Color Superconductivity in Dense Quark Matter • Basics Of

Riezlern, August 5, 2009 1

Andreas Schmitt

Institut f¨ur Theoretische Physik Technische Universit¨at Wien 1040 Vienna, Austria

Color superconductivity in dense quark matter

For a review, see M. Alford, K. Rajagopal, T. Sch¨afer, A. Schmitt, Rev. Mod. Phys. 80, 1455 (2008)

Basics of (color) superconductivity • Color-ﬂavor locking (CFL) – highest densities • Stressed pairing – below CFL densities • Transport properties – quark matter in compact stars • Riezlern, August 5, 2009 2

Part I •

Basics of (color) superconductivity (page 1/5) •

free energy Ω = E µN µ E F = − no interactions: add fermion • at E = µ without cost E(p) Ε attractive interaction: • p add pair with gain pairs condense • superconductivity →

This Bardeen-Cooper-Schrieﬀer (BCS) argument holds for electrons in metal, 3He atoms, . . ., and quarks in quark matter Riezlern, August 5, 2009 4

Basics of (color) superconductivity (page 2/5) • Electromagnetic vs. color superconductor

Where? What? Attractive Cooper Meissner force pairs eﬀect “usual” metals, ion lattice phonons electrons photon superconductor alloys & electrons color neutron quarks gluons quarks gluons ( ) superconductor stars & gluons (and photon) ∗

( ) ∗ Most color superconductors are not electromagnetic superconductors (“rotated electromagnetism”) Exception: Spin-1 color superconductors A. Schmitt, Q. Wang, D. H. Rischke, PRL 91, 242301 (2003) Riezlern, August 5, 2009 5

Basics of (color) superconductivity (page 3/5): • attractive quark-quark interaction one-gluon exchange •

a attractive in antisymmetric antitriplet channel [3¯]c a s SU(3)c :[3]c [3]c = [3¯] [6] ⊗ c ⊕ c ﬂavor space • SU(3) :[3] [3] = [3¯]a [6]s f f ⊗ f f ⊕ f order parameter (for spin-0 pairing): • ψαCγ ψβ ǫαβA ǫ φA [3¯]a [3¯]a h i 5 j i ∝ ijB B ∈ c ⊗ f Riezlern, August 5, 2009 6

Basics of (color) superconductivity (page 4/5): • energy gap

Fermion dispersions

1.5 1.0 1.0 hole 0.5 0.5 Μ 2D k k 0.0 0.0 Ε Ε

-0.5 -0.5 particle -1.0 -1.0 -1.5 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 k k 2 2 ǫk = (k µ) ǫk = (k µ) + ∆ ± − ±q − suppression of speciﬁc heat, viscosity, neutrino emissivity, etc. → Critical temperature (in BCS theory): Tc 0.57∆ ≃ Riezlern, August 5, 2009 7

Basics of (color) superconductivity (page 5/5): • Meissner eﬀect B normal conductor superconductor

magnetic field

0 λ penetration depth

spontaneous symmetry breaking: U(1)em Z • → 2 Anderson-Higgs mechanism: photon Meissner mass m • M 1 m = ,B e mM r M λ ∝ − gluon Meissner masses in color superconductors • Riezlern, August 5, 2009 8

QCD phase diagram (page 1/3): • Known and unknown territories

~170 MeV Quark−Gluon Plasma

Hadrons Color Superconductivity ~10 MeV CFL Compact ?Stars 308MeV µ

High densities: Moderate densities: rigorous theoretical control perturbative QCD not valid • • no nonperturbative gaps in strange mass & neutrality: • our understanding • stress on Cooper pairing Riezlern, August 5, 2009 9

QCD phase diagram (page 2/3): • Validity of perturbative QCD

~170 MeV Quark−Gluon Plasma

Hadrons Color Superconductivity ~10 MeV CFL Compact ?Stars 308 MeV 1 GeV 10 8 MeV µ

perturbative QCD g( µ ) < 0.8

here: always N = 3 (ignore heavy quarks) • f N > 3: T. Sch¨afer, NPB 575, 269 (2000) • f Riezlern, August 5, 2009 10

QCD phase diagram (page 3/3): 2 Possible scenarios •

T T

"non−CFL"

CFL CFL Compact Stars Compact Stars ? µ ? µ

CFL superseded by CFL superseded by nuclear matter: “non-CFL” matter: eﬀective theory for CFL in complicated phase structure? • • strongly-coupled regime rely on Nambu-Jona-Lasinio- • CFL matter in the core of type models • compact stars? Riezlern, August 5, 2009 11

Question: What is the ground state of deconﬁned quark matter at moderate densities (in the interior of compact stars)?

1. Theoretical approach: start from CFL and ask “what is next phase down in density?” (if not hadronic matter) 2. Phenomenological approach: “guess” possible phase, compute its properties and compare with astrophysical observations 3. Tabletop approach: learn from parallels to cold fermionic atoms in optical trap Riezlern, August 5, 2009 12

Part II •

On safe grounds: Asymptotically large density •

0 ms mu m µ all quark masses negligible ≃ ≃ ≃ d ≪ “color-ﬂavor locked phase (CFL)”

M. Alford, K. Rajagopal, F. Wilczek, NPB 537, 443 (1999)

φA = δA ψαCγ ψβ ǫαβA ǫ B B ⇒ h i 5 j i ∝ ijA

SU(3)c SU(3) SU(3) U(1) SU(3) Z ⇒ × L × R × B → c+L+R × 2 U(1) U(1) ⊃ Q ⊃ Q˜ | {z } | {z } s d s d

d u s u u Riezlern, August 5, 2009 14

Properties of CFL (page 1/3) • (1) chiral symmetry breaking

usual chiral symmetry breaking: LR pairing ψ¯ ψ • h R Li here: LL, RR pairing ψ ψ , however • h R Ri

SU(3)c SU(3) SU(3) U(1) SU(3) × L × R × B → c+L+R U(1) U(1) ⊃ Q ⊃ Q˜ | {z } | {z } chiral symmetry broken through “locking” to color • octet of pseudo-Goldstone modes K0, K , π0, . . . • ± quark-hadron continuity? T. Sch¨afer, F. Wilczek, PRL 82, 3956 (1999) • Riezlern, August 5, 2009 15

Properties of CFL (page 2/3) • (2) superﬂuidity SU(3)c SU(3) SU(3) U(1) SU(3) × L × R × B → c+L+R U(1) U(1) ⊃ Q ⊃ Q˜ exactly massless| Goldstone{z mode} φ | {z } • vortices in rotating CFL M. M. Forbes, A. R. Zhitnitsky, PRD 65, 085009 (2002) • (3) rotated electromagnetism SU(3)c SU(3) SU(3) U(1) SU(3) × L × R × B → c+L+R U(1) U(1) ⊃ Q ⊃ Q˜ | {z } | {z } Cooper pairs neutral under Q˜ = Q + 2 T • √3 8 photon-gluon mixing with (small) mixing angle • 3g2 cos2 θ = 1 3g2 +4e2 ≃ Riezlern, August 5, 2009 16

Why CFL is favored (page 1/2) • general order parameter • ψαCγ ψβ ǫαβA ǫ φA h i 5 j i ∝ ijB B complex 3 3 matrix φA huge conﬁguration space?! • × B → – cf. superﬂuid 3He SO(3) SU(2) U(1) L × S × – 3 3 order parameter in angular momentum× L, spin S A A3 – A phase: φB = δ (δB1 + iδB2) A A – B phase: φB = δB Riezlern, August 5, 2009 17

Why CFL is favored (page 2/2) • simple at high densities! full SU(3)c SU(3) symmetry • → × f can restrict to diagonal φA • B T φ U SU(3)c , V SU(3) : U φ V diagonal ∀ ∃ ∈ ∈ f then φA = δA is only phase where all quarks pair • B B for instance 2SC phase φA = δA3δ • B B3 paired: unpaired:

d u s d u s d u s Riezlern, August 5, 2009 18

Part III •

Going down in density: • Large, but not asymptotically large, densities

strange mass Ms 120MeV no longer µ 400 MeV • ≃ ≪ ≃ T

~170 MeV Quark−Gluon Plasma

Hadrons

~10 MeV CFL Compact Stars 308MeV µ

"switch on" strange mass Ms s at given chemical potentials µ, µe: Ms = 0 reduces p • 6 F electric neutrality: increase in pd to compensate • F Fermi momenta “try” to split apart • Riezlern, August 5, 2009 20

p ’s splitting apart • F unpaired 2SCpairing CFLpairing p p p F F F

d d 2 M 4µ d s u u u 2 s Ms 4µ s s

red green blue red green blue red green blue

any pairing pattern most “comfortable” with Ms and neutrality? • stressed pairing is unavoidable! • K. Rajagopal, A. Schmitt, PRD 73, 045003 (2006) Riezlern, August 5, 2009 21

Stressed Cooper pairing: • a general phenomenon (page 1/2)

BCS-pairing: split Fermi momenta:

k d kF

kF s −k kF

for instance: two species (spin states) of cold fermionic atoms • – Experiments: Y. Chin, M.W. Zwierlein, C.H. Schunck, A. Schirotzek, W. Ketterle, PRL 97, 030401 (2006) G.B. Partridge, W. Li, R.I. Kamar, Y. Liao, R.G. Hulet, Science 311, 503 (2006) – Theory (review): D.E. Sheehy, L. Radzihovsky, Ann. Phys. 322, 1790 (2007) Riezlern, August 5, 2009 22

Stressed Cooper pairing: • a general phenomenon (page 2/2)

M.W. Zwierlein, A. Schirotzek, C.H. Schunck, W. Ketterle, Science 311, 492 (2006)

phase separation of superﬂuid and normal components • phase separation unlikely in quark matter (local color neutrality!) • M. Alford, C. Kouvaris, K. Rajagopal, PRL 92, 222001 (2004) Riezlern, August 5, 2009 23

CFL pairing with small stress • create common Fermi surface: • cost in free energy 2 2 4 kd δp µ M F ∼ F ∼ s

s form pairs: kF • gain in free energy ∆2µ2 ∼ 2 CFL survives for ∆ & Ms • µ 1.0 1.0

0.5 0.5 2D 2HD-∆ΜL k k 0.0 0.0 Ε Ε

-0.5 -0.5

-1.0 -1.0

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 k k ǫ = (k µ)2 +∆2 ǫ = (k µ¯)2 +∆2 δµ k ± − k ± − ± p p Riezlern, August 5, 2009 24

Large stress: gapless CFL? • “gapless CFL” for δµ > ∆ → M. Alford, C. Kouvaris, K. Rajagopal, PRL 92, 222001 (2004)

1.0 1.0

0.5 0.5 2HD-∆ΜL k k 0.0 0.0 Ε Ε

-0.5 -0.5

-1.0 -1.0

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 k k or: “breached” pairing → E. Gubankova, W. V. Liu and F. Wilczek, PRL 91, 032001 (2003) Riezlern, August 5, 2009 25

Large stress: gapless CFL? No! • “gapless CFL” for δµ > ∆ → M. Alford, C. Kouvaris, K. Rajagopal, PRL 92, 222001 (2004)

1.0 1.0

0.5 0.5 2HD-∆ΜL k k 0.0 0.0 Ε Ε -0.5 -0.5 unstable -1.0 -1.0

0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 k k or: “breached” pairing → E. Gubankova, W. V. Liu and F. Wilczek, PRL 91, 032001 (2003) chromomagnetic instability (at T = 0) → M. Huang, I. A. Shovkovy, PRD 70, 051501 (2004) R. Casalbuoni, R. Gatto, M. Mannarelli, G. Nardulli, M. Ruggieri, PLB 605, 362 (2005) ground state must be diﬀerent currents → → Riezlern, August 5, 2009 26

Less (and less symmetric) pairing (page 1/4) • Kaon condensation “CFL-K0” P. F. Bedaque and T. Sch¨afer, NPA 697, 802 (2002) chiral ﬁeld pure CFL: Σ = 1 • • kaon condensation Σ= eiϕT6 Σ= φ† φR • ⇒ L (relative L/R rotation)

K 0 = us du in other words: • E 2 create kaon with mass Ms 2 2 p m = am (ms+mu) ∆ F K0 d ≪ from 0 s¯ + u +¯u + d → (a ∆2/µ2) ∼ u d s Riezlern, August 5, 2009 27

Less (and less symmetric) pairing (page 2/4) • (Super)currents in CFL: “curCFL-K0” T. Sch¨afer, PRL 96, 012305 (2006); A. Kryjevski, PRD 77, 014018 (2008) A. Schmitt, NPA 820, 49C (2009)

i x T8 i(ϕ/2)T6 φL(x) = ∆ e K· e i x T8 i(ϕ/2)T6 φR(x) = ∆ e K· e− “anisotropic breach” • stable and unstable Fermi surface topologies: •

"breach" (unstable) curCFL−K0 (stable) Riezlern, August 5, 2009 28

Less (and less symmetric) pairing (page 3/4) • More currents in CFL: crystalline structures (LOFF) M. Alford, J. Bowers, K. Rajagopal, PRD 63, 074016 (2001) M. Mannarelli, K. Rajagopal and R. Sharma, PRD 73, 114012 (2006) ud ∆ exp(2iqa x) h i ∼ 3 3 · Xa us ∆ exp(2iqa x) h i ∼ 2 2 · Xa ds 0 h i ≃ here: “CubeX” • q3 , q2 each contain 4 • vectors,{ } { together} pointing to the corners of a cube ∆3(x), ∆2(x) Riezlern, August 5, 2009 29

Less (and less symmetric) pairing (page 4/4) • Last resort: single ﬂavor pairing

need J = 1 Cooper pairs • φ [3¯]a [3]s ∈ c ⊗ J kd diﬀerent possible phases: F • s Color-spin locking (CSL), kF A-phase, polar phase . . . T. Sch¨afer, PRD 62, 094007 (2000) A. Schmitt, PRD 71, 054016 (2005) preferred phase at high densities: CSL • gap much smaller than in spin-0 phases: • 2 ∆J=1 . 10− ∆J=0 Riezlern, August 5, 2009 30

Stressed pairing: free energy comparison •

] g2SC

4 unpaired 0 2PW MeV 6 -10

-20 2SC 2Cube45z gCFL CubeX -30 curCFL-K0

0 -40 CFL CFL-K Energy Difference [10 Energy Difference -50 0 50 100 150 200 2 µ MS/ [MeV] here: ∆CFL = 25 MeV (pert. QCD: ∆ 20 MeV, NJL: ∆ (20 100) MeV). CFL ≃ CFL ≃ − Riezlern, August 5, 2009 31

Part VI •

Color superconductivity in compact stars (I) • F. Weber, Prog. Part. Nucl. Phys. 54, 193 (2005) 36.0 M=1.4 M M=1.40M sun

standard cooling 6.5 (modified Urca process) Cooling of the star standard 34.0 Crab

3C58 6.0 −1 1055-52 intermediateVela ) -1 s

32.0 0656+14 s /erg s delayed s L by super− log T / K

log( fluidity

Geminga 5.5 log L /erg s neutrino emissivity (quark matter,enhancedenhanced meson condensates, • direct Urca process)

T 10MeV 0.1 keV 30.0 ∼ → 5.0 (t 0 106 yr) ∼ → 1929+10

28.0 4.5 -1.0 1.0 3.0 5.0 7.0 τ log Time/yearslog( /yr) unpaired quark matter Iwamoto, PRL 44, 1637 (1980) CFL Jaikumar, Prakash, Sch¨afer, PRD 66, 063003 (2002) 2SC Jaikumar, Roberts, Sedrakian, PRC 73, 042801 (2006) spin-1 Schmitt, Shovkovy, Wang, PRD 73, 034012 (2006) LOFF Anglani, Nardulli, Ruggieri, Mannarelli, PRD 74, 074005 (2006) Riezlern, August 5, 2009 33

Neutrino emissivity • Non-CFL CFL direct Urca process (pseudo-)Goldstone • • u + e d + νe processes → d u + e +¯νe →

π , K e +¯νe ⇒ ± ± → ± 0 457 2 6 π νe +¯νe ǫν αsG T µeµuµ → ≃ 630 F d ϕ + ϕ ϕ + νe +¯νe → very small emissivity (unpaired quark matter) ⇒ non-CFL • 2 15 (2SC, LOFF, spin-1, ...): GF T ǫν 2 4 ungapped quarks ∼ f µ → possible exception (?): CSL • Riezlern, August 5, 2009 34

Color superconductivity in compact stars (II) •

magnetic fields “magnetic” CFL • E.J. Ferrer, V. de la Incera, C. Manuel, spin-0 no Meissner effect PRL 95, 152002 (2005) • M.G. Alford, J. Berges, K. Rajagopal, de Haas-van Alphen oscillations NPB 571, 269 (2000) • K. Fukushima and H. J. Warringa, spin-1 Meissner effect PRL 100, 032007 (2008) • J. L. Noronha and I. A. Shovkovy, A. Schmitt, Q. Wang, D.H. Rischke, PRD 76, 105030 (2007) PRL 91, 242301 (2003)

ferromagnetism in the curCFL-K0 phase • D. T. Son and M. A. Stephanov, PRD 77, 014021 (2008) Riezlern, August 5, 2009 35

Color superconductivity in compact stars (III) • glitches

need vortex pinning at some crystal • maybe: neutron superﬂuidity + ion lattice in crust • however: inconsistent with precession of star on yr time scale • ∼ B. Link, arXiv:astro-ph/0608319 better (?): crystalline color superconductivity • large shear modulus of CubeX (and 2Cube45z) • M. Mannarelli, K. Rajagopal, R. Sharma, PRD 76, 074026 (2007) ∆ 2 µ 2 ν =3.96 1033erg/cm3 × 10 MeV 400 MeV 20 – 100 times more rigid than neutron star crust • Riezlern, August 5, 2009 36

Color superconductivity in compact stars (IV) •

r-mode instability

r-modes: È ÓÐa Ö ÎieÛ EÕÙaØÓ ÖiaÐ ÎieÛ • non-radial pulsation modes L. Lindblom, arXiv:astro-ph/0101136 grow unstable spin down • in a perfect-ﬂuid rotating star • the star drastically and emission of gravitational waves quickly (within days) →

mass shedding limit fast rotating stars are observed! • ω 1ms 1 max shear viscosity − bulk dominates ≃ c viscosity must be some damping mecha- Ω / Ω dominates • nism bulk/shear viscosity

→ T ( K ) Riezlern, August 5, 2009 37

What is bulk viscosity? •

volume oscillation • µ µ chemical V0 d s →non-equilibrium

µ µs =0 d − 6 re-equilibration via µ µ • V0 − δ V d s u + d u + s ↔ resonance • phenomenon: u s W− external oscillation µ µ d s vs. microscopic rate d u Riezlern, August 5, 2009 38

Compute rate for u + d u + s in 2SC • ↔

u ∆ s u ∆ s u s u − s − − W− W Γ = W + 2 W + 2 + 2SC 4 d ∆ ∆ u d ∆ u d ∆ u d u

u ∆ s u ∆ s − − W + 2 W + 4 d ∆ ∆ u d ∆ u

small temperatures, paired: unpaired: T Tc 30MeV ≪ ≃ d u s d u 1 Γ2SC = Γunpaired s 9 d u s due to exponential suppression exp( ∆/T ) of gapped modes − Riezlern, August 5, 2009 39

Results for bulk viscosity

• 8

6 unpaired E 1 s

A Γ 2Π =1.0ms-1 γ Γ 4 H L ζ = α log - 2 2 ΓH2ΠL=0.2ms 1 γ + ω 2 2SC

0.1 0.2 0.5 1 2 5 10 20 T@MeVD

29 29

2SC 2SC E E 28 28 g g cm s cm s 27 27 A A Ζ Ζ unpaired unpaired 26 26 log log

25 25 -1 -1 ΩH2ΠL=0.2 ms ΩH2ΠL=1.0 ms 0.1 0.2 0.5 1 2 5 10 20 0.1 0.2 0.5 1 2 5 10 20 T@MeVD T@MeVD Riezlern, August 5, 2009 40

Quark matter bulk viscosity: diﬀerent phases • 30 10 unpaired 2SC

1024

LD MeVL H∆m=+0.5 18 CFL 1 10 ω/(2π)=1ms− cm s

H µ = 400MeV g 1012 @ 0 0 Ζ L δm m µ Φ®Φ+Φ ≡ K − K 0.5 MeV 106 =- 0 H∆m -K 1 CFL 0.01 0.05 0.10 0.50 1.00 5.00 10.00 TMeV unpaired J. Madsen, PRD 46, 3290 (1992) 2SC M.G. Alford, A. Schmitt, JPG 34, 67-101 (2007) CFL from K0 φ + φ M.G. Alford, M. Braby, S. Reddy, T. Schafer, PRC 75, 055209 (2007) ↔ CFL-K0 from K0 φ + φ M.G. Alford, M. Braby, A. Schmitt, JPG 35, 115007 (2008) ↔ CFL from φ φ + φ C. Manuel, F. Llanes-Estrada, JCAP 0708, 001 (2007) See also: Spin-1↔B.A. Sa’d, I.A. Shovkovy, D.H. Rischke, PRD 75, 065016 (2007) Vortices in CFL M. Mannarelli, C. Manuel, B. A. Sa’d, PRL 101, 241101 (2008) Riezlern, August 5, 2009 41

Summary • 3-flavor quark matter at asymptotically high densities is in • the Color-Flavor locked (CFL) state CFL quark matter • – breaks chiral symmetry – is a superfluid – is a transparent insulator phase(s) between CFL and hadronic matter (if there is/are any) • is/are unknown – large coupling – stressed pairing renders ground state complicated transport properties dominated by Goldstone modes in • CFL ( small ν-emissivity, specific heat, bulk viscosity ...) and ungapped→ quarks in non-CFL ( large emissivity ...) → Riezlern, August 5, 2009 42

Outlook/open questions •

QCD at high densities: BEC – BCS crossover? • (however: quark bound states contain 3 quarks, not 2)

insight from large-Nc QCD: • competition/coexistence with quarkyonic phase? L. McLerran, R. D. Pisarski, NPA 796, 83 (2007) second critical endpoint at low T in QCD phase diagram? • T. Hatsuda, M. Tachibana, N. Yamamoto, G. Baym, PRL 97, 122001 (2006) superﬂuid properties of CFL-K0, curCFL-K0; • multicomponent superﬂuid? M.G. Alford, A. Schmitt, in preparation