Physics Reports 407 (2005) 205–376 www.elsevier.com/locate/physrep NJL-model analysis of dense quark matter Michael Buballa∗ Institut für Kernphysik, Technische Universität Darmstadt, Schlossgartenstr. 9, D-64289 Darmstadt, Germany Accepted 15 November 2004 Editor W. Weise Available online 20 January 2005 Abstract Investigations of deconfined quark matter within NJL-type models are reviewed, focusing on the regime of low temperatures and “moderate” densities, which is not accessible by perturbative QCD. Central issue is the interplay between chiral symmetry restoration and the formation of color superconducting phases. In order to lay a solid ground for this analysis, we begin with a rather detailed discussion of two- and three-flavor NJLmodels and their phase structure, neglecting the possibility of diquark pairing in a first step. An important aspect of this part is a comparison with the MIT bag model. The NJLmodel is also applied to investigate the possibility of absolutely stable strange quark matter. In the next step the formalism is extended to include diquark condensates. We discuss the role and mutual influence of several conventional and less conventional quark–antiquark and diquark condensates. As a particularly interesting example, we analyze a spin-1 diquark condensate as a possible pairing channel for those quarks which are left over from the standard spin-0 condensate. For three-flavor systems, we find that a self-consistent calculation of the strange quark mass, together with the diquark condensates, is crucial for a realistic description of the 2SC–CFLphase transition. We also study the effect of neutrality constraints which are of relevance for compact stars. Both, homogeneous and mixed, neutral phases are constructed. Although neutrality constraints generally tend to disfavor the 2SC phase we find that this phase is again stabilized by the large values of the dynamical strange quark mass which follow from the self-consistent treatment. Finally, we combine our solutions with existing hadronic equations of state to investigate the existence of quark matter cores in neutron stars. © 2004 Published by Elsevier B.V. PACS: 12.39.−Ki; 12.38.AW; 11.30.Qc; 25.75.Nq; 26.60.+c Keywords: Dense quark matter; QCD phase diagram; Color superconductivity; Compact stars ∗ Tel.: +49 6151 16 3272; fax: +49 6151 16 6076. E-mail address: [email protected] (M. Buballa). 0370-1573/$ - see front matter © 2004 Published by Elsevier B.V. doi:10.1016/j.physrep.2004.11.004 206 M. Buballa / Physics Reports 407 (2005) 205–376 Contents 1. Introduction .........................................................................................207 1.1. Basics of QCD ..................................................................................209 1.2. The QCD phase diagram ..........................................................................212 1.3. Scope and outline of this report.....................................................................215 2. NJL-model description of color non-superconducting two-flavor quark matter ..................................217 2.1. MIT bag model ..................................................................................217 2.1.1. Hadron properties .........................................................................218 2.1.2. Thermodynamics .........................................................................220 2.1.3. Equation of state at zero temperature .........................................................223 2.2. Nambu–Jona-Lasinio model in vacuum ..............................................................225 2.2.1. Constituent quarks and mesons ..............................................................226 2.2.2. Parameter fit .............................................................................229 2.3. Non-zero densities and temperatures ................................................................231 2.3.1. Thermodynamic potential ..................................................................232 2.3.2. Chiral phase transition and stable quark matter solutions at zero temperature ........................236 2.3.3. Comparison with the bag model .............................................................242 2.3.4. Phase diagram ............................................................................246 2.3.5. 1/Nc corrections ..........................................................................248 2.4. Asymmetric matter ...............................................................................250 3. Three-flavor systems ..................................................................................255 3.1. Vacuum properties ...............................................................................256 3.1.1. Lagrangian...............................................................................256 3.1.2. Gap equation and meson spectrum ...........................................................257 3.2. Thermodynamics ................................................................................259 3.2.1. Formalism ...............................................................................259 3.2.2. Quark masses and effective bag constants at non-zero density ....................................260 3.3. Stability of strange quark matter ....................................................................263 3.3.1. The strange quark matter hypothesis .........................................................263 3.3.2. NJL-model analysis .......................................................................265 4. Two-flavor color superconductors .......................................................................270 4.1. Diquark condensates .............................................................................271 4.1.1. Pauli principle ............................................................................271 4.1.2. Scalar color-antitriplet diquark condensate ....................................................272 4.2. Interaction ......................................................................................275 4.2.1. Asymptotic densities ......................................................................275 4.2.2. NJL-type interactions ......................................................................277 4.3. Interplay with other condensates ....................................................................280 4.3.1. Hartree–Fock approach ....................................................................281 4.3.2. Thermodynamic potential ..................................................................284 4.3.3. Dispersion laws and gapless solutions ........................................................287 4.3.4. Numerical results .........................................................................289 4.4. Spin-1 pairing of the “blue” quarks .................................................................295 4.4.1. Condensation pattern and symmetries ........................................................295 4.4.2. Gap equation .............................................................................297 4.4.3. Thermal behavior .........................................................................300 4.4.4. Spin-1 pairing in the red–green sector ........................................................303 4.4.5. MJ =±1 ...............................................................................305 5. Three-flavor color superconductors ......................................................................306 5.1. Three degenerate flavors ..........................................................................306 M. Buballa / Physics Reports 407 (2005) 205–376 207 5.1.1. Condensation patterns .....................................................................306 5.1.2. Properties of the CFLphase ................................................................307 5.2. Realistic strange quark masses .....................................................................309 5.2.1. Formalism ...............................................................................311 5.2.2. Numerical results without ’t Hooft interaction .................................................314 5.2.3. Influence of the ’t Hooft interaction ..........................................................318 6. Neutral quark matter ..................................................................................322 6.1. Formalism ......................................................................................323 6.1.1. Conserved charges and chemical potentials ....................................................323 6.1.2. Thermodynamic potential for non-uniform quark chemical potentials ..............................324 6.2. Numerical results ................................................................................326 6.2.1. Equal chemical potentials ..................................................................326 6.2.2. Phase structure ...........................................................................328 6.2.3. Homogeneous neutral solutions .............................................................333 6.3. Mixed phases ...................................................................................336 6.3.1. Formalism ...............................................................................336 6.3.2. Numerical results .........................................................................337 6.3.3. Surface and Coulomb effects ................................................................340