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 $SU(N_c)$ 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.
We predict the existence of the gapless Carlson-Goldman modes in the color-flavor locked color superconducting phase of dense QCD. These modes resemble Nambu-Goldstone modes of the superconducting phase below the critical temperature where the Anderson-Higgs mechanism takes place. These modes exist in the broken phase in the vicinity of the critical line. Their presence does not eliminate the Meissner effect. The effect of Landau damping on the width of the Carlson-Goldman modes is discussed.
The vNRQCD Lagrangian for colored heavy scalar fields in the fundamental representation of QCD and the renormalization group analysis of the corresponding operators are presented. The results are an important ingredient for renormalization group improved computations of scalar-antiscalar bound state energies and production rates at next-to-next-to-leading-logarithmic (NNLL) order.
An introductory review of physics of color superconducting state of matter is presented. Comparison with superconductivity in electron systems reveals difficulties involved in formulating color superconductivity theory at moderately ultra-nuclear density.
Under the assumption that the color charge can be written in a BRST exact form, the color confinement mechanism proposed by Kugo and Ojima (KO) explains the confinement of any colored particles including dynamical quarks and gluons. This mechanism, however, is known to break down in the Abelian gauge which treats the maximal Abelian subgroup of the gauge group in a special manner. In order to study whether the failure of the KO mechanism is particular only to the Abelian gauge or whether this failure occurs in a wide class of gauges including the ordinary Lorentz type gauge, we carry out a ren…
We derive a generalized Ward identity for color-superconducting quark matter via the functional integral approach. The identity implies the gauge independence of the color-superconducting gap parameter on the quasi-particle mass shell to subleading order in covariant gauge.
We derive an Abelian-like Ward identity in color superconducting phase and calculate vertex corrections to the color superconducting gap. Making use of the Ward identity, we show that subleading order contributions to the gap from vertices are absent for gapped excitations.
We consider the phenomenon of the Andreev reflection of "hadrons" at the interface between hadronic and color superconducting phases, which are expected to appear in the neutron star interior. Here, hadrons are defined as a superposition of constituent quarks, each of which is Andreev-reflected. We study what kind of reflections are possible to come out of incident mesons and baryons in the hadronic phase, attached to different color superconducting phases. Then, some peculiar patterns of the reflections are obtained.
I review recent progress in our understanding of the color superconducting phase of matter above nuclear density, giving particular emphasis to the effort to find observable signatures of the presence of this phase in compact stars.
Color superconductivity in quark matter is studied for electrically charge neutral neutron star matter in $β$-equilibrium. Both bulk quark matter and mixed phases of quark and nuclear matter are treated. The electron chemical potential and strange quark mass affect the various quark chemical potentials and therefore also the color superconductivity due to dicolor pairing or color-flavor locking.
The properties of plasmons, Nambu-Goldstone bosons and gapless Carlson-Goldman collective modes in color-flavor locked phase of color superconducting dense quark matter at finite temperature are reviewed. A possibility of a kaon condensation with an abnormal number of the NG bosons is also discussed.
Using generalized Nielsen identities a formal proof is given that the fermionic quasiparticle dispersion relations in a color superconductor are gauge independent. This turns out to involve gluonic tadpoles which are calculated to one-loop order in a two-flavor color superconductor. Regarding the appearance of gluon tadpoles, we argue that in QCD the color superconducting phase is automatically color neutral.
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.
The renormalization group is used to resum leading logarithmic contributions of the form alpha_s^{n+1} beta_0^n log^n (Delta/mu) to the gap equation appropriate for high density QCD. The scale dependence of the strong coupling constant alpha_s increases the gap by a factor of exp(33/16 (pi^2/4 - 1)) ~ 20 in the small coupling limit.
We study the Bethe-Salpeter equations for spin zero diquark composites in the color superconducting phase of N_f=2 cold dense QCD. The explicit form of the spectrum of the diquarks, containing an infinite tower of narrow (at high density) resonances, is derived. It is argued that there are five pseudo-Nambu-Goldstone bosons (pseudoscalars) that remain almost massless at large chemical potential. These five pseudoscalars, along with the massless quarks of the third color should play an important role in the infrared dynamics of N_f=2 dense QCD.
The color matrix elements of the ghost triangle diagram that appears in the triple gluon vertex and the ghost-ghost-gluon triangle diagram that appears in the ghost-gluon-ghost vertex are calculated. The ghost-ghost-gluon triangle contains a loop consisting of two color diagonal ghosts and one gluon and a loop consisting of two color antisymmetric ghosts and one gluon. Consequently, the pQCD argument in the infrared region based on the one particle irreducible diagram should be modified. Implications for the Kugo-Ojima color confinement and the QCD running coupling are discussed.
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.
Realistic nonstrange hybrid compact stars with color superconducting quark matter in their interior are constructed. It is shown that a positively charged two-flavor color superconducting phase could naturally appear in the core of a hybrid star as one of the components of a globally neutral mixed phase. The negatively charged normal quark phase is the other component of the mixed phase. The quark core of the star is surrounded by another mixed phase made of hadronic and normal quark matter. The two mixed phases are separated by a sharp interface. Finally, the lowest density regions of the sta…
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.
We investigate the effects of the dynamical formation of the chiral condensates on color superconducting phases under the electric and color neutrality constraints at vanishing temperature. We shall show that the phase appearing next to the color-flavor-locked (CFL) phase down in density depends on the strength of the diquark coupling. In particular, the gapless CFL (gCFL) phase is realized only in a weak coupling regime. We give a qualitative argument on why the gCFL phase in the weak coupling region is replaced by some other phases in the strong coupling, once the competition between dynamic…
Lack of Lorentz invariance of QCD at finite quark chemical potential in general implies the need of Lorentz non-invariant condensates for the self-consistent description of the color-superconducting ground state. Moreover, the spontaneous breakdown of color SU(3) in this state naturally leads to the existence of SU(3) non-invariant non-superconducting expectation values. We illustrate these observations by analyzing the properties of an effective 2-flavor Nambu-Jona-Lasinio type Lagrangian and discuss the possibility of color-superconducting states with effectively gapless fermionic excitation…
The non-leptonic weak processes $s+u\to u+d$ and $u+d \to s+u$ are known to dominate the dissipation mechanism responsible for the viscosity of strange quark matter in its normal phase. The rates of such processes remain unknown for many color superconducting phases of quark matter. In this paper, we partially fill up the gap by calculating the difference of the rates of the two non-leptonic weak processes in four transverse spin-one color superconducting phases of quark matter (slightly) out of $β$-equilibrium. The four phases studied are the color-spin locked phase, the polar phase, the plan…
The role of the strange quark mass for the phase structure of QCD at non-vanishing densities is studied by employing a recently developed self-consistent truncation scheme for the Dyson-Schwinger equations of the quark propagators in Landau gauge. Hereby the medium modification of the effective quark interaction by the polarization of gluons is implemented. Taking into account this effect results in significantly smaller dynamical quark masses at the Fermi surface. Due to this reduction the color-flavor locked phase is always the preferred color-superconducting phase at zero temperature and fo…
The masses of the pseudo-Nambu-Goldstone bosons in the color superconducting phase of dense QCD with two light flavors are estimated by making use of the Cornwall-Jackiw-Tomboulis effective action. Parametrically, the masses of the doublet and antidoublet bosons are suppressed by a power of the coupling constant as compared to the value of the superconducting gap. This is qualitatively different from the mass expression for the singlet pseudo-Nambu-Goldstone boson, resulting from non-perturbative effects. It is argued that the (anti-) doublet pseudo-Nambu-Goldstone bosons form colorless [with …
A more general expression for the quark propagator including both chiral and diquark condensates has been derived by using energy projectors. This makes it possible to study the phase transition from hadron phase to color superconductivity phase in the moderate baryon density region by using Feynman diagrammatic method or Green-function method.
A two-flavor color superconducting (2SC) Nambu--Jona-Lasinio (NJL) model is introduced at finite temperature T, chemical potential mu and in the presence of a constant magnetic field eB. The effect of (T,mu,eB) on the formation of chiral and color symmetry breaking condensates is studied. The complete phase portrait of the model in T-mu, mu-eB, and T-eB phase spaces for various fixed eB, T, and mu is explored. A threshold magnetic field eB_t~ 0.5 GeV^2 is found above which the dynamics of the system is solely dominated by the lowest Landau level (LLL) and the effects of T and mu are partly com…
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 kno…
We present an explicit and simple form of the renormalization group equation which governs the quantum evolution of the effective theory for the Color Glass Condensate (CGC). This is a functional Fokker-Planck equation for the probability density of the color field which describes the CGC in the covariant gauge. It is equivalent to the Euclidean time evolution equation for a second quantized current-current Hamiltonian in two spatial dimensions. The quantum corrections are included in the leading log approximation, but the equation is fully non-linear with respect to the generally strong backg…
We use a Landau theory and the epsilon expansion to study the superfluid phase transition of two-color QCD at nonzero temperature, T, and baryonic chemical potential, mu. At low T, and for N_f flavors of massless quarks, the global SU(N_f) \times SU(N_f) \times U(1) symmetry is spontaneously broken by a diquark condensate down to Sp(N_f) \times Sp(N_f) for any mu > 0. As the temperature increases, the diquark condensate melts, and at sufficiently large T the symmetry is restored. Using renormalization group arguments, we find that in the presence of the chiral anomaly term there can be a secon…
We revisit the large-$N_c$ phase diagram of $\cal{N}$ = 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 va…
The phase diagram of a three-flavor Polyakov-loop Nambu-Jona-Lasinio model is analyzed for the case of isospin symmetric matter with color superconducting phases. The coexistence of chiral symmetry breaking and two-flavor color superconductivity (2SC phase) and a thermodynamic instability due to the implementation of a color neutrality constraint is observed. It is suggested to use a universal hadronization pressure to estimate the phase border between hadronic and quark-gluon plasma phases. Trajectories of constant entropy per baryon are analyzed for conditions appropriate for heavy-ion colli…
We show that within a recently developed nonlocal chiral quark model the critical density for a phase transition to color superconducting quark matter under neutron star conditions can be low enough for these phases to occur in compact star configurations with masses below 1.3 M_sun. We study the cooling of these objects in isolation for different values of the gravitational mass and argue that, if the quark matter phase would allow unpaired quarks, the corresponding hybrid stars would cool too fast. The comparison with observational data puts tight constraints on possible color superconductin…
Several relevant issues in computing neutrino emissivity in Urca processes in color superconducting quark matter are addressed. These include: (1) The constraint on $u$ quark abundance is given from electric neutrality and the triangle relation among Fermi momenta for participants. (2) The phase space defined by Fermi momentum reduction of quarks is discussed in QCD and NJL model. (3) Fermi effective model of weak interaction is reviewed with special focus on its form in Nambu-Gorkov basis.
The dense core of compact stars is the natural medium for the realization of color superconductivity. A common characteristic of such astrophysical objects is their strong magnetic fields, especially those of the so called magnetars. In this talk, I discuss how a color superconducting core can generate or/and enhance the stellar magnetic field independently of a magnetohydrodynamic dynamo mechanism. The magnetic field generator is in this case a gluonic current which circulates to stabilize the color superconductor in the presence of a strong magnetic field or under the pairing stress produced…
The properties of magnetized color superconducting cold dense quark matter under compact star conditions are investigated using a $SU(2)_f$ Nambu Jona-Lasinio (NJL)-type model in which the divergences are treated using a magnetic field independent regularization scheme in order to avoid unphysical oscillations. We study the phase diagram for several model parametrizations. The features of each phase are analyzed through the behavior of the chiral and superconducting condensates together with the different particle densities for increasing chemical potential or magnetic field. While confirming …
We study effects of two-flavor color superconductivity on the structure of strange dwarfs, which are stellar objects with similar masses and radii with ordinary white dwarfs but stabilized by the strange quark matter core. We find that unpaired quark matter is a good approximation to the core of strange dwarfs.
We discuss the Meissner effect for a color superconductor formed by cold dense quark matter. Though color and ordinary electromagnetism are broken in a color superconductor, there is a linear combination of the photon and a gluon that remains massless. Consequently, a color superconducting region may be penetrated by an external magnetic field. We show that at most a small fraction of the magnetic field is expelled, and if the screening distance is the smallest length scale in the problem there is no expulsion at all. We calculate the behavior of the magnetic field for a spherical geometry rel…
Fluctuations of the color superconducting order parameter in dense quark matter at finite temperatures are investigated in terms of the phenomenological Ginzburg - Landau approach. Our estimates show that fluctuations of the di-quark gap may strongly affect some of thermodynamic quantities even far below and above the critical temperature. If the critical temperature of the di-quark phase transition were rather high one could expect a manifestation of fluctuations of the di-quark gap in the course of heavy ion collisions.
We calculate the neutrino mean free path in color superconducting quark matter, and employ it to study the cooling of matter via neutrino diffusion in the superconducting phase as compared to a free quark phase. The cooling process slows when quark matter undergoes a second order phase transition to a superconducting phase at the critical temperature $T_c$. Cooling subsequently accelerates as the temperature decreases below $T_c$. This will directly impact the early evolution of a newly born neutron star should its core contain quark matter. Consequently, there may be observable changes in the…
The color neutral two-flavor superconducting (2SC) phase of cold and dense quark matter is studied in the presence of constant magnetic fields and at moderate baryon densities. In the first part of the paper, a two-flavor effective Nambu--Jona-Lasinio model consisting of a chiral symmetry breaking mass gap sigma_{B}, a color superconducting mass gap Delta_{B} and a color neutrality coefficient mu_{8} is introduced in the presence of a rotated U(1) magnetic field B. To study the phenomenon of magnetic catalysis in the presence of strong magnetic fields, the gap equations corresponding to sigma_…
We discuss gapless color superconductivity for neutral quark matter in beta equilibrium at zero as well as at nonzero temperature. Basic properties of gapless superconductors are reviewed. The current progress and the remaining problems in the understanding of the phase diagram of strange quark matter are discussed.
We have shown, that the possibility of the existence of the mixed phase of the non-uniform chiral (NCh) and the color superconducting (2SC) ground state depends significantly on the choice of the parameters and type of the regularization scheme. Our calculations indicates, that in the 3d cut-off regularization scheme, the mixed region of the NCh and the 2SC phases exists for a broad set of NJL model parameters. However, in the Schwinger regularization scheme, if parameters are fitted to the physical quantities in the vacuum, then, the mixed region of the NCh and the 2SC phases does not exists.
The stability conditions of a renormalization group improved effective potential have been discussed in the case of scalar QED and QCD with a colorless scalar. We calculate the same potential in these models assuming the existence of non-perturbative fixed points associated to a conformal phase. In the case of scalar QED the barrier of instability found previously is barely displaced as we approach the fixed point, and in the case of QCD with a colorless scalar not only the barrier is changed but the local minimum of the potential is also changed.
The influence of intense magnetic fields on the behavior of color superconducting cold quark matter is investigated using an SU(2)$_f$ NJL-type model for which a novel regulation scheme is introduced. In such a scheme the contributions which are explicitly dependent on the magnetic field turn out to be finite and, thus, do not require to be regularized. As a result of this, nonphysical oscillations that might arise in the alternative regularization schemes previously used in the literature are naturally removed. The sensitivity of our results to the model parametrization is analyzed.
We study the interplay of the chiral and the color superconducting phase transitions in an extended Nambu--Jona-Lasinio model with a multi-quark interaction that produces the nonlinear chiral-diquark coupling. We observe that this nonlinear coupling adds up coherently with the ω^2 interaction to produce the chiral-color superconductivity coexistence phase or cancel each other depending on its sign. We discuss that large coexistence region in the phase diagram is consistent with the quark-diquark picture for the nucleon whereas its smallness is the prerequisite for the applicability of the Ginz…
Transport coefficients of dense quark matter are needed to study properties of compact stars. They can tell us about the cooling, vibrational and rotational properties of the star. We report below a computation of the shear viscosity in the CFL phase at low temperature T. CFL quark matter behaves as a superfluid, and at low T its transport properties are dominated by the collisions of superfluid phonons.
In the preceding paper(Prog.Theor.Phys.108(2002)929 or hep-ph/0207255), we have shown that the critical line of the first order chiral transition of QCD can have two endpoints. In this addendum, we elucidate the mechanism to realize the two-endpoint structure in the QCD phase diagram and argue the robustness for the appearance of such an interesting phase structure.
The gap parameter for color superconductivity is expected to be a gauge invariant quantity, at least on the appropriate mass shell. Computing the gap to subleading order in the QCD coupling constant, g, we show that the prefactor of the exponential in 1/g is gauge dependent off the mass shell, and independent of gauge on the mass shell.
When nuclear matter reaches a high enough density, we expect that the nucleons will overlap so much as to lose their separate identities, and merge into quark matter. In this talk I will review some theoretical expectations and speculations about quark matter, focusing on the phenomenon of quark pair condensation (color superconductivity) and its application to compact stars.
We study junctions of vortices, or boojums, at the interface between color and hadronic superconducting/superfluid phases. This type of interface could be present in the interior of neutron stars, where an inner core made of quark matter in the color-flavor-locked phase is surrounded by an outer shell of superconducting protons and superfluid neutrons. We study the fate of magnetic (proton) and superfluid (neutron) vortices as they enter the color-flavor locked phase. We find that proton vortices terminate on Dirac monopoles of the massless magnetic field, and magnetic fluxes of massive gauge …
We consider color superconductivity with two flavors of massless quarks which form Cooper pairs with total spin zero. We solve the gap equation for the color-superconducting gap parameter to subleading order in the QCD coupling constant $g$ at zero temperature. At this order in $g$, there is also a previously neglected contribution from the real part of the quark self-energy to the gap equation. Including this contribution leads to a reduction of the color-superconducting gap parameter $\f_0$ by a factor $b_0'=\exp \big[ -(\p ^2+4)/8 \big]\simeq 0.177$. On the other hand, the BCS relation $T_c…
We calculate the specific heats of quasi-particles of two-flavor QCD in its crystalline phases for low temperature. We show that for the different crystalline structures considered here there are gapless modes contributing linearly in temperature to the specific heat. We evaluate also the phonon contributions which are cubic in temperature. These features might be relevant for compact stars with an inner shell in a color superconducting crystalline phase.
We examine electrically and color neutral quark matter in beta-equilibrium focusing on the possibility of mixed phases between different color superconducting phases. To that end we apply the Gibbs criterion to ensure phase equilibrium and discuss the external conditions under which these mixed phases can occur. Neglecting surface and Coulomb effects we find a rich structure of different mixed phases with up to four components, including 2SC and CFL matter as well as more ``exotic'' components, like a phase with us- and ds-pairing but without ud-pairing. Preliminary estimates indicate, however…
The McLerran-Venugopalan (MV) model is a Gaussian effective theory of color charge fluctuations at small-$x$ in the limit of large valence charge density, {\it i}.{\it e}., a large nucleus made of uncorrelated color charges. In this work, we explore the effects of the first non-trivial (even C-parity) non-Gaussian correction on the color charge density to the MV model ("quartic" term) in SU(2) and SU(3) color group in the non-perturbative regime. We compare our (numerical) non-perturbative results to (analytical) perturbative ones in the limit of small or large non-Gaussian fluctuations. The c…
We construct the nuclear and quark matter equations of state at zero temperature in an effective quark theory (the Nambu-Jona-Lasinio model), and discuss the phase transition between them. The nuclear matter equation of state is based on the quark-diquark description of the single nucleon, while the quark matter equation of state includes the effects of scalar diquark condensation (color superconductivity). The effect of diquark condensation on the phase transition is discussed in detail.
