Color Superconductivity
Condensed matter physics of quark matter Contents
1. Introduction 2. Color superconductivity at an ultra-high baryon density ------First principle calculation of QCD 3. Color superconductivity at a realistic baryon density ------NJL effective action approach 4. Concluding remarks I. Introduction • Quarks of different color and flavors: up down strange charm bottom top u d s c b t u d s c b t u d s c b t Mass 1.5-4MeV 4-8MeV 100-200MeV 1.3GeV 5GeV 170GeV Charge 2/3 -1/3 -1/3 2/3 -1/3 2/3 Baryon no. 1/3 1/3 1/3 1/3 1/3 1/3 • Quark contents of hadrons: p = (uud) n = (udd), etc.
The relevant quark flavors for color superconductivity are u, d & s. • Dynamics of quarks ---- QCD (quantum chromodynamics) ψ Yang-Mills theory µνof SU(3) γcolor gauge group: µν µ ψ 1 ∂ ψ L = − trF F − − igA + mψ µ µ 2 ∂x +µνgauge fixingν terms + renormalization counterterms where µ µ ∂A ∂A F = − ν − ig[]Aµ , A ∂x ∂x ν
µ 1 A = Al λ stands for gluon field with µ = 1,2,3,4 and l = 1,...,8 2 µ l
denotes the quark 4-spinor. It carries three color indexes andN f
flavor indexes. For CSC N f = 2 ~ 3 m is the mass matrix in the flavor space
λ's are 3x3 Gell-Mann matrices given by 0 1 0 0 − i 0 1 0 0 λ1 = 1 0 0 λ2 = i 0 0 λ3 = 0 -1 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 − i 0 0 0 λ4 = 0 0 0 λ5 = 0 0 0 λ6 = 0 0 1 1 0 0 i 0 0 0 1 0
0 0 0 1 0 0 1 λ7 = 0 0 − i λ8 = 0 1 0 i 0 0 3 0 0 - 2 Quark field ψ supports the fundamental representation, 3 of the color SU(3)
l Gluon field Aµ supports the adjoint representation, 8. • Properties of QCD Asymptotic freedom ( anti-screening ): α g 2 2π QCD running coupling = π= s E 4 9ln Λ E = max(T, µ) for T >> Λ, or µ >> Λ T = temperature µ = chemcal potential Λ ≅ 200MeV
Baryon density Fermi energy(MeV) αS (0) 10nB 442 0.88 (0) 20550 0.151 1000000nB • Quark confinement ------Elementary excitations of QCD at low temperature and chemical potential are hadrons (nucleons and mesons). • Quark deconfinement ------Elementary excitations at high temperature or high chemical potential are quarks and gluons.
MIT bag model predicts the crossover baryon density from nuclear matter to quark matter
(0) nB ~ 10nB
nucleons loses identities under squeezing QCD phase diagram
T RHIC
Quark-gluon plasma
Nuclear matter
2SC CFL µ
Nuetron star
RHIC: relativistic heavy ion collision 2SC & CFL ------two color superconducting phases Experimental tools:
• RHIC: high temperature, low chemical potential; • Neutron star observation: low temperature high chemical potential.
Theoretical tools:
• Perturbation theory: high temperature or high chemical potential; • Lattice simulation: intermediate temperature and low chemical potential; • NJL effective action: low temperature and intermediate chemical potential; • AdS/CFT correspondence: decomfined phase. II. Color superconductivity at an ultra-high baryon density • Zero-th order
Chemical potential µ >> Λ such that α S << 1 A fermi sea of u, d, s quarks of equal Fermi momenta The whole systemξ is electrically neutral and is ulta-relativistic.
Kinetic energy relative to the Fermi energy (=chemical potential) µ 2 2 p = p + m − ≅ p − µ Quark number density for each color and flavor: k 3 n = F c, f 3π 2 Baryon number density = (1/3) X quark number density • Pairing via one-gluon-exchange λ Barrois; Frautschi; Bailin & Love λ δ δ ′ ′ 2λc1c1 c2c2 δ ∝ g l λl gluon δ
quark quark δ δ c c ′ c c ′ 4 ′ ′ ′ ′ 2 ′ ′ δ ′ ′ ()1 1 ()2 2 = − ( c1 c1 c2 c2 − c1 c2 c2 c1 )+ ( c1 c1 c2 c2 + c1 c2δ c2 c1 ) l l 3 3 3 ⊗ 3 = 3 ⊕ 6
color antisymmetric color symmetric attractive repulsive
Favorite pairing channel: different colors, different flavors, s-wave • Temperature of Pairing Instability Son; Schaefer & Wilczek; Hong, Miransky &Shovkovy; Brown, Liu & Ren; Wang & Rischke;π Manuel 3 2 γ π 2 +4 2048 2 µ − + E − T = e 2g 8 []1+ O(g) p 3 3 g 5 1 1 ---- Non BCS scaling: lnT ~ instead of lnT ~ p g p g 2 ---- Strong type I. • Transition Temperature π 2 Giannakis, Hou, Ren & Rischke Tc = 1+ g Tp 12 2
---- Gauge filed fluctuation renders the phase transition first order, the Mechanism of Halperin, Lubinsky and Ma. • Energy gap at zero temperature Schaeferω & Wilczek; Pisarski & Rischke; Reuter; Feng, Hou, Li & Ren ---- Solution to the Eliashberg equation:
ω ∆ 0 π π for 0 ω< < ∆ 0 ∆()= g | | ∆ 0 cos ln − i for ω > ∆ 0 ∆ 2 3 2 0 1 − 3 −γ E ∆ 0 = 2 πe Tp
----ω quasi particle pole: µ 2 − ( p − ) 2 − ∆2 (ω) = 0 • Calculation details
Schwinger-Dyson equation:
= +
= proper vertex of diquark scattering = Γ(P′ | P)
= 2PI vertex of diquark scattering = γ (P′ | P)
= dressed quark propagator = S(Q)
Incoming quark momenta - energies = P ≡ (±p,±iν ) Outgoing quark momenta - energies = P′ ≡ (±p′,±iν ′) ν and ν ′ = Matsubara energies • Calculation details
Schwinger-Dyson equation:
= +
T Γ()P′ | P γ= (P′ | P) + ∑ (P′ | Q)S(Q)S(−Q)Γ(Q | P) Ω Q = γ (P′ | P) + ∑ Kγ(P′ | Q)Γ(Q | P) Q
Fredholm integral equation for Γ(P′ | P) D(T, µ) ≡ the Fredholm determinant
The temperature of pairing instability = the highest temperature such that D(T, µ) = 0 • Calculation details
Schwinger-Dyson equation:
= +
Perturbative expansion:
= + • • •
= + + • • •
= hard dense loop (HDL) resummed gluon propagator HDL gluon propagator:
Dµν (K) = = + σ δ + + … σ −1 k k µ − i j for ν , = i, j 2 2 ij 2 k + k0 + M (k,k0 ) k = −1 µ for =ν = 0 2 k + E (k,k0 ) σ For pairing, k0 << k << µ π k0 Color magnetic self energy M (k,k0 ) ≅ 2 Landau damping 4lD k Color electric self energy 1 σ E (k, k0 ) = 2 lD 1 3gµ2 lD is the Debye length 2 = 2 lD 2π Magnetic gluon dominate the pairing Forward singularity, non BCS scaling
(p′,iν ′) (−p′,-iν ′)
1 Pairing potential, V ~ d cosθ ( 2 1 ∫ )~ g ln ν −1 | −ν ′ | (p,iν ) (−p,−iν ) where cosθ ≡ pˆ ⋅pˆ ′ Pairing instability : ∞ 1 1 1 1 ~ T V d( p − ) ~ g 2T ln ∑ µ 2 2 ∑ ν ∫ −∞ ( p − ) + | | | | ν µ d 1 µ 1 ~ g 2 ν ln ~ g 2 ln 2 ν ∫ ν T Tp p µ ν 3πν 2 ν Careful analysis yields T = b exp− Son p 5 g 2g Collecting constant pertaining to the logarithm Schaerfer & Wilczek; 2048 2 b = eγ E b′ Pisarski & Rischke 3 3 Non Fermi liquid behaviorπ γ ν Quark self energy ν 2 3 g kc Σ()P = −i 2 4 ln 2 = O(g) for ν ~ Tp 9 mD | |
where kc >> Tp ---- Generalization of the QED results of Holstein, Norton, Picus and Reizer µ Contribution to the pairing temperature: π Brown, Liu & Ren 2048 2 b′′ 3 2 γ π 2 + 4 T = exp− + − p 5 E 3 3 g 2g ------8
Careful examination of other higher order diagrams leads to b′′ = 1+ O(g) Brown, Liu & Ren Done! • Color-flavor locking (CFL) below the transition temperature ψ Alford, Rajagopalψ & Wilczek φ ---- Energetically favored condensateδ δ < c1 c2 >= ( c1 c2 − δ c1 δ c2 ) f1 f2 f1 f2 f2 f1 ----Symmetry breaking
SU (3) ⊗ SU (3) ⊗ SU (3) ⊗ U (1) → SU (3) ⊗ Z c f R f L B c+ f R + f L 2
----Analogous to the AB phase of the superfluid of 3 He where orbital angular momentum is locked with spin, the symmetry breaking is
(orbital rotation) ⊗ (spin rotation) → (simultaneous rotation) III. Color superconductivity at a realistic baryon density
• The density accessible inside the core of a neutron star
µ = 400 ~ 500MeV and α S ≅ 0.76 ~ 1.00 ------One-gluon-exchange gives transition temperature around 3.7MeV (=43,000,000,000K) at 500MeV, but the perturbative result may not be reliable; ------Nonperturbative effects may end up with higher transition temperature; ------The mass of strange quark cannot be neglected; ------Crossover from 3 flavor (u, d, s) color superconductivity to 2 flavor (u, d) color superconductor ------Fermi momentum mismatch between different flavors suggest a number of exotic superconductivity, e.g. Sarma state, LOFF, phase separation. • Fermi momentum mismatch Consider an ideal gas of three quark favors and electrons at T = 0 The energy density 3 ku 3 kd 3 ks E = dpp 2 p 2 + m2 + dpp 2 p 2 + m2 + dpp 2 p 2 + m2 π 2 ∫ u 2 ∫ d 2 ∫ s 0 0 0
ku 1 2 2 2 + 2 dpp p + meπ π ∫ 3 0 1 k The baryon number density n = ()n + n + n ≡ 0 B 3 u d s π 2 π 2 1 1 π The electric charge density n = n − n − n − n = 0 Q 3 u 3 d 3 s e k 2 k 2 where n = f f = u,d, s n = e f 2 e 3π 2 Charge neutrality: nQ = 0 Pressure µ µ p = µbb + µqq − E Equilibrium condition: µ µ
∂p ∂p ∂p µ µ ∂p = = = = 0 ∂ku ∂kd ∂ks ∂ke µ b , q b , q b , q b ,µq • Fermi momentum mismatch 1 The solution at k >> m 2 3 0 s k0 = ()π nB
k u ≅ k 0 2 m s k d ≅ k 0 + 4 k 0 2 3 flavor color superconductivity m s k s ≅ k 0 − 4 k 0 2 m s k e ≅ 4 k 0
The solution at mu , md << k0 << ms
ku ≅ 0.87k0
kd ≅ 1.09k0 2 flavor color superconductivity (2SC)
ke ≅ 0.22k0 ψψ • Two flavor NJL(Nambu-Jona-Lasinioψτψ ) effective Lagrange Rapp, Schaefer, Shuryakψ &Velkovskyγ ε τ ψ ψ 2 ∂ψ 2 c γ ε c L = −ψγµ + G ( ) + ( ) + G ()( τ ψ ) S D C 5 2 C 5 2 ∂xµ ε γ * c ab cab where ψ C = 2ψ , the three 3x3 matrices ( ) = ε
act on color indexes (red, green and blue) and theτ second2 Pauli matrix acts on flavor indexes (u, d).
-2 -2 GS ≅ 5.32 GeV , GD ≅ 4 GeV and UV cutoff Λ f ≅ 653 MeV. (estimated by fitting the hadron masses at zero chemical potential.) ---- Effective four fermion point interaction inspired by QCD instantons; ---- The last term ( diquark coupling ) gives rise to CSC; ---- Pairing channel is color antisymmetric, like the one-gluon exchange; ---- s-wave pairing between different colors and different flavors, e.g. (u, d) and (u, d); ---- Mean field theory solution; ---- Share some features of a cold atom system. • Exotic superconductivity under mismatch µ ---- Pairing betweenµ Fermi Momenta and ---- Point attractionµ (BCS like). 1 1 δ 1 = + , = − µ 3 B 6 Q 3 Q Homogeneous condensate <ψψ >= constant
BCS state A C
A = (0, ∆ 0 ) ∆ Sarma state B = 0 , 0 2
C ≅ ()∆ 0 , ∆ 0 ∆ 0 ≡ the solution at T = 0. B
Solution to the gap equation LOFF ( Larkin-Ovchinnikov-Fulde-Ferrell) condensate: ---- Single plane wave <ψψ >∝ e 2iq⋅r 2q Choose q to minimize the thermodynamic potential Energetically more favored than the normal, BCS and Sarma phases within the LOFF window
0.706∆ 0 < δ < 0.754∆ 0 q ≅ 1.2∆ 0
---- Multi-plane wave 2iq⋅r <ψψ >∝ ∑ ∆ q e q May lower the thermodynamic potential further, difficult to treat. µ µ µ ε 1 1 δ 1 • 2SC-BCS = B + Q , = − µQ µ 3 6 3 ---- Excitation spectrum δ ± 2 2 p = ( p − ) + ∆ ± ∆ > δ Gapped ---- Thermodynamic potential µ µ H − B − µ Q T δ µ B Q Γ ≡ − ln exp− = − p Ω δT π2 2 2 2 2 2ω0 1 ∆ ≅ Γn ( , ) + 2 − ∆ ln + + µ ∆ 2 4G δ D where µ 1 4 µ 4 Γn ( ,δ ) = − [()()− + + δ ] µ 2 µ 4π ---- Equilibrium condition µ π 2 δ µ − ∂Γ 2 = 0 ⇒ ∆ ≅ 2ω e 8GDµ ≡ ∆ ∂∆ π δ 0 0 B , Q ---- At the equilibrium 2 2 2 1 Γ ≅Γ n ( , ) + ()2 − ∆ 0 Γ > Γn for δ > ∆ 0 2 2 ---- Charge neutrality may be restored by unpaired quarks Liu & Wilczek, • g2SCε (Sarma state) Huang, Zhuang & Zhao, µ Shovkovy & Huang ---- Excitationδ spectrumµ µ δ δ ± 2 π 2 Gapless p = ( p − ) + ∆ ± ∆ < δδ
---- Thermodynamic potential δ δ 2 2 + 2 − ∆2 1 Γ ≅ Γ ( , ) + ∆2 ln − 2 − ∆2 − ∆2 + δ 2 n 2 ∆ 2 0
---- Equilibrium condition ∂Γ = 0 ⇒ ∆ = ∆ 0 ()2δ − ∆ 0 ∂∆ µ B ,µQ ---- At the equilibrium δ µ δ µ 2 Γ = Γ (), + (2 − ∆ ) 2 > Γ ()µ,δ n π 2 0 n ∂ 2Γ < 0 Ustable! 2 ∂∆ µ B ,µQ Stabilization by the charge neutrality Shovkovy & Huang
BCS state A C D Charge neutrality ∂Γ nQ = − = 0 G • ∂µ Sarma Q µ,∆ state F B E ---- Along the red line µ 2 ∂nQ 2 2 µ ∂∆µ ∂ W ∂ Γ µ µ , = + B Q > 0 2 2 Γ < Γ ∂∆ ∂∆ ∂nQ G F B ,nQ B , Q ∂µQ µB ,∆
where W ≡ Γ − µQ nQ = Γ ---- Generalization to 3 flavors ---> gCFL Alford, Bowers & Rajagopal
---- Good candidate, but … Chromomagnetic instability Huang & Shovkovy Color Magnetic field A l , l = 1, 2, ..., 8, ∇ ⋅ A l = 0
Polarization function l′l Π ij (k,k0 ) = (k,k0 )
1 k k = lim δ − i j Π ll (k,0) Inverse penetration depth square 2 k→0 ij 2 ij dl k For (u,d) or (u,d ) pairing 1 µ = 0 2 π δ d1,2,3 δ δ 1 g 2 2 ∆2 − 2 2 2 − ∆2 = − ( − ∆) 2 2 µ 2 2 θ δ d 4,5,6,7 π 2∆ ∆ δ 1 ()3g 2 + e 2 2 = 1− δ θ ()δ − ∆ 2 2 2 2 d8 27 − ∆
2 2 Note that d 4,5,6,7 < 0 and d8 < 0 for g2SC (δ > ∆) ! ------• 2SC-LOFF (one way out of the instability) µ δ µ 2iq⋅r Alford, Bowers & Rajagopal; <ψψ >∝ ∆e Giannakis & Ren π ---- Thermodynamic potential: δ 3 3 2 2 ∆ 1 (q + ) (q − ) Γ ≅ Γ ( , ) + ∆2 ln − + f ()x + f ()x n 2 + − ∆ 0 2 4q δ 4q 1+ x 2 where f (x) = ()1− x 2 ln + ()2x3 − 3x +1 1− x 3 θ ∆2δ ∆ x = 1− 1− ± 2 ()q ± q ± δ
∂Γ µ δ ∂Γ ---- Equilibrium condition: = = 0 ∂∆ , ,q ∂q µ,δ ,∆ ---- Excitation spectrum: gapless ∂ 2Γ ∂ 2Γ ∂∆2 ∂∆∂q ---- Within the LOFF window, the matrix 2 2 ∂ Γ ∂ Γ 2 ∂∆∂q ∂q is positive definite at equilibrium. ---- The inverse penetrationµ depth square: Giannakis & Ren π 1 = 0 δ δ 2 δ d1,2,3 ij µ 2 π 2 q q q q 1 g ∆ q i j δ δ∆ q i j = A , ij − + B , d 2 6 2 δ δ q 2 q 2 4,5,6,7 ij δ
2 2 2 1 ()3g + e ∆ q qi q j ∆ q qi q j = C , ij − + D , d 2 27 2 q 2 δ δ q 2 8 ij
The functions A, B, C, D are all positive within the LOFF window No chromomagnetic instability there.
---- A charge neutral LOFF state can be constructed within the LOFF window. Giannakis, Hou & Ren ---- Generalization to 3 flavors by Casalbuoni et. al. • Other Candidates i) Gluon condensation: Gorbar, Hashimoto & Miransky The condensation induces additional contribution to the inverse penetration depth square through the non Abelian gluon coupling. ii) Heterotic structure of 2SC and normal phases: ---- similar to the phase separation of cold atom system; ---- microscopic charge fluctuation, macroscopic neutral.
2SC 2SC 2SC
NN Bedaque, Caldas & Rupak
2SC 2SC 2SC
iii) Equal flavor pairing: Schaefer ---- (u,u) pair or (d,d) pair; ---- non s-wave pairing, beyond NJL model; ---- one-gluon-exchange gives 9 − 2 Brown, Liu & Ren; Schaefer Tc (p - wave) ≅ e Tc (s - wave) • Observing CSC in a Neutron Star Alford, Bowers & Rajagopal
---- If there is a quark matter core inside a neutron star, the impact of its equation of state may show up in the accretion process in a binary system. But the change of the equation of state by the color superconductivity may be too small to be observed with the present technique. ---- Color superconductivity may be detected by the cooling process via neutrino emission. The change of the cooling rate during the history of a neutron star may indicate formations of various type of energy gaps ---- A neutrino burst from a supernova may be expected because of the CSC transition in a newly born neutron star. ---- The pinning of the rotational vortices by the crystalline of LOFF may contribute to the glitch phenomena of a neutron star. ---- R-mode instability may be triggered by the CSC transition because of the suppression of the shear and bulk viscosities by the gap. IV. Concluding Remarks
• A cold quark matter of sufficiently high baryon density will become color superconducting at sufficiently low temperature. The transition temperature and the energy gap can be calculated with perturbative QCD. The long range nature of the pairing force gives rise to a non BCS scaling. • At realistic baryon density, the Fermi momentum mismatch may lead to an exotic superconducting ground state of a quark matter. But a consensus has not been reached. • It is necessary to pin point a number of clear cut prediction to observe the color superconductivity in nature, especially in a neutron star. • Color superconductivity is a subject that integrates high energy physics, condensed matter physics, nuclear physics and astrophysics. Review Articles
• D. Bailin and A. Love, Phys. Rep. 107, 325 (1984). • K. Rajagopal and F. Wilczek, hep-ph/0011333. • D. K. Hong, Acta Phys. Polon. B32, 1253 (2001). • M. Alford, Ann. Rev. Nucl. Part. Sci., 51, 131 (2001). • T. Schaefer, hep-ph/0304281. • D. Rischke, Prog. Part. Nucl. Phys. 52, 197 (2004). • R. Casalbuoni and G. Nardulli, Rev. Mod. Phys., 76, 263 (2004). • M. Ruballa, Phys. Rept. 407, 205 (2005). • Hai-cang Ren, CCAST lecture, hep-ph/0404074. • Mei Huang, CCAST lecture, Int. J. Mod. Phys. E14, 675 (2005). • I. Shovkovy, Found. Phys. 35, 1309 (2005). • M. Alford, K. Rajagopal, T. Schaefer and A. Schmitt, arXiv:0709.4635. Thank you!