Color Superconductivity
Ra´ulSantos
Nov. 8 2010
Ra´ulSantos () Color Superconductivity Nov. 8 2010 1 / 17 Outline
1 1. Introduction
2 2. Normal Superconductivity
3 3. QCD at finite Baryon density
4 4. Color Superconductivity in Neutron Stars
5 5. Summary
Ra´ulSantos () Color Superconductivity Nov. 8 2010 2 / 17 Introduction
QCD is asymptotically free at large energies (T , µ) T µ → Quark Gluon Plasma (QGP), Appropriate regime for pQCD, testeable with current experiments, RHIC and LHC. µ T → New phase, Condensation of quarks, forming a Color superconductor (CSC) Why it is important? Natural scenario in Neutron Stars, → Consequences of CSC in their evolution
Ra´ulSantos () Color Superconductivity Nov. 8 2010 3 / 17 Phase diagram in QCD
Figure: Different phases of nuclear matter (Schematic) [Alford]
Ra´ulSantos () Color Superconductivity Nov. 8 2010 4 / 17 Phenomenology of (usual) Superconductivity A superconductor can behave as if it had no measurable DC electrical resistivity & 2 years. A superconductor can behave as a perfect diamagnet → Meissner Effect. A superconductor usually behaves as if there were a gap in energy of width 2∆ centered about the Fermi energy, in the set of allowed one-electron levels. The energy gap ∆ increases in size as the temperature drops.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 5 / 17 BCS superconductivity revisited
Coupling between phonons and electrons can create an overall attractive interaction between fermions → Cooper pairs.
X g X † † H = n − c c c 0 c 0 (1) k kσ Ld k+q↑ −k↓ k +q↓ −k ↑ σk kk0 g P Defining the order parameter ∆ = Ld k hΩ|ck↑c−k↓|Ωi, and after some manipulations we can diagonalize the Hamiltonian, obtaining
X † † H = λk αk↑αk↑ − α−k↓α−k↓ (2) σk q 2 2 with λk = k + ∆ , and the self-consistency condition for ∆ gives
ωD −1 ∆ = ≈ ωD exp sinh 1/gν(EF ) gν(Ef )
Ra´ulSantos () Color Superconductivity Nov. 8 2010 6 / 17 Inevitability of Color Superconductivity
Why we expect a similar phenomena in QCD at high µ? Atractive interaction in the antitriplet channel.
A A Nc + 1 Nc + 1 TabTa0b0 = − (δabδa0b0 − δaa0 δbb0 ) + (δabδa0b0 + δaa0 δbb0 ). 4Nc 4Nc Effective interaction mediated by Instanton vaccum.
G T n ¯ n ¯T T n ¯ n ¯T L = − [(ψ Cτ2λAψ)(ψτ2λACψ ) + (ψ Cτ2λAγ5ψ)(ψCτ2λAγ5Cψ )] 16Nc (Nc − 1) G T n ¯ n ¯T + (ψ Cτ2λS σµν ψ)(ψτ2λS σµν Cψ ) 32Nc (Nc + 1)
Ra´ulSantos () Color Superconductivity Nov. 8 2010 7 / 17 But, who pairs with whom?
Structure of the condensate
When µ mu, md , ms we can treat them in equal footing.
aα bβ aα bβ ab αβρ hψiL (p)ψjL (−p)i = −hψiR (p)ψjR (−p)i = ∆(p) ijρ
Solution of the consistency equation which minimizes the energy. Local minimum of the ground state energy functional. Global minimum? not rigorously proved (yet). This pattern involves quarks of all 9 color-flavor combinations, providing a gap for all of them. This makes the energy of such a ground state lower than that of the other possibilities, which leave some quarks ungapped.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 8 / 17 Color Flavor Locked Phase
aα bβ ab αβρ ab α β α β hψi (p)ψj (−p)i ∝ ∆(p) ijρ = ∆(p) (δi δj − δj δi ).
Color and Flavor are ’Locked’ together. Breaks chiral symmetry (up to arbitrarily high densities). It is a Color superconductor (Meissner Effect). It is a superfluid. Structure of the symmetry breaking
[SU(3)c ] × SU(3)R × SU(3)L × U(1)B → SU(3)c+L+R × Z2
Ra´ulSantos () Color Superconductivity Nov. 8 2010 9 / 17 Gauge symmetry breaking and electromagnetism
[SU(3)c ] × SU(3)R × SU(3)L × U(1)B → SU(3)c+L+R × Z2
One of the generators of SU(3)L+R is the electric charge, which generates the U(1)Q gauge symmetry. This means
SU(3)L+R+c ⊃ [U(1)Q¯ ] which is unbroken and correspond to a simultaneous electromagnetic and color rotation. 7 gluons and one gluon-photon linear combination become massive via the Meissner effect. The mixing angle is g cos θ = . pg 2 + 4e2/3
Ra´ulSantos () Color Superconductivity Nov. 8 2010 10 / 17 Consecuences of Sym. Breaking.
g cos θ = , e g → θ ∼ 0. pg 2 + 4e2/3 Q¯ photon ∼ original photon with a small admixture of gluon. The Q¯ -electric and magnetic fields satisfy Maxwell’s equations with a dielectric constant and index of refraction
e2 cos θ2 µ 2 n = 1 + 2 . 9π ∆CFL CFL phase is a transparent insulator, Massive Gluons → Color Superconductor.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 11 / 17 Intermediate densities
What about µ below ms ?. 2 Color pairing is the most symmetrical form of nuclear matter.
α 5 β αβ3 hψi Cγ ψj i ∝ ∆2SC ij3
[SU(3)c ] × SU(2)R × SU(2)L × U(1)B × U(1)S ⊃ [U(1)Q ] | {z } → [SU(2)rg ] × SU(2)R × SU(2)L × U(1)B¯ × U(1)S ⊃ [U(1)Q¯ ]. | {z } 2SC quark matter is therefore a color superconductor but is neither a superfluid nor an electromagnetic superconductor.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 12 / 17 Color Superconductivity in Neutron Stars
Figure: Neutron Star expected structure
14 −3 ρ0 ∼ 2.8 × 10 g cm density of atomic nucleus.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 13 / 17 Mass- Radius Relation
Ra´ulSantos () Color Superconductivity Nov. 8 2010 14 / 17 Cooling
Neutrino Emissivity Ordinary dense matter (proton fraction higher than 0.1).
α µ 2 T 6 I ' (4 · 1025 erg cm−3 s−1) s . ν 0.5 500MeV 109K
Ordinary dense matter → proton fraction lower than 0.1.
2/3 8 II 20 −3 −1 n T ν ' (1.2 · 10 erg cm s ) × 9 n0 10 K CFL phase T 15 III ∝ ν 109K
Ra´ulSantos () Color Superconductivity Nov. 8 2010 15 / 17 Summary
High Density in nuclear matter → new phases of matter: Color Superconductivity.
µ ms , Color flavor locked phase
µ ∼ ms , 2 Flavor Color Superconductivity Effects in Neutron star evolution. Open problems Nuclear equation of state. Explicit determination of transition density. Lattice approach, new algorithms.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 16 / 17 References
M. G. Alford, A. Schmitt, K. Rajagopal and T. Schafer, “Color superconductivity in dense quark matter,” Rev. Mod. Phys. 80, 1455 (2008) [arXiv:0709.4635 [hep-ph]].
“The Phases of Quantum Chromodynamics: From Confinement to Extreme Environments”; John B. Kogut and Mikhail A. Stephanov. Cambridge University Press “Quark matter and the masses and radii of compact stars”; M. Alford, D. Blaschke, A. Drago, , Kl hn, T.2, G. Pagliara, J. Schaffner-Bielich, [astro-ph/0606524]
“Soft equations of state for neutron-star matter ruled out by EXO 0748 - 676”; F. Ozel,¨ Nature 441, 1115-1117.
“Hybrid stars that masquerade as neutron stars”; M. Alford, M. Braby , M. Paris, S. Reddy.
“High-Density QCD and Instantons”; R. Rapp, T. Schfer, E. V. Shuryak and M. Velkovsky, Annals of Physics Volume 280, Issue 1.
Ra´ulSantos () Color Superconductivity Nov. 8 2010 17 / 17