Color Superconductivity in dense quark matter

1. ColorSuperconductivity in densequarkmatter „The Condensed Matter Physics of QCD“ Markus Fasel 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 1

2. Outline • Introduction • Superconductivity: The BCS theory • Color superconductivity • 2SC Phase • CFL Phase • Color superconductivity in compact stars • Summary 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 2

3. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 3

4. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 4

5. Cooper Pairs • 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 5

6. 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: 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 6

7. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 7

8. Energy gap • Ground state energy Minimization • Gap parameter • Gap equation • Ground state Energy 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 8

9. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 9

10. Diquark Condensate: Diquark Condensate • Requirement • 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 10

11. 2SC-Phase • 2 Flavors  Symmetry reduces to SU(2) • 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 11

12. Weak Coupling • Assuming small coupling • Nambu-Gorkov spinors with • Inverse free Fermion propagator with • Dyson-Schwinger equations • Fermion Part: • Gluon Part nucl-th/0410092v1 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 12

13. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 13

14. NJL-Model • Mean Field Model • Assumption: • Gluon exchange described by four-quark interaction • Lagrangian • Free Lagragian • Quark-Antiquark interaction • Coupling GS • Quark-Quark Interaction • Coupling GD • Gap Equation: • Solution: 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 14

15. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 15

16. CFL Phase • Diquark Condensate • Three Flavours: • SU(3)  Gell-Mann matrices • A{2,5,7} • Three Colors: • SU(3)  Gell-Mann matrices • 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 16

17. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 17

18. 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<<Δfor all quasiparticles • Small neutrino emissivity • No electromagnetic superconductivity • Pressure 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 18

19. 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) 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 19

20. 2SC-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 Symmetries 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 20

21. Color Superconductivity in Compact Stars • Requirements: Charge neutrality Vanishing electric charge density β-equilibrium: All weak processes should be in equilibrium Astro-ph/0407155v2 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 21

22. Model for charge neutral color superconducting phases • Effective Model (NJL-Model) chosen • Chemical Potential: μ • Baryonchemicalpotential • Electricchemicalpotential • Color chemical potential • Grand Canonical Potential • Charge neutrality • Gap equations and and 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 22

23. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 23

24. Gapless 2SC phase • Three phases • Normal matter: • Gapless mode: • Gapped mode: where nucl-th/0410092v1 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 24

25. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 25

26. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 26

27. Meissner masses in the gCFL-Phase • 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? a=8 a=3 a=1,2 CFL gCFL a=5,6 CFL gCFL a=4,5 Phys. Lett. B 605, 362-368 Phys. Lett. B 605, 362-368 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 27

28. Phase Diagram GD≈3/4GS hep-ph/0503184v2 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 28

29. Phase Diagram GD≈GS hep-ph/0503184v2 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 29

30. Quark masses (T=0) GD≈3/4 GS hep-ph/0503184v2 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 30

31. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 31

32. 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 3. Januar 2020 | Relativistic Heavy Ion Physics Seminar | 32