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

2. Outline Introduction Superconductivity: The BCS theory
Color superconductivity
2SC Phase
CFL Phase
Color superconductivity in compact stars
Summary
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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/ 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
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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/
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5. 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
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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:
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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
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8. Energy gap Ground state energy Minimization Gap parameter
Gap equation
Ground state Energy
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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/ v1
Cooper Pairs: Diquark Condensate
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10. 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:
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11. 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
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12. Weak Coupling Assuming small coupling Nambu-Gorkov spinors with
Inverse free Fermion propagator
Dyson-Schwinger equations
Fermion Part:
Gluon Part
nucl-th/ v1
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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/ v1
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14. 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:
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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
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16. 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
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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
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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<<Δ for all quasiparticles
Small neutrino emissivity
No electromagnetic superconductivity
Pressure
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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)
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20. 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
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21. Color Superconductivity in Compact Stars
Requirements:
Charge neutrality
Vanishing electric charge density
β-equilibrium:
All weak processes should be in equilibrium
Astro-ph/ v2
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22. 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
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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/ v1
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24. Gapless 2SC phase Three phases where Normal matter: Gapless mode:
Gapped mode:
nucl-th/ v1
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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/
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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,
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27. Meissner masses in the gCFL-Phase
Phys. Lett. B 605,
Phys. Lett. B 605,
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?
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28. Phase Diagram GD≈3/4GS hep-ph/0503184v2
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29. Phase Diagram GD≈GS hep-ph/0503184v2
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30. Quark masses (T=0) GD≈3/4 GS hep-ph/0503184v2
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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
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32. References Two Lectures on color superconductivity: nucl-th/0410092v1
Color superconductivity in dense quark matter : hep-ph/ v2
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, (1999)
Charge neutrality effects on two-flavor color superconductivity: Phys. Rev. D, (2003)
Screening masses in neutral two-flavor color superconductor: hep-ph/
Gapless Color-Flavor-Locked Quark Matter: Phys. Rev. Lett. 92, (2004)
Meissner masses in the gCFL phase of QCD, Phys. Lett. B 605, (2005)
The phase diagram of neutral quark matter: Self consistent treatment of quark masses: hep-ph/ v2
Theory of Superconductivity: Phys. Rev. 108, (1957)
Strange Quark Matter and Compact Stars: Astro-ph/ v2
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