Color Superconductivity Research Articles

Color superconductivity in a self-consistent NJL-type model

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 factor of quark-antiquark interaction channels. Besides, the susceptibilities show that the color superconducting phase transition is of the second-order and takes place before the chiral crossover transition as quark number density increases. In the end, we study the critical temperatures ${T}_{c}$ of the color superconductivity and it agrees with the perturbative result of diquark condensate $\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}0.57{T}_{c}$.

Extension to imaginary chemical potential in a holographic model

We extend a bottom up holographic model, which has been used in studying the color superconductivity in QCD, to the imaginary chemical potential ($\mu_I$) region, and the phase diagram is studied on the $\mu_I$-temperature (T) plane. The analysis is performed for the case of the probe approximation and for the background where the back reaction from the flavor fermions are taken into account. For both cases, we could find the expected Roberge-Weiss (RW) transitions. In the case of the back-reacted solution, a bound of the color number $N_c$ is found to produce the RW periodicity. It is given as $N_c\geq 1.2$. Furthermore, we could assure the validity of this extended model by comparing our result with the one of the lattice QCD near $\mu_I=0$.

The dual properties of chiral and isospin asymmetric dense quark matter formed of two-color quarks

In this paper the phase structure of dense baryon matter composed of u and d quarks with two colors has been investigated in the presence of baryon μB , isospin μI and chiral isospin μI5 chemical potentials in the framework of Nambu-Jona-Lasinio model with quark-antiquark and quark-quark interaction channels. In the chiral limit, it has been shown in the mean-field approximation that the duality between phases with spontaneous chiral symmetry breaking and condensation of charged pions, found in the three color case, remains valid in the two color case. In addition, it has been shown that there are two more dualities in the phase diagram in two color case, namely (as in the case with μI5 = 0), at μI5 ≠ 0 the general (μ, μI, μI5)-phase portrait of the model has dual symmetry between the phase with condensation of charged pions and the phase with diquark condensation. This duality stays exact even in the physical point, m0 ≠ 0. And at m0 = 0 the (μ, μI, μI5)- phase portrait becomes even more symmetrical, since dual symmetry between phases with spontaneous chiral symmetry breaking and diquark condensation appears. It is shown that due to the dualities the phase diagram is extremely symmetric and has interlacing structure. One can show that the phase portrait of two-color NJL model can be obtained just by duality properties from the results of investigations of three-color NJL model (it was noticed only after the numerical calculations have been performed). Three-color case shares only one duality of the two color one, and one can only see a facet of this enormously symmetric picture in the case of three colors. Using dualities only, it is possible to show that there are no mixed phases (phases with two non-zero condensates). This prediction of dualities is of great use, because for sure it can be shown by the direct calculations but it would be enormously more complicated and time-consuming numerically.

Dense two-color QCD towards continuum and chiral limits

We study two-color QCD with two flavors of Wilson fermion as a function of quark chemical potential mu and temperature T, for two different lattice spacings and two different quark masses. We find that the quarkyonic region, where the behaviour of the quark number density and the diquark condensate are described by a Fermi sphere of almost free quarks distorted by a BCS gap, extends to larger chemical potentials with decreasing lattice spacing or quark mass. In both cases, the quark number density also approaches its non-interacting value. The pressure at low temperature is found to approach the Stefan-Boltzmann limit from below.

Fierz-complete NJL model study. III. Emergence from quark-gluon dynamics

Our understanding of the dynamics and the phase structure of dense strong-interaction matter is to a large extent still built on the analysis of low-energy models, such as those of the Nambu-Jona-Lasinio-type. In this work, we analyze the emergence of the latter class of models at intermediate and low energy scales from fundamental quark-gluon interactions. To this end, we study the renormalization group flow of a Fierz-complete set of four-quark interactions and monitor their strength at finite temperature and quark chemical potential. At small quark chemical potential, we find that the scalar-pseudoscalar interaction channel is dynamically rendered most dominant by the gauge degrees of freedom, indicating the formation of a chiral condensate. Moreover, the inclusion of quark-gluon interactions leaves a significant imprint on the dynamics as measured by the curvature of the finite-temperature phase boundary which we find to be in accordance with lattice QCD results. At large quark chemical potential, we then observe that the dominance pattern of the four-quark couplings is changed by the underlying quark-gluon dynamics, without any fine-tuning of the four-quark couplings. In this regime, the scalar-pseudoscalar interaction channel becomes subleading and the dominance pattern suggests the formation of a chirally symmetric diquark condensate. In particular, our study confirms the importance of explicit $U_{\mathrm{A}}(1)$ breaking for the formation of this type of condensate at high densities.

Two-colour QCD phases and the topology at low temperature and high density

We delineate equilibrium phase structure and topological charge distribution of dense two-colour QCD at low temperature by using a lattice simulation with two-flavour Wilson fermions that has a chemical potential μ and a diquark source j incorporated. We systematically measure the diquark condensate, the Polyakov loop, the quark number density and the chiral condensate with improved accuracy and j → 0 extrapolation over earlier publications; the known qualitative features of the low temperature phase diagram, which is composed of the hadronic, Bose-Einstein condensed (BEC) and BCS phases, are reproduced. In addition, we newly find that around the boundary between the hadronic and BEC phases, nonzero quark number density occurs even in the hadronic phase in contrast to the prediction of the chiral perturbation theory (ChPT), while the diquark condensate approaches zero in a manner that is consistent with the ChPT prediction. At the highest μ, which is of order the inverse of the lattice spacing, all the above observables change drastically, which implies a lattice artifact. Finally, at temperature of order 0.45Tc, where Tc is the chiral transition temperature at zero chemical potential, the topological susceptibility is calculated from a gradient-flow method and found to be almost constant for all the values of μ ranging from the hadronic to BCS phase. This is a contrast to the case of 0.89Tc in which the topological susceptibility becomes small as the hadronic phase changes into the quark-gluon plasma phase.

Functional renormalization group approach to color superconducting phase transition

We investigate the order of the color superconducting phase transition using the functional renormalization group approach. We analyze the Ginzburg-Landau effective theory of color superconductivity and more generic scalar SU(Nc) gauge theories by calculating the β function of the gauge coupling in arbitrary dimension d based on two different regularization schemes. We find that in d = 3, due to gluon fluctuation effects, the β function never admits an infrared fixed point solution. This indicates that, unlike the ordinary superconducting transition, color superconductivity can only show a first-order phase transition.

Spectrum of a supersymmetric color superconductor

We have recently shown that the ground state of mathcal{N} = 4, SU(Nc) super Yang- Mills coupled to Nf ≪ Nc flavors, in the presence of non-zero isospin and R-symmetry charges, is a supersymmetric, superfluid, color superconductor. The holographic description consists of Nf D7-brane probes in AdS5×S5 with electric and instantonic fields on their worldvolume. These correspond to fundamental strings and D3-branes dissolved on the D7-branes, respectively. Here we use this description to determine the spectrum of mesonic excitations. As expected for a charged superfluid we find non-relativistic, massless Goldstone modes. We also find extra ungapped modes that are not associated to the breaking of any global symmetries but to the supersymmetric nature of the ground state. If the quark mass is much smaller than the scale of spontaneous symmetry breaking a pseudo-Goldstone boson is also present. We highlight some new features that appear only for Nf> 2. We show that, in the generic case of unequal R-symmetry charges, the dissolved strings and D3-branes blow up into a D5-brane supertube stretched between the D7-branes.

Gluon propagator in two-color dense QCD: Massive Yang-Mills approach at one loop

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

Novel color superconducting phases of mathcal{N} = 4 super Yang-Mills at strong coupling

We revisit the large-Nc phase diagram of mathcal{N} = 4 super Yang-Mills theory at finite R-charge density and strong coupling, by means of the AdS/CFT correspondence. We conjecture new phases that result from a black hole shedding some of its charge through the nucleation of probe color D3-branes that remain at a finite distance from the black hole when the dual field theory lives on a sphere. In the corresponding ground states the color group is partially Higgsed, so these phases can be identified as having a type of color superconductivity. The new phases would appear at intermediate values of the R-charge chemical potential and we expect them to be metastable but long-lived in the large-Nc limit.

Phenomenological theory of Superconductivity in the frameworks of QCD and RCD

Carrying out the study of phenomenological theory of superconductivity at microscopic level in the framework of QCD and restricted chromodynamics, we have investigated the topology of dyons in magnetic gauge space and showed that RCD vavuum containing plasma of dyons and anti-dyons behave like a chromomagnetic superconductor within which the chromomagnetic gluons propagate in the non-perturbative vacuum. Study of the superconductivity, dual superconductivity and color superconductivity have shown to automatically incorporate the condensation of monopoles or dyons ultimately leading to superconducting confining phases of the QCD vacuum.

Phenomenological theory of Superconductivity in the frameworks of QCD and RCD

Carrying out the study of phenomenological theory of superconductivity at microscopic level in the framework of QCD and restricted chromodynamics, we have investigated the topology of dyons in magnetic gauge space and showed that RCD vavuum containing plasma of dyons and anti-dyons behave like a chromomagnetic superconductor within which the chromomagnetic gluons propagate in the non-perturbative vacuum. Study of the superconductivity, dual superconductivity and color superconductivity have shown to automatically incorporate the condensation of monopoles or dyons ultimately leading to superconducting confining phases of the QCD vacuum.

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.

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.

Color superconductivity in a holographic model

A holographic bottom-up model used in studying the superconducting system is applied to search for the color superconducting phase of supersymmetric Yang-Mills theory. We apply the probe analysis of this model to the supersymmetric Yang-Mills theory in both the confinement and deconfinement phases. In this analysis, we find the color superconductivity in both phases when the baryon chemical potential exceeds a certain critical value. This result implies that, above the critical chemical potential, a color non-singlet diquark operator, namely the Cooper pair, has its vacuum expectation value even in the confinement phase. In order to improve this peculiar situation, we proceed the analysis by taking account of the full back-reaction from the probe. As a result, the color superconducting phase, which is observed in the probe approximation, disappears in both the confinement and deconfinement phases when parameters of the theory are set within their reasonable values.

A supersymmetric color superconductor from holography

We use holography to study d = 4, mathcal{N}=4 , SU(Nc) super Yang-Mills coupled to Nf ≪ Nc quark flavors. We place the theory at finite isospin density nI by turning on an isospin chemical potential μI = Mq, with Mq the quark mass. We also turn on two R-symmetry charge densities n1 = n2. We show that the ground state is a supersymmetric, superfluid, color superconductor, namely a finite-density state that preserves a fraction of supersymmetry in which part of the global symmetries and part of the gauge symmetries are spontaneously broken. The holographic description consists of Nf D7-brane probes in AdS5 × S5. The symmetry breaking is due to the dissolution of some D3-branes inside the D7-branes triggered by the electric field associated to the isospin charge. The massless spectrum contains Goldstone bosons and their fermionic superpartners. The massive spectrum contains long-lived, mesonic quasi-particles if nI ≪ μI3, and no quasi-particles otherwise. We discuss the possibility that, despite the presence of mass scales and charge densities in the theory, conformal and relativistic invariance arise as emergent symmetries in the infrared.

Lattice study of static quark-antiquark interactions in dense quark matter

In this paper we study the interactions among a static quark-antiquark pair in the presence of dense two-color quark matter with lattice simulation. To this end we compute Polyakov line correlation functions and determine the renormalized color averaged, color singlet and color triplet grand potentials. The color singlet grand potential allows us to elucidate the number of quarks induced by a static quark antiquark source, as well as the internal energy of such a pair in dense quark matter. We furthermore determine the screening length, which in the confinement phase is synonymous with the string breaking distance. The screening length is a decreasing function of baryon density, due to the possibility to break the interquark string via a scalar diquark condensate at high density. We also study the large distance properties of the color singlet grand potential i a dense medium and find that it is well described by a simple Debye screening formula, parameterized by a Debye mass and an effective coupling constant. The latter is of order of unity, i.e. even at large density two-color quark matter is a strongly correlated system.

Chiral symmetry breaking, color superconductivity, and equation of state for magnetized strange quark matter

We investigate the vacuum structure of dense quark matter in strong magnetic fields at finite temperature and densities in a 3 flavor Nambu Jona Lasinio (NJL) model including the Kobayashi-Maskawa-t'Hooft (KMT) determinant term using a variational method. The method uses an explicit structure for the `ground' state in terms of quark-antiquark condensates as well as diquark condensates. The mass gap equations and the superconducting gap equations are solved self consistently and are used to compute the thermodynamic potential along with the charge neutrality conditions. We also derive the equation of state for the charge neutral strange quark matter in the presence of strong magnetic fields which could be relevant for neutron stars.

Third family of compact stars within a nonlocal chiral quark model equation of state

A class of hybrid compact star equations of state is investigated that joins by a Maxwell construction a low-density phase of hadronic matter, modeled by a relativistic meanfield approach with excluded nucleon volume, with a high-density phase of color superconducting two-flavor quark matter, described within a nonlocal covariant chiral quark model. We find the conditions on the vector meson coupling in the quark model under which a stable branch of hybrid compact stars occurs in the cases with and without diquark condensation. We show that these hybrid stars do not form a third family disconnected from the second family of ordinary neutron stars unless additional (de)confining effects are introduced with a density-dependent bag pressure. A suitably chosen density dependence of the vector meson coupling assures that at the same time the $2~$M$_\odot$ maximum mass constraint is fulfilled on the hybrid star branch. A twofold interpolation method is realized which implements both, the density dependence of a confining bag pressure at the onset of the hadron-to-quark matter transition as well as the stiffening of quark matter at higher densities by a density-dependent vector meson coupling. For three parametrizations of this class of hybrid equation of state the properties of corresponding compact star sequences are presented, including mass twins of neutron and hybrid stars at 2.00, 1.39 and 1.20 $M_\odot$, respectively. The sensitivity of the hybrid equation of state and the corresponding compact star sequences to variations of the interpolation parameters at the 10% level is investigated and it is found that the feature of third family solutions for compact stars is robust against such a variation. This advanced description of hybrid star matter allows to interpret GW170817 as a merger not only of two neutron stars but also of a neutron star with a hybrid star or of two hybrid stars.

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.