Color Superconductivity In High Density
Quark Matters
Haiming Deng
Physics and Astronomy, Stony Brook University
Outline
1.  Conventional Superconductivity in metal (BCS)
2.  Comparing free electron gas and high density
quark matter
3.  Basics of Quantum Chromodynamics, color
interaction
4.  Color coupling of quarks in Fermi-sea
5.  Other coupling and phase diagram
Conventional Superconductivity
-BCS theory
Ø  Two electron with opposite spin
are coupled into a cooper pair
Ø  Spin 0 -> bosonic, able to
condensate into single state
2-
Forming cooper pair require
o  Free particles
o  Attractive potential (no matter
how weak)
2+
Normal Metallic state
(Pauli exclusive principle)
Conduction band
Ef
Overlap
Superconducting state
Condensate into
bosonic state
gap ∆E
Valance band
Lattice kinetic energy goes down as T lowering…
Energy barrier (∆E) protects
SC state from perturbations
Ø  Stay paired
Ø  No scattering from lattices
(energy not enough to kick
the state)
Electron and Quark Fermi liquid
Electron at Fermi surface
Quarks
q  Carry color charge
§  Red, green or blue
q  Have six flavors
§  u, d, s, c, t and b
Fermi Sea of Quarks
q  High density, “sea of free
electrons”
q  0.5MeV
q  Carry electric charge
q  Carry spin ½
Ø  Fermions
Gravitational, electromagnetic
interaction with each other
(High density & low temperature, < 10^12K)
q  Almost “free quarks”
q  c, t and b short life, very
massive excluded in low T
limit
q  u, d, s has small to moderate
mass
q  Spin ½ fermions, carry
fractional electric charge as
well
Interact with four fundamental
forces
Ø  Can create various
attractive potential!
Interactions Between Quarks
The dominating interaction is the “color” interaction,
described by SU(3) non abelian gauge theory
(Quantum Chromodynamics)
Properties of QCD:
𝑉(𝑟)=𝑘𝑟
S. Bethke, E. Tests of Asymptotic Freedom, arXiv:hep-ex/0606035
Ø  Short distance 𝑟≪1 𝑓𝑚, Asymptotically free.
o  Coupling is weak.
Ø  Long distance, 𝑟≪1 𝑓𝑚, color confinement.
o  Interaction strength increases as quarks
separate; Energy to separate a quark from other
quark is high enough to create another quark
pair
𝛼=0.1187±0.0010
Gluon and Color Quark Interactions
8 type of color gluon state
r,g,b = red, green, blue
¯ˉ𝑟 , ¯ˉ𝑔 , ¯ˉ𝑏 = Anti- (red, green, blue)
Bouncing of gluon and
Conservation of color charge
q  Blue quark emit a blue-antigreen
gluon to green quark.
o  Blue quark à green
o  Green quark à blue
Ø  Color and anti-color cancel each
other
Ø  Color Charge is conservated
q  Diff. types of gluon exchange between
color quarks
Picture source: Wikipedia
BCS for Quark Fermi liquid: Color Superconductivity
Normal state
-High density, low temperature
Quark matter form bands like ordinary lattice
Ø  Linear energy-momentum dispersion à Zero
effective mass
Ø  Behave like “free”
o  T decreases or
o  u increases
For a attractive quark-quark interaction (QCD)
o  Single gluon exchange
o  Color antisymmetric (i.e red-blue, red-green)
Ø  Pairing of two quarks, spin 0, ±1 à Bosonic,
Cooper Pairs!
Color superconducting state
o  Energy gap
o  Linear energy-momentum dispersion (otherwise
gravitational force could create resistant)
o  Similar to conventional superconductivity
D. Ivanenko and D.F. Kurdgelaidze, Astrofiz. 1, 479 (1965); Lett.
Nuovo Cim
Color superconducting phases
Quark Cooper pair: <​𝑄↓𝑖𝑙↑𝛼 ​𝑄↓𝑗𝑚↑𝛽 >
o  Color 𝛼,𝛽=𝑟𝑒𝑑,𝑔𝑟𝑒𝑒𝑛,𝑏𝑙𝑢𝑒
o  Flavor 𝑖, 𝑗=𝑢,𝑑,𝑠
o  Spin 𝑙, 𝑚=↑,↓
Special pairing
Color-Flavor-Locked (CFL) cooper pair
à9x9 possible BCS pairing patterns
q  C-C, C-F, C-S, C-F-S coupling
q  Color + flavor or spin
Attractive potential requirement:
Ø  Color antisymmetric
Ø  Spin antisymmetric
Ø  Flavor antisymmetric
v  Spin coupling is too weak compare
to other phases
à Pairing with different flavor is
flavored
𝛼,𝛽=𝑟𝑒𝑑, 𝑔𝑟𝑒𝑒𝑛, 𝑏𝑙𝑢𝑒 (1,2,3
𝑖,𝑗=𝑢,𝑑,𝑠(1,2,3)
For any two choice of quarks
their color are locked to
either + or –
o  i.e ud or -ud
Ø  Spontaneous chiral
symmetry breaking
Ø  Superfluid
Ø  EM insulator
Color-flavor-locked (CFL) pairing
Color Superconductivity Phase diagram of quark matter
Ø  Low density quark matter form nuclear liquid for
sufficiently low T. Weak coupling allowed (i.e color-spin)
Ø  Intermediate density ( i.e neutron stars) à Quark Fermi
liquid. color coupling dominates à many possible way of
forming Cooper pairs
Ø  Highest density à asymptotic freedom between quarks
flavor color-flavor-locked pairing
Summary
q  BCS theory for Conventional superconductivity
q  Quarks Fermi liquid properties and analogy to
electron Fermi liquid
q  Color integrations between quarks(QCD)
q  Color Superconductivity and their phases
q  Color-flavor-locked pairing
q  Phase diagram of Quark matter
Thank you for your attentions…
Reference
1.  R. Nave. "Confinement of Quarks". HyperPhysics. Georgia State University,
Department of Physics and Astronomy (2008)
2.  T. Muta (2009).
Foundations of quantum chromodynamics: an introduction to perturbative
methods in gauge theories (3rd ed.)
3.  M. Alford, K. Rajagopal, T. Schafer, A. Schmitt, “Superconducting Quarks:
Condensed Matter in the Heavens”arXiv:0709.4635
4.  Cooper, Leon (November 1956). "Bound Electron Pairs in a Degenerate
Fermi Gas". Physical Review 104 (4)