Phase Structure of 2-Flavor Quark Matter: Heterogeneous Superconductors

We analyze the free energy of charge and color neutral 2-flavor quark matter within the BCS approximation. We consider both the homogeneous gapless superconducting phase and the heterogeneous mixed phase where normal and BCS superconducting phases coexist. We calculate the surface tension between normal and superconducting phases and use it to compare the free energies of the gapless and mixed phases. Our calculation, which retains only the leading order gradient contribution to the free energy, indicates that the mixed phase is energetically favored over an interesting range of densities of relevance to 2 flavor quark matter in neutron stars.Comment: 11 pages, 4 figures. Major Revisions. Includes a detailed discussion of the kinetic terms of the effective theory, instabilities of the gapless phase and the charge neutral phase diagra

CFL Phase of High Density QCD at Non Zero Strange Quark Mass

We compute free energy of quark matter at asymptotically high baryon number density in the presence of non zero strange quark mass including dynamics of pseudo Nambu-Goldstone bosons due to chiral symmetry breaking, extending previously existing analysis based on perturbative expansion in $m_s^2/4\mu\Delta.$ We demonstrate that the CFL$K^0$ state has lower free energy than the symmetric CFL state for $0<m_s^2/4\mu\Delta<2/3$. We also calculate the spectrum of the fermionic quasiparticle excitations about the kaon condensed ground state in the regime $m_s^2/4\mu\Delta \sim 1$ and find that $(m_s^2/4\mu\Delta)_{crit}=2/3$ for the CFL-gCFL phase transition, the leading order result reported in [1], is not modified.Comment: 16 pages, 3 figure

The Isoperimetric Profile of a Noncompact Riemannian Manifold for Small Volumes

In the main theorem of this paper we treat the problem of existence of minimizers of the isoperimetric problem under the assumption of small volumes. Applications of the main theorem to asymptotic expansions of the isoperimetric problem are given.Comment: 33 pages, improved version after the referee comments, (Submitted

Evaluating the Gapless Color-Flavor Locked Phase

In neutral cold quark matter that is sufficiently dense that the strange quark mass M_s is unimportant, all nine quarks (three colors; three flavors) pair in a color-flavor locked (CFL) pattern, and all fermionic quasiparticles have a gap. We recently argued that the next phase down in density (as a function of decreasing quark chemical potential mu or increasing strange quark mass M_s) is the new ``gapless CFL'' (``gCFL'') phase in which only seven quasiparticles have a gap, while there are gapless quasiparticles described by two dispersion relations at three momenta. There is a continuous quantum phase transition from CFL to gCFL quark matter at M_s^2/mu approximately equal to 2*Delta, with Delta the gap parameter. Gapless CFL, like CFL, leaves unbroken a linear combination "Q-tilde" of electric and color charges, but it is a Q-tilde-conductor with gapless Q-tilde-charged quasiparticles and a nonzero electron density. In this paper, we evaluate the gapless CFL phase, in several senses. We present the details underlying our earlier work which showed how this phase arises. We display all nine quasiparticle dispersion relations in full detail. Using a general pairing ansatz that only neglects effects that are known to be small, we perform a comparison of the free energies of the gCFL, CFL, 2SC, gapless 2SC, and 2SCus phases. We conclude that as density drops, making the CFL phase less favored, the gCFL phase is the next spatially uniform quark matter phase to occur. A mixed phase made of colored components would have lower free energy if color were a global symmetry, but in QCD such a mixed phase is penalized severely.Comment: 18 pages, RevTeX; Version to appear in Phys Rev D. Minor rewording, references adde

Neutrino emission from compact stars and inhomogeneous color superconductivity

We discuss specific heat and neutrino emissivity due to direct Urca processes for quark matter in the color superconductive Larkin-Ovchinnikov-Fulde-Ferrell (LOFF) phase of Quantum-Chromodynamics. We assume that the three light quarks $u, d, s$ are in a color and electrically neutral state and interact by a four fermion Nambu-Jona Lasinio coupling. We study a LOFF state characterized by a single plane wave for each pairing. From the evaluation of neutrino emissivity and fermionic specific heat, the cooling rate of simplified models of compact stars with a quark core in the LOFF state is estimated.Comment: 16 pages, 5 figures, revtex4 style. Version accepted for publication in Phys. Rev.

BCS BEC crossover and phase structure of relativistic system: a variational approach

We investigate here the BCS BEC crossover in relativistic systems using a variational construct for the ground state and the minimization of the thermodynamic potential. This is first studied in a four fermion point interaction model and with a BCS type ansatz for the ground state with fermion pairs. It is shown that the antiparticle degrees of freedom play an important role in the BCS BEC crossover physics, even when the ratio of fermi momentum to the mass of the fermion is small. We also consider the phase structure for the case of fermion pairing with imbalanced populations. Within the ansatz, thermodynamically stable gapless modes for both fermions and anti fermions are seen for strong coupling in the BEC regime. We further investigate the effect of fluctuations of the condensate field by treating it as a dynamical field and generalize the BCS ansatz to include quanta of the condensate field also in a boson fermion model with quartic self interaction of the condensate field. It is seen that the critical temperature decreases with inclusion of fluctuations.Comment: 18 pages, 13 figures, one more section added, title modified, version to appear in Phys Rev

On the ground state of gapless two flavor color superconductors

This paper is devoted to the study of some aspects of the instability of two flavor color superconductive quark matter. We find that, beside color condensates, the Goldstone boson related to the breaking of $U(1)_A$ suffers of a velocity instability. We relate this wrong sign problem, which implies the existence of a Goldstone current in the ground state or of gluonic condensation, to the negative squared Meissner mass of the $8^{th}$ gluon in the g2SC phase. Moreover we investigate the Meissner masses of the gluons and the squared velocity of the Goldstone in the multiple plane wave LOFF states, arguing that in such phases both the chromo-magnetic instability and the velocity instability are most probably removed. We also do not expect Higgs instability in such multiple plane wave LOFF. The true vacuum of gapless two flavor superconductors is thus expected to be a multiple plane wave LOFF state.Comment: 16 pages, RevTe3X4 styl

Kaon Condensation in a Nambu--Jona-Lasinio (NJL) Model at High Density

We demonstrate a fully self-consistent microscopic realization of a kaon-condensed colour-flavour locked state (CFLK0) within the context of a mean-field NJL model at high density. The properties of this state are shown to be consistent with the QCD low-energy effective theory once the proper gauge neutrality conditions are satisfied, and a simple matching procedure is used to compute the pion decay constant, which agrees with the perturbative QCD result. The NJL model is used to compare the energies of the CFLK0 state to the parity even CFL state, and to determine locations of the metal/insulator transition to a phase with gapless fermionic excitations in the presence of a non-zero hypercharge chemical potential and a non-zero strange quark mass. The transition points are compared with results derived previously via effective theories and with partially self-consistent NJL calculations. We find that the qualitative physics does not change, but that the transitions are slightly lower.Comment: 21 pages, ReVTeX4. Clarified discussion and minor change

Analytical and numerical evaluation of the Debye and Meissner masses in dense neutral three-flavor quark matter

We calculate the Debye and Meissner masses and investigate chromomagnetic instability associated with the gapless color superconducting phase changing the strange quark mass $M_s$ and the temperature $T$. Based on the analytical study, we develop a computational procedure to derive the screening masses numerically from curvatures of the thermodynamic potential. When the temperature is zero, from our numerical results for the Meissner masses, we find that instability occurs for $A_1$ and $A_2$ gluons entirely in the gapless color-flavor locked (gCFL) phase, while the Meissner masses are real for $A_4$, $A_5$, $A_6$, and $A_7$ until $M_s$ exceeds a certain value that is larger than the gCFL onset. We then handle mixing between color-diagonal gluons $A_3$, $A_8$, and photon $A_\gamma$, and clarify that, among three eigenvalues of the mass squared matrix, one remains positive, one is always zero because of an unbroken U(1)_\tilde{Q} symmetry, and one exhibits chromomagnetic instability in the gCFL region. We also examine the temperature effects that bring modifications into the Meissner masses. The instability found at large $M_s$ for $A_4$, $A_5$, $A_6$, and $A_7$ persists at finite $T$ into the $u$-quark color superconducting (uSC) phase which has $u$-$d$ and $s$-$u$ but no $d$-$s$ quark pairing and also into the two-flavor color superconducting (2SC) phase characterized by $u$-$d$ quark pairing only. The $A_1$ and $A_2$ instability also goes into the uSC phase, but the 2SC phase has no instability for $A_1$, $A_2$, and $A_3$. We map the unstable region for each gluon onto the phase diagram as a function of $M_s$ and $T$.Comment: 17 pages, 18 figure