Color Superconductivity in dense quark matter „The Condensed Matter Physics of QCD“ Markus Fasel 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 1 28. November 2018 |

Outline Introduction Superconductivity: The BCS theory Color superconductivity 2SC Phase CFL Phase Color superconductivity in compact stars Summary 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 2

QCD QCD-Lagrangian: : Gauge Field : Field strength : Color symmetry transformation g : Coupling constant Asymptotic freedom Dependence of the coupling constant on the momentum transfer Small scales: quarks get asymptotically free hep-ph/0211012 v1 QCD Symmetry: SU(3)c X SU(3)L X SU(3)R X U(1)B SU(3)C: local SU(3)L: global SU(3)R: global 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 3

Phase Diagram Several phases for the QCD Hadronic Phase Quark-Gluon Plasma Color-Superconductivity Restoration of chiral symmetry in the Quark-Gluon Plasma Confinement in the Hadronic Phase hep-ph/0812.283 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 4

Cooper Pairs Definition Cooper pairs: Opposite sign momentum Interaction between Electrons repulsive Electron moving through a lattice causes vibration Second Electron in the road of the first one sees attractive interaction caused by the lattice vibration Phonon exchange Repulsive Interaction screened by attractive Interaction Definition Cooper pairs: Opposite sign momentum Opposite spin charge 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 5

Effective Electron-Electron Interaction Hamilton Operator for the Electron-Phonon-Interaction T(q): Matrix Element of the Electron-Phonon-Interaction b+,b : Creation and annihilation operators for the phonons a+, a: Creation and annihilation operators for the electrons Model Hamilton Operator Transformation leads to a Hamilton operator for the effective electron-electron interaction Interaction Repulsive: Attractive: 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 6

BCS Ground State Variational approach for the calculation of the ground state BCS Ground State Ansatz Model Hamilton Operator with Calculation of the Ground state energy by variational approximation uk and vk variational parameter See Talk from S. Huber 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 7

Energy gap Ground state energy Minimization Gap parameter Gap equation Ground state Energy 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 8

Color-Superconductivity Screening of the repulsive Coulomb interaction One-Gluon Exchange Color antitriplet Additional degrees of Freedom Color Charge Flavor Variety of superconducting phases 2SC Phase CFL Phase … nucl-th/0410092v1 Cooper Pairs: Diquark Condensate 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 9

Diquark Condensate Requirement Gell-Mann-Matrices Diquark Condensate: Pauli-Principle: Operator totally antisymmetric Dirac Space: Charge-Conjugate-Operator Flavor Space: Nf = 2: Pauli-Matrices Anti-symmetric: A = 2 Nf= 3: Gell-Mann-Matrices Anti-symmetric: A∈{2,5,7} Color Space: Gell-Mann Matrices Gell-Mann-Matrices Diquark Condensate: 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 10

2SC-Phase 2 Flavors  Symmetry reduces to SU(2) 3 Colors  SU(3) Pauli matrices A Anti-symmetric: A = 2 3 Colors  SU(3) Gell-Mann matrices B Anti-symmetric: B{2,5,7} Diquark-Condensate i,j: Flavor space a,b: Color space ->Pairing of Red and Green quarks to Anti-Blue Up and Down Spin-Up and Spin-Down u d u d 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 11

Weak Coupling Assuming small coupling Nambu-Gorkov spinors with Inverse free Fermion propagator Dyson-Schwinger equations Fermion Part: Gluon Part nucl-th/0410092v1 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 12

Gap Equation Ansatz for the Quark Propagator with off-diagonal elements due to Cooper-pairing with off-diagonal terms and Ansatz for self energy Solution of the Dyson Schwinger equation leads to the gap equation Solution Gaps in quasiparticle excitation spectrum Dispersion Relation nucl-th/0410092v1 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 13

NJL-Model Mean Field Model Assumption: Lagrangian Gluon exchange described by four-quark interaction Lagrangian Free Lagragian Quark-Antiquark interaction Coupling GS Quark-Quark Interaction Coupling GD Gap Equation: Solution: 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 14

Properties of the 2SC Phase Blue particles Gapless quasiparticles in the low energy spectrum Dominant contribution to Specific Heat Electrical Conductivity Heat Conductivity Large Neutrino emissivity Quasiparticle pairs Contribution to the transport of thermodynamic quantities suppressed by exp(-Δ/T) at T<<Δ Pressure Pauli Pressure Diquark pairing 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 15

CFL Phase Diquark Condensate Three Flavours: Three Colors: SU(3)  Gell-Mann matrices A{2,5,7} Three Colors: B{2,5,7} Non-vanishing condensates: s22,s55,s77 Only special combinations of flavors and color pair u d s22 d s s55 s u s77 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 16

Energy gap NJL-Model describes interaction General Ansatz: Assumptions on the strange mass: ms = 0: ms= ∞:  2SC phase Gap Parameter for the CFL phase Results for ΔCFL: ~10-100MeV 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 17

Properties of the CFL Phase No gapless quasiparticles in low energy spectrum Contribution to transport of thermodynamic quantities suppressed by exp(-Δ/T) at T<<Δ for all quasiparticles Small neutrino emissivity No electromagnetic superconductivity Pressure 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 18

Symmetries Goldstone Theorem: For each generator of a broken global symmetry exists a massless Spin-0 (pseudoscalar) Boson Color-Meissner-Effect: Gauge bosons become massive due to coupling to the Higgs field when the gauge symmetry is broken (Higgs-Anderson mechanism) 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 19

Symmetries 2SC-Phase  -conductor CFL-Phase Color symmetry: reduces to SU(2) 5 of 8 gluons become massive Chiral symmetry: restored Unbroken symmetry: Rotated Baryon Number Rotated Charge  -conductor CFL-Phase Color and Chiral symmetry SU(3)C X SU(3)L X SU(3)R X U(1)B reduces to SU(3)C+R+L X Z2  “Color-Flavor-Locking” 8 massive gluons Flavor symmetry: broken 8 Goldstone Bosons Baryon number symmetry breaking Additional Goldstone Boson Superfluid with respect to the Baryon number Rotated Charge  -insulator 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 20

Color Superconductivity in Compact Stars Requirements: Charge neutrality Vanishing electric charge density β-equilibrium: All weak processes should be in equilibrium Astro-ph/0407155v2 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 21

Model for charge neutral color superconducting phases Effective Model (NJL-Model) chosen Chemical Potential: μ Baryon chemical potential Electric chemical potential Color chemical potential Grand Canonical Potential Charge neutrality Gap equations and and 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 22

Dispersion Relation Definitions Mean Chemical potential Deviation of the chem. Potential Implications on the dispersion relation 2 Gaps in quasiparticle spectrum Case: normal 2SC phase and nucl-th/0410092v1 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 23

Gapless 2SC phase Three phases where Normal matter: Gapless mode: Gapped mode: nucl-th/0410092v1 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 24

Meissner effect in the g2SC phase Meissner mass of the gluons Generators of the unbroken SU(2) stay massless Imaginary Meissner mass Of the 8th gluon only in the gapless 2SC phase Of the gluons 3-7 also in the gapped phase Instable Ground State Gluon condensation? x ε {3,4,5,6} x=8 Gluons 0-2 Gluons 3-6 Gluon 7 g2SC hep-ph/04082268 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 25

Gapless CFL Phase Mass of the strange quark Assumption 100MeV ≤ ms ≤ 500MeV Charge neutrality condition Automatically fulfilled by CFL phase (nu ≈ nd ≈ ns) Color charge neutrality Compensation term to avoid violation Gapless modes for Breakup of the degeneracy in Δ CFL gCFL Phys. Rev. Lett. 92, 222001 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 26

Meissner masses in the gCFL-Phase Phys. Lett. B 605, 362-368 Phys. Lett. B 605, 362-368 Partial breaking of the degeneracy of the Meissner masses at the onset of the gCFL Phase Masses of gluons 1 and 2 become imaginary Instability  Gluon Condensation? 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 27

Phase Diagram GD≈3/4GS hep-ph/0503184v2 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 28

Phase Diagram GD≈GS hep-ph/0503184v2 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 29

Quark masses (T=0) GD≈3/4 GS hep-ph/0503184v2 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 30

Conclusions Color superconductivity possible due to attractive 1-gluon exchange Cooper pairing described by the diquark condensate Energy gap in the quasi-particle spectrum Interaction model by effective Mean-Field model Compact stars Neutrality conditions Charge neutrality Color charge neutrality β-Equilibrium Phases with gapless modes in the quasi-particle energy spectrum Imaginary Meissner masses Chromomagnetic instable 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 31

References Two Lectures on color superconductivity: nucl-th/0410092v1 Color superconductivity in dense quark matter : hep-ph/08122831v2 Color superconductivity in dense quark matter: Czech. J. Phys. 55, 521 – 539 (2005) Color-flavor locking and chiral symmetry breaking in high density QCD: Nucl. Phys. B 537, 443-458 (1999) Charge neutrality effects on two-flavor color superconductivity: Phys. Rev. D, 065015 (2003) Screening masses in neutral two-flavor color superconductor: hep-ph/04082268 Gapless Color-Flavor-Locked Quark Matter: Phys. Rev. Lett. 92, 222001 (2004) Meissner masses in the gCFL phase of QCD, Phys. Lett. B 605, 362-368 (2005) The phase diagram of neutral quark matter: Self consistent treatment of quark masses: hep-ph/0503184v2 Theory of Superconductivity: Phys. Rev. 108, 1175-1204 (1957) Strange Quark Matter and Compact Stars: Astro-ph/0407155v2 28. November 2018 | Relativistic Heavy Ion Physics Seminar | 32