Superconducting phases of quark matter Igor A. Shovkovy Frankfurt Institute for Advanced Studies Johann W. Goethe-Universität Max-von-Laue-Str. 1 60438 Frankfurt am Main, Germany
Outline I. Introduction into color superconductivity Phase diagram of baryonic matter Dense matter in nature Color superconductivity Color superconductivity of 2 quark flavors (2SC) Color superconductivity of 3 quark flavors (CFL) Spin-1 color superconductivity (1 quark flavor) II. Color superconductivity in stars Neutrality vs. color superconductivity Unconventional pairing in color superconductors Current status and Summary Review: Igor A. Shovkovy, Two lectures on color superconductivity nucl-th/0410091, Found. Phys. 35 (2005) 1309-1358
T µ phase diagram of QCD
Dense baryonic matter in Nature Compact (neutron) stars Radius: R 10 km Mass: 1.25M < M< 2M? Core temperature: 10 kev< T< 10 MeV Surface magnetic field: 10 8 G< B< 10 14 G Rotational period: 1.6 ms < P < 12 s What is the state of matter at the highest stellar densities, ρ c > 5ρ 0?
hyperon star Possible phases of matter inside stars strange star quark hybrid star s u p e r c o n,p,e, µ n d color superconducting (non )strange quark matter (u, d and s quarks) (g)2sc 2SC+s Σ,Λ,Ξ, u,d,s normal quark matter H (g)cfl CFL K + CFL K 0 CFL π 0 N+e N+e+n u c t K i n,p,e, µ n π g p r o t o n traditional neutron star n superfluid s R ~ 10 km crust neutron star with pion condensate Fe 10 6 g/cm 3 10 11 10 14 Hydrogen/He atmosphere nucleon star g/cm 3 g/cm 3 [figure from F. Weber, astro-ph/0407155 (modified)]
Very dense baryonic matter Baryons at high density quark matter 0.5 α s (Q) 0.4 Data Theory Deep Inelastic Scattering e + e - Annihilation Hadron Collisions Heavy Quarkonia NLO NNLO Lattice 0.3 Λ(5) MS α s(μ Z) { 245 MeV 0.1210 QCD O(α 4 211 MeV 0.1183 s) 181 MeV 0.1156 Asymptotic freedom: α s (µ) 1 µ Λ QCD [Gross&Wilczek; Politzer, 73] Weakly interacting regime [Collins&Perry, 75] 0.2 0.1 1 10 100 Q [GeV] Note: realistic densities in stars may not be sufficiently large: ρ< 10ρ 0, where ρ 0 0.15 fm 3 µ< 500 MeV
Ground state of dense quark matter Noninteracting quarks: k z (i) Deconfined quarks (µ Λ QCD ) (ii) Pauli principle (s = 1 2 ) } k y (i) Effective models (µ> Λ QCD ) Interacting quarks: (ii) One-gluon exchange (µ Λ QCD ) p p k p k k = 3 + 6 a Attractive Repulsive s k x Cooper instability Color superconductivity ( ΨC ) α i γ 5Ψ β j 0
N f = 2 color superconductivity (2SC) Simplest case, 2SC phase [Barrois, 78; Bailin&Love, 84] N f = 2, i.e., up and down quarks N c = 3, i.e., red, green and blue up d p = u q d q =0 Gluon-exchange interaction: p k Repulsive u p F q F u p k p k = 3 + 6 a Attractive s d q F p F d Cooper pairs: ( ) 3 ( ) 0 ( u,d d,u )
Diquark condensate: Symmetries of 2SC state ( ΨC ) α i γ 5Ψ β j ε3αβ ǫ ij [Alford et al, hep-ph/9711395], [Rapp et al, hep-ph/9711396] baryon number U(1) B Ũ(1) B with B = B 2 3 T 8 (quark matter is not superfluid) gauge symmetry U(1) em Ũ(1) em with Q = Q 1 3 T 8 (there is no Meissner effect) chiral SU(2) L SU(2) R intact approximate axial U(1) A is broken 1 pseudo-ng boson color gauge symmetry SU(3) c SU(2) c (Higgs mechanism)
N f = 3 color superconductivity Cooper pair: ( ) 3 ( ) J=0 ( u,d d,u ) 3 Diquark condensate [Alford, Rajagopal, & Wilczek, Nucl. Phys. B 537 (1999) 443]: ( ΨC L ) α i (Ψ L) β j = ( ΨC R ) α i (Ψ R) β j I ε αβi ǫ iji The ground state symmetry is the same as in hypernuclear matter, i.e., no phase transition between hadronic and quark matter is required! Hadron gas Quark gluon plasma 2SC Hypernuclear matter crossover CFL
Symmetries of CFL ground state chiral SU(3) L SU(3) R is broken down to SU(3) L+R+c 8 (pseudo-)ng bosons, i.e., π 0, π ±, K ±, K 0, K 0, η (almost like in vacuum QCD) baryon number U(1) B is broken 1 NG boson (φ) (quark matter is superfluid) gauge symmetry U(1) em Ũ(1) em with Q = Q + 2 3 T 8 (there is no Meissner effect) approximate axial U(1) A is broken 1 pseudo-ng boson (η ) color gauge symmetry SU(3) c is broken by Anderson-Higgs mechanism 8 massive gluons
N f = 1 color superconductivity Cooper pair: ( ) 3 J=1 Diquark condensate: ( Ψ C) α γ5 Ψ β ε αβc (ˆkδ cδ sinθ + γ ( δ ) k) cosθ [Iwasaki & Iwado, 1995], [Schäfer, hep-ph/0006034], [Alford et al, hep-ph/0210106] Many possibilities, e.g., see [A.Schmitt, nucl-th/0412033]: Color-spin-locked phase: cδ = δ cδ largest pressure (?) Planar phase: cδ = δ cδ δ c3 δ δ3 Polar phase: cδ = δ c3 δ δ3 A-phase: cδ = δ c3 (δ δ1 + iδ δ2 ) unusual neutrino emission Many similarities with superfluidity in 3 He...
Color superconductivity inside stars Matter in the bulk of a star must be (i) in β-equilibrium: µ d = µ u + µ e, i.e., the weak processes u + e d + ν & d u + e + ν? have equal rates; (ii) electrically and color neutral: n el Q = 0, ncolor Q = 0 If n Q 0, the Coulomb energy is ( E Coulomb n 2 QR 5 M c 2 n Q 10 15 e/fm 3 ) 2 ( ) 5 R 1 km e.g., if 10 2 < n Q < 10 1 e/fm 3 E 2SC Coulomb M c 2
Unconventional Cooper pairing, N f = 2 The best Cooper pairing occurs when µ d µ u, i.e., n d µ3 d π 2 µ3 u π 2 n u Neutral matter, however, appears when n d 2n u There are no enough electrons in β equilibrium to help... µ e = µ d µ u 2 1/3 µ u µ u 1 4 µ u n e 1 192 n u n u Therefore, Cooper pairing is unavoidably distorted by the mismatch parameter: u p q u δµ pdown F 2 p up F = µ e 2 0 d q p d What happens then?
Gapless 2SC phase Competition: δµ vs. 0 (where 0 is the gap at δµ = 0) The winner is determined by the diquark coupling strength [Shovkovy&Huang, Phys. Lett. B 564 (2003) 205] 1. δµ> 0 the mismatch does not allow Cooper pairing: normal phase is the ground state 2. δµ< 1 2 0 coupling is strong enough to win over the mismatch: 2SC is the ground state 3. 1 2 0 < δµ < 0 regime of intermediate coupling strength: the ground state is the gapless 2SC phase
Quasiparticle spectrum in g2sc phase Intermediate coupling [Huang & Shovkovy, Nucl. Phys. A 729 (2003) 835] E (p) d E (p) u + p p δµ+ u δµ δµ+ δµ d δµ- µ ur µ - µ _ µ + µ dg The energy gaps in the quasiparticle spectra are 0 & + δµ
Chromomagnetic instability Abnormal results for gluon screening masses [Huang & Shovkovy, Phys. Rev. D 70 (2004) 051501(R)]: m 2 2 D m g 8 6 4 2 1 2 3 4 Μ m 2 2 M m g 0.4 0.2 0.2 0.4 0.6 2 1 2 3 4 Μ m M 2 0 A = 1, 2, 3 red solid line A = 4, 5, 6, 7 green long-dash line A = 8 blue short-dash line
½Ô ¾ Æ ½ Ô ¾ Tachyons as the origin of instability Collective modes with quantum numbers of gluons [Gorbar et al, hep-ph/0602221] 4-7th: p 2 0 = m 2 + v 2 p 2 m 2 > 0 for > 2δµ ÈË Ö Ö ÔÐ Ñ ÒØ Å ¾ ¾ 1 0.8 0.6 0.4 0.2 0 (p 0 / ) 2 0.04 0.02 0-0.02 Æ ¼ Æ ¼ 0 0.2 0.4 0.6 0.8 1 (p/ ) 2 m 2 < 0 for < 2δµ -0.2 0 0.5 1 1.5 2 8th: p 2 0 = v 2 p 2 v 2 < 0 for < δµ v 2 0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 Æ magnetic mode 1.2 1.4 1.6 1.8 2 Μ
N f = 2 + 1 color superconductivity, 0 < m s < Fermi momentum of strange quarks is lowered: E(k) µ k s F µ m2 s 2µ Then, the ground state is defined by: (?) only condensates of same flavor (spin-1 channel) m s k F s ~ µ m s 2 2µ k F ~ µ k (?) only superconductivity of up and down quarks (2SC or g2sc) (?) gapless CFL phase ( yet unknown stabilization mechanism) [Alford, Kouvaris & Rajagopal, Phys. Rev. Lett. 92 (2004) 222001] (?) crystalline pairing (nonzero momentum pairing, LOFF) [Alford, Bowers & Rajagopal, Phys. Rev. D 63 (2001) 074016] (?) P-wave kaon condensates [Schäfer, Phys. Rev. Lett. 96 (2006) 012305]
Crystalline phase (LOFF) [Alford, Bowers & Rajagopal, Phys. Rev. D 63 (2001) 074016] Cooper pairs with nonzero momenta: p z p-q = µ d p z B d µ d µ u 2 q p x p x B u (a) p+q = µ u (b) [Bowers, hep-ph/0305301]
Baryon spectrum in CFL phase [Kryjevski & Schafer, hep-ph/0407329] P-wave kaon condensation Within the framework of effective theory: 80 Ò ¼ ¼ P-wave meson condensation Å Î 60 40 20 20 40 60 80 100 Ô Ô Ò ¼ ¼ Ò ¼ ¼ ÈË Ö Ö ÔÐ Ñ ÒØ 0.0005 0.0004 0.0003 0.0002 0.0001-0.0001-0.0002 Æ Å ¾ ¾ 0.05 0.1 0.15 0.2 Ñ ¾ ¾Ô µ Å Î [Kryjevski, hep-ph/0508180], [Schafer, hep-ph/0508190] See also [Son & Stephanov, cond-mat/0507586] No instabilities
Observational data as a tool Cooling of neutron stars: (i) thermal relaxation: t 100 yr (ii) neutrino cooling: 100 t 10 6 yr (ii) surface cooling: t 10 6 yr [Alford et al, astro-ph/0411560], [Blaschke et al, astro-ph/0411619] log 10 (T s [K]) 6.4 6.2 6 5.8 HJ (Y - 3P 2 *0.1) with K = 240 MeV with Med. effects, our crust, Gaussian FF Crab PSR J0205+64 in 3C58 RX J0822-43 CTA 1 1E 1207-52 RX J0002+62 Vela PSR 0656+14 PSR 1055-52 Geminga 1.00 1.10 1.21 c 1.42 1.50 1.60 1.71 1.79 Other potential observables R-mode instabilities Magnetic field decay Glitches... Ω RX J1865-3754 5.6 0 1 2 3 4 5 6 7 log 10 (t[yr]) B Time
Down to earth... Dense quark matter may be mimicked in tabletop experiments by studying trapped cold gases of fermionic atoms (e.g., 6 Li or 40 K) First experimental results: BEC limit BCS limit Zwierlein et. al.,, cond-mat/0511197 Partridge et. al.,, cond-mat/0511752
Summary At µ Λ QCD QCD dynamics is weakly coupled, but nonperturbative In this limit, QCD can be studied from first principles Under conditions in stars, Cooper pairing is unconventional There may exist many different phases in the QCD phase diagram as well as in stars Physics of stars and physics of matter around us might be closer related than one might naively expect... Many problems remain: (i) instabilities of gapless phases (ii) inhomogeneous ground states (iii) search for observables, etc.