarks are not opposite. The superconducting gaps and the critical couplings are calculated both at high and intermediate densities.
We argue that in quark matter at high densities, the color magnetic field remains unscreened and leads to the phenomenon of color superconductivity. Using the renormalization group near the Fermi surface, we find that the long-range nature of the magnetic interaction changes the asymptotic behavior of the gap $Δ$ at large chemical potential $μ$ qualitatively. We find $Δ\simμg^{-5}\exp(-{3π^2\over\sqrt{2}}{1\over g})$, where $g$ is the small gauge coupling. We discuss the possibility of breaking rotational symmetry by the formation of a condensate with nonzero angular momentum, as well as inter…
We review what is different and what is similar in a color superconductor as compared to an ordinary BCS superconductor. The parametric dependence of the zero-temperature gap on the coupling constant differs in QCD from that in BCS theory. On the other hand, the transition temperature to the superconducting phase is related to the zero-temperature gap in the same way in QCD as in BCS theory.
We calculate the energy release associated with first-order phase transition between different types of single flavor color superconductivity in a magnetic field.
We demonstrate that crystalline color superconductivity may arise as a result of pairing between massless quarks and quarks with nonzero mass m_s. Previous analyses of this phase of cold dense quark matter have all utilized a chemical potential difference δμto favor crystalline color superconductivity over ordinary BCS pairing. In any context in which crystalline color superconductivity occurs in nature, however, it will be m_s-induced. The effect of m_s is qualitatively different from that of δμin one crucial respect: m_s depresses the value of the BCS gap Δ_0 whereas δμleaves Δ_0 unchanged. …
Cold dense quark matter is in a crystalline color superconducting phase wherever pairing occurs between species of quarks with chemical potentials whose difference δμlies within an appropriate window. If the interaction between quarks is modeled as point-like, this window is rather narrow. We show that when the interaction between quarks is modeled as single-gluon exchange, the window widens by about a factor of ten at accessible densities and by much larger factors at higher density. This striking enhancement reflects the increasingly (1+1)-dimensional nature of the physics at weaker and weak…
We show that the pseudogap of the quark density of states is formed in hot quark matter as a precursory phenomenon of the color superconductivity on the basis of a low-energy effective theory. We clarify that the decaying process of quarks near Fermi surface to a hole and the diquark soft mode (qq)_{soft} is responsible for the formation of the pseudogap. Our result suggests that the pseudogap is a universal phenomenon in strong coupling superconductors.
Four of the five expected Goldstone modes, which will be eaten up by gauge fields, in neutral two-flavor color superconductor are actually pseudo-Goldstone modes, and their degenerated mass is exactly the magnitude of the color chemical potential, which is introduced to guarantee the color neutrality at moderate baryon density.
We study the gauge field fluctuations in dense quark matter and determine the temperature of the induced first-order phase transition to the color-superconducting phase in weak coupling. We find that the local approximation of the coupling between the gauge potential and the order parameter, employed in the Ginzburg-Landau theory, has to be modified by restoring the full momentum dependence of the polarization function of gluons in the superconducting phase.
We discuss the possibility that in finite density QCD an anisotropic phase is realized. This case might arise for quarks with different chemical potential and/or different masses. In this phase crystalline structures may be formed. We consider this possibility and we describe, in the context of an effective lagrangian, the corresponding phonons as the Nambu-Goldstone bosons associated to the breaking of the space symmetries.
We investigate neutral quark matter with homogeneous and inhomogeneous color condensates at finite temperature in the frame of an extended NJL model. By calculating the Meissner masses squared and gap susceptibility, the uniform color superconductor is stable only in a temperature window close to the critical temperature and becomes unstable against LOFF phase, mixed phase and gluonic phase at low temperatures. The introduction of the inhomogeneous phases leads to disappearance of the strange intermediate temperature 2SC/g2SC and changes the phase diagram of neutral dense quark matter signific…
At ultra-high density, matter is expected to form a degenerate Fermi gas of quarks in which there is a condensate of Cooper pairs of quarks near the Fermi surface: color superconductivity. In these proceedings I review some of the underlying physics, and discuss outstanding questions about the phase structure of ultra-dense quark matter.
We construct the effective Lagrangian describing the light glueballs associated with the unbroken and confining SU(2) color subgroup for the 2 flavor superconductive phase of QCD. This Lagrangian constitutes a key ingredient for understanding the non perturbative physics of 2 flavor color superconductivity. We estimate the two photon decay process of the light glueballs using the saturation of the electromagnetic trace anomaly at the effective Lagrangian level. The present results are particularly relevant to our model of Gamma Ray Bursts based on color superconductivity in Quark Stars (R. Ouy…
The problem of color superconductivity in dense QCD is reconsidered in the improved rainbow approximation to the Schwinger-Dyson equation. The effect of the unscreened magnetic modes of gluons on the value of the color condensate is studied. In particular, it is shown that, at sufficiently large values of the chemical potential, these modes lead to the enhancement of the superconducting order parameter. The interplay between the instanton induced interaction and the one-gluon induced one in color superconductivity is discussed.
Because of a logarithmic enhancement from soft, collinear magnetic gluons, in dense quark matter the gap for a color superconducting condensate with spin zero depends upon the QCD coupling constant g not as exp(-1/g^2), like in BCS theory, but as exp(-1/g). In weak coupling, the ratio of the transition temperature to the spin-zero gap at zero temperature is the same as in BCS theory. We classify the gaps with spin one, and find that they are of the same order in g as the spin-zero gap.
The color superconductivity of a dense quark matter is reviewed with emphasis on the long range nature of the pairing force and the multiplicity of the order parameter. The former gives rise to a non BCS behavior of the superconducting energy scale and the latter modifies the critical value of the Ginzburg-Landau parameter that separates the superconductivity of type I and that of type II.
I discuss recent developments in our understanding of the color-superconducting phases of cold, dense quark matter. I describe the phase diagram as a function of density and the strange quark mass, and outline some ideas about possible observational consequences of these exotic phases.
We give an introduction crystalline color superconductivity, arguing that it is likely to occur wherever quark matter in which color-flavor locking does not occur is found. We survey the properties of this form of quark matter, and argue that its presence in a compact star may result in pulsar glitches, and thus in observable consequences. However, elucidation of this proposal requires an understanding of the crystal structure, which is not yet in hand.
Two aspects of quark matter at high density are addressed: one is color superconductivity and the other is ferromagnetism. We are mainly concerned with the latter and its relation to color superconductivity, which we call "color magnetic superconductivity". The relation of ferromagnetism and chiral symmetry restoration is also discussed.
We discuss a possibility of transition from color superconductivity of the standard BCS type at high density, to Bose-Einstein Condensation (BEC) of Cooper pairs at lower density. Examining two-flavor QCD over a wide range of baryon density, we found the size of a Cooper pair becomes small enough to be comparable to the averaged quark-quark distance at lower density. We also consider the same problem in two-color QCD.
The first lecture provides an introduction to the physics of color superconductivity in cold dense quark matter. The main color superconducting phases are briefly described and their properties are listed. The second lecture covers recent developments in studies of color superconducting phases in neutral and beta-equilibrated matter. The properties of gapless color superconducting phases are discussed.
In this paper, the NJL-type model is used to investigate the color superconductivity. The four-fermion interactions of the NJL-type model are Fierz-transformed into two different classes, i.e., the quark-antiquark and the quark-quark interaction channels, associated with the chiral symmetry breaking and color superconductivity respectively. We conclude that the weighting factor between quark-antiquark and quark-quark interaction channels has significance on the phase structure when the mean-field approximation is employed, and the baryon number density gives a tight constraint on the weighting…
The chiral phase transition and color superconductivity in an extended NJL model with eight-quark interactions are studied. The scalar-type nonlinear term hastens the chiral phase transition,the scalar-vector mixing term suppresses effects of the vector-type linear term and the scalar-diquark mixing term makes the coexisting phase wider.
Present computations of the gap of color superconductivity in weak coupling assume that the quarks which participate in the condensation process are infinitely long-lived. However, the quasiparticles in a plasma are characterized by having a finite lifetime. In this article we take into account this fact to evaluate its effect in the computation of the color gap. By first considering the Schwinger-Dyson equations in weak coupling, when one-loop self-energy corrections are included, a general gap equation is written in terms of the spectral densities of the quasiparticles. To evaluate lifetime …
We study a model for color superconductivity with both three colors and massless flavors including quark pairing. By using the Hamiltonian in the color-flavor basis we can calculate the quantum entropy. From this we are able to further investigate the phases of the color superconductor, for which we find a rather sharp transition to color superconductivity above a chemical potential around $290 $MeV.
The effects of chiral imbalance and external magnetic field on pion superfluidity and color superconductivity are investigated in extended Nambu--Jona-Lasinio models. We take Schwinger approach to treat the interaction between charged pion condensate and magnetic field at finite isospin density and include simultaneously the chiral imbalance and magnetic field at finite baryon density. For the superfluidity, the chiral imbalance and magnetic field lead to catalysis and inverse catalysis effects, respectively. For the superconductivity, the chiral imbalance enhances the critical baryon density,…
In any context in which color superconductivity arises in nature, it is likely to involve pairing between species of quarks with differing chemical potentials. For suitable values of the differences between chemical potentials, Cooper pairs with nonzero total momentum are favored, as was first realized by Larkin, Ovchinnikov, Fulde and Ferrell (LOFF). Condensates of this sort spontaneously break translational and rotational invariance, leading to gaps which vary periodically in a crystalline pattern. Unlike the original LOFF state, these crystalline quark matter condensates include both spin z…
We present a detailed study of the recently proposed gapless color superconductivity in neutral two-flavor quark matter in beta-equilibrium at zero as well as at finite temperature. We clarify the nature of the fermionic quasiparticles, and discuss the physical properties of this gapless phase of quark matter. The appearance of two "secondary" spin-1 condensates in the gapless ground state is pointed out. The gluon screening properties are also discussed. It is found that finite temperature properties of the gapless color superconductor are rather unusual. One of the most striking results is t…
We investigate the possible existence of spin polarization and color superconductivity in the Nambu--Jona-Lasinio model with a tensor-type interaction at finite density and temperature. The thermodynamic potential is calculated by the functional integral method. Numerical results indicate that at low temperature and quark chemical potential the chiral condensed phase exists, and at intermediate chemical potential the color superconducting phase appears. In addition, depending on the magnitude of the tensor coupling, at large chemical potential and low temperature, a color superconducting phase…
We investigate the single flavor color superconductivity in a magnetic field. Because of the absence of the electromagnetic Meissner effect, forming a nonspherical CSC phase, polar, A or planar, does not cost energy of excluding magnetic flux. We found that these nonspherical phases do occupy a significant portion of the phase diagram with respect to magnetic field and temperature and may be implemented under the typical quark density and the magnetic field inside a compact star.
The status of our understanding of the properties and manifestations of color superconductivity at zero and low density is briefly reviewed. Future possibility is highlighted.
Recent developments in weak-coupling color superconductivity are reviewed. These developments are as follows. The mean field gap equation is solved for most common superconducting phases up to subleading order; BCS relation is found to be violated in color-flavor-locking and color-spin-locking phases due to the two-gaps structure of the order parameter; The Debye and Meissner masses of gluons and photon with their rotated partners are calculated for these phases; We found that there is no electromagnetic Meissner effect in spin-one color superconductor; A proof of gauge parameter independence …
We investigate the vacuum realignment for chiral symmetry breaking and color superconductivity at finite density in Nambu-Jona-Lasinio model in a variational method. The treatment allows us to investigate simultaneous formation of condensates in quark antiquark as well as in diquark channels. The methodology involves an explicit construction of a variational ground state and minimisation of the thermodynamic potential. Color and electric charge neutrality conditions are imposed through introduction of appropriate chemical potentials. Color and flavor dependent condensate functions are determin…
The phase structure of hadronic matter at high density relevant to the physics of compact stars and relativistic heavy-ion collisions is studied in a low-energy effective quark theory. The relevant phases that figure are (1) chiral condensation, (2) diquark color condensation (color superconductivity) and (3) induced Lorentz-symmetry breaking (ISB). For reasonable strengths for four-Fermi interactions implied by one-gluon exchange quark interactions as well as instanton-mediated interactions, we find that the ISB phase sets in at a moderate density while the quark condensate vanishes, followed…
In crystalline color superconductivity, quark pairs form at non-zero total momentum. This crystalline order potentially enlarges the domain of color superconductivity in cold dense quark matter. We present a perturbative calculation of the parameters governing the crystalline phase and show that this is indeed the case. Nevertheless, the enhancement is modest, and to lowest order is independent of the strength of the color interaction.
Baryonic matter at high density and low temperature is a color superconductor. In real world, this state of matter may naturally appear inside compact stars. A construction of a hybrid compact star with two flavor color superconducting quark matter in its interior is presented.
At high nuclear density and small temperature, due to the asymptotic freedom property of Quantum ChromoDynamics and to the existence of an attractive channel in the color interaction, diquark condensates might be formed. Since these condensates break the color gauge symmetry, this phenomenon has the name of color superconductivity. In the last few years this has become a very active field of research. While a direct experimental test is still missing, color superconductivity might have implications in astrophysics because for some compact stars, e.g. pulsars, the baryon densities necessary for…
A coexistent phase of spin polarization and color superconductivity in high-density QCD is studied at zero temperature. The axial-vector self-energy stemming from the Fock exchange term of the one-gluon-exchange interaction has a central role to cause spin polarization. As a significant feature, the Fermi surface is deformed by the axial-vector self-energy and then rotation symmetry is spontaneously broken down. The gap function results in being anisotropic in the momentum space in accordance with the deformation. It is found that spin polarization barely conflicts with color superconductivity…
Motivated by the SO(5) theory of high-temperature superconductivity and antiferromagnetism, we ask if an SO(10) theory unifies color superconductivity and chiral symmetry breaking in QCD. The transition to the color superconducting phase would then be analogous to a spin flop transition. While the spin flop transition generically has a unified SO(3) description, the SO(5) and SO(10) symmetric fixed points are unstable, at least in (4 - epsilon) dimensions, and require the fine-tuning of one additional relevant parameter. If QCD is near the SO(10) fixed point, it has interesting consequences fo…
A brief introduction into the properties of dense quark matter is given. Recently proposed gapless color superconducting phases of neutral and beta-equilibrated dense quark matter are discussed. The current status in the field is described, and the promising directions of the future research are outlined.
The simplest pattern of color superconductivity involves BCS pairing between up and down quarks. We argue that this ``2SC'' phase will not arise within a compact star. A macroscopic volume of quark matter must be electrically neutral and must be a color singlet. Satisfying these requirements imposes a significant free energy cost on the 2SC phase, but not on color-flavor locked (CFL) quark matter, in which up, down and strange quarks all pair. As a function of increasing density, therefore, one may see a single phase transition from hadronic matter directly to CFL quark matter. Alternatively, …
We survey the non-locked color-flavor-spin channels for quark-quark (color superconducting) condensates in QCD, using an NJL model. We also study isotropic quark-antiquark (mesonic) condensates. We make mean-field estimates of the strength and sign of the self-interaction of each condensate, using four-fermion interaction vertices based on known QCD interactions. For the attractive quark pairing channels, we solve the mean-field gap equations to obtain the size of the gap as a function of quark density. We also calculate the dispersion relations for the quasiquarks, in order to see how fully g…
We investigate possible precursory phenomena of color superconductivity in quark matter at finite temperature T with use of a simple Nambu-Jona-Lasinio model. It is found that the fluctuating pair field exists with a prominent strength even well above the critical temperature T_c. We show that the collective pair field has a complex energy located in the second Riemann sheet, which approaches the origin as T is lowered to T_c. We discuss the possible relevance of the precursor to the observables to be detected in heavy ion collisions.
A scaling analysis is made of the order parameter describing monopole condensation at the deconfining transition of N_f=2 QCD around the chiral point. In accordance with scaling properties of the specific heat, studied in a previous paper, scaling is consistent with a first order transition. The status of dual superconductivity of the vacuum as a mechanism of color confinement is reviewed.
In dense quark matter, the response of the color superconducting gaps to a small variation, $δμ$, in the chemical potential of the strange quark was studied. The approximation of three massless flavors of quarks and a general ansatz for the color flavor structure of the gap matrix was used. The general pole structure of the quasi-particle propagator in this ansatz is presented. The gap equation was solved using both an NJL interaction model and perturbative single gluon exchange at moderate densities and results are presented for varying values of $δμ$. Quantitative and qualitative differences…
Under the combination effect of the recommencement heating due to spin-down of strange stars and the heat perseveration due to weak conduct heat of the crust, the Cooper pair breaking and formation(PBF) in color superconduction quark matter arises. We investigated the cooling of the strange stars with a crust in color superconductivity phase including both decomfinement heating and PBF process. We find that deconfinement heating can delay the thermal evolution of strange stars and the PBF process suppresses the early temperature rise of the stars. The cooling strange stars behave within the br…
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.
After a brief review of the phenomena expected in cold dense quark matter, color superconductivity and color-flavor locking, we sketch some implications of recent developments in our understanding of cold dense quark matter for the physics of compact stars. We give a more detailed summary of our recent work on crystalline color superconductivity and the consequent realization that (some) pulsar glitches may originate in quark matter.
We investigate possible precursory phenomena of color superconductivity at finite temperature $T$ with an effective theory of QCD. It is found that the fluctuation of the diquark pair field exists with a prominent strength even well above the critical temperature $T_c$. We show that such a fluctuaiton forms a collective mode, the corresponding pole of which approaches the origin as $T$ is lowered to $T_c$ in the complex energy plane. We discuss the possible relevance of the precursor to the observables to be detected in heavy-ion collisions.
A formal proof is given that the fermionic quasiparticle dispersion laws in a color superconductor are gauge independent. It is shown that the gluon (photon) field acquires a non-vanishing expectation value in a color superconductor, which is related to color (electric) neutrality.
We present a derivation of the gap equation for the crystalline color superconducting phase of QCD which begins from a one-loop Schwinger-Dyson equation written using a Nambu-Gorkov propagator modified to describe the spatially varying condensate. Some aspects of previous variational calculations become more straightforward when rephrased beginning from a diagrammatic starting point. This derivation also provides a natural base from which to generalize the analysis to include quark masses, nontrivial crystal structures, gluon propagation at asymptotic densities, and nonzero temperature. In thi…
