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Color and the strange Mark Alford Washington University Saint Louis, USA

1 Outline

I High density QCD Color-superconducting quark , Color-avor locking (CFL) II Quark matter in compact stars

weak interactions, neutrality, Ms; gapless CFL III Signatures of color superconductivity in compact stars Transport properties, mass-radius relationship IV Looking to the future

2 I. High density QCD QCD phase diagram

Conjectured form NJL model calculation T 60

RHIC 50 NQ

40 2SC QGP uSC guSC→ 30 T [MeV] non−CFL χ 20 SB g2SC

CFL 10 gas liq CFL NQ ←gCFL 0 320 340 360 380 400 420 440 460 480 500 µ [MeV] nuclear µ (Ruster, Werth, Buballa, Shovkovy, Rischke, hep-ph/0503184) heavy ion collisions: chiral critical point and rst-order line compact stars: color superconducting quark matter core

3 Cooper pairing of at high density At suciently high density and low temperature, there is a Fermi sea of almost free quarks.

µ = EF E F = E N p But quarks have attractive QCD interactions.

Any attractive quark-quark interaction causes pairing instability of the . This is the Bardeen-Cooper-Schrieer (BCS) mechanism of superconductivity.

4 High-density QCD calculations Guess a color-avor-spin pairing pattern P ; mini- mize free energy wrt : gap equation for . = hqiaqjbi1P I = Pij ab

1. Weak-coupling methods. First-principles calculations = direct from QCD Lagrangian, valid in the asymptotic regime, currently & 106 MeV. 2. Nambu–Jona-Lasinio models, ie quarks with four- = fermion coupling based on instanton vertex, single exchange, etc. This is a semi-quantitative guide to physics in the compact star regime 400 MeV, not a systematic approximation to QCD. NJL gives 10100 MeV at 400 MeV. Both methods agree on the favored pairing pattern.

5 Color superconductivity in three avor quark matter: Color-avor locking (CFL) Equal number of colors and avors gives a special pairing pattern (Alford, Rajagopal, Wilczek, hep-ph/9804403)

N hqi qj i (+1)i j + (1)j i = ijN + ( ) color , This is invariant under equal and opposite avor i, j rotations of color and (vector) avor

SU(3) SU(3) SU(3) U(1) → SU(3) + + Z2 color | L {z R} B | {zC L R}

U(1)Q U(1)Q˜ Breaks chiral symmetry, but not by a hqqi condensate. There need be no phase transition between the low and high density phases: (“quark-hadron continuity”) Unbroken “rotated” electromagnetism, Q˜, -gluon mixture.

6 Color-avor-locked (“CFL”) quark pairing

Q˜ 0 0 0 1 +1 1 +1 0 0 u d s d u s u s d u d s d u s u s d

7 II. Quark matter in compact stars Where in the universe is color-superconducting quark matter most likely to exist? In compact stars. A quick history of a compact star.

A star of mass M & 10M burns Hydrogen by fusion, ending up with an Iron core. Core grows to Chandrasekhar mass, collapses ⇒ supernova. Remnant is a compact star:

mass radius density initial temp

1.4M O(10 km) > nuclear 30 MeV

The star cools by neutrino emission for the rst million years.

8 The real world: Ms and neutrality In the real world there are three complications to the simple account given so far. 1. Strange quark mass is not innite nor zero, but intermediate. It depends on density, and ranges between about 500 MeV in the vacuum and about 100 MeV at high density. 2. Neutrality requirement. Bulk quark matter must be neutral with respect to all gauge charges: color and electromagnetism. 3. Weak interaction equilibration. In a compact star there is time for weak interactions to proceed: neutrinos escape and avor is not conserved. So quark matter in a compact star might be CFL, or something else: gapless CFL; kaon-condensed CFL, 2SC, 1SC, crystalline (LOFF),. . .

9 Color superconductivity with realistic strange quark mass

With Ms 120500 MeV, the pure CFL pairing pattern is distorted and perhaps broken. Approaches to studying this are: 1. Use a more general pairing ansatz, eg allowing d-u pairing to dier from d-s and u-s. This is how we get gapless CFL (gCFL). 2. Use an eective theory of the low-energy degrees of freedom, including avor rotations of the CFL condensate (“pions” and “kaons”). 3. Allow condensates with momentum: crystalline color superconductivity (“LOFF”). Guiding principle: species a and b start unpairing when energy gained from a → b is greater than cost of breaking a-b , ie 2 Ms b a > 2. For CFL, unpairing starts when > 2CF L.

10 Distortions of CFL with rising Ms

Keep only the = 0 Q˜ 0 0 0 1 +1 1 +1 0 0 (3A, 3A) part. u d s d u s u s d Generalize to allow 3 dif- u ud us ferent gap parameters. Plug it into an NJL d ud ds model and minimize the s us ds free energy, imposing d ud neutrality and weak equi- u ud librium. s us In each 2 2 block, chemical potentials and u us Ms try to pull the Fermi s ds momenta apart; pairing d ds holds them together.

11 Gapless phases

Quasiparticle dispersion relations for Free energy as a function of 2 the ds sector. Ms / in an NJL model. E µ 10 ]

4 g2SC blocking 2 unpaired region Ms/µ <2∆ 0 MeV 2 6 Ms/µ =2∆ -10 2 2SC Ms/µ >2∆ -20 gCFL

-30 ∆ -40 CFL p Energy Difference [10 ∆ gapless -50 modes 0 25 50 75 100 125 150 2 µ MS/ [MeV] Alford, Kouvaris, Rajagopal, hep-ph/0311286 2 When Ms / reaches 2ds, un- pairing begins: a “blocking” or “breached pairing” region opens up, bordered by gapless modes.

12 Problem: magnetic instability of gCFL The gCFL phase appears to have an imaginary Meissner mass for the rg (Huang and Shovkovy, hep-ph/0408268; Casalbuoni, Gatto, Mannarelli, Nardulli, Ruggieri, hep-ph/0410401; K. Fukushima hep-ph/0506080). This is a generic consequence of gapless quasiquark dispersion relations: Meissner mass, charged condensate 0.2 T=0 2 charged quark species, chem 0.1 T=0.3 ∆=δµ pots , form Cooper pairs

0

2 Meissner with gap parameter . M T=0.1 The Meissner mass goes imagi- -0.1 nary when the gap in the disper- T=0 -0.2 sion relation reaches zero. 0 0.5 1 1.5 2 ∆

(Alford and Wang, hep-ph/0501078)

13 IV. Signatures of color superconductivity in compact stars Gaps in spectra aect Transport properties. Pairing energy aects Equation of state.

Transport properties, mean free paths, conductivities, viscosities, etc. 1. Cooling by neutrino emission, neutrino pulse at birth (Page, Prakash, Lattimer, Steiner, hep-ph/0005094; Carter and Reddy, hep-ph/0005228 Reddy, Sadzikowski, Tachibana, nucl-th/0306015; Alford, Jotwani, Kouvaris, Rajagopal, astro-ph/0411560) 2. Gravitational waves: r-mode instability (Madsen, astro-ph/9912418) 3. Glitches and crystalline (“LOFF”) pairing (Alford, Bowers, Rajagopal, hep-ph/0008208 ) 4. Pulsar kicks and spin-1 pairing (Schmitt, Shovkovy, Wang, hep-ph/0502166)

14 Equation of state (mass-radius relationship) Pressure of quark matter relative to hadronic vacuum

4 2 2 2 2 pressure p = (1 c) () () Ms + () B

Parameters: c: Correction due to QCD interactions between quarks. Estimated c 0.3-0.4 (Fraga, Pisarski, Schaner-Bielich, hep-ph/0101143)

Ms: Strange quark mass. Actually depends on . Ms = 150–400 MeV. : Color-superconducting gap. Depends on . 10–100 MeV B: Bag constant. May depend on . B1/4 150–200 MeV

15 Inuence of c parameter on EoS

200_ 10000_

APR

) 3 150_ QM: c=0 ) 100_ 3 QM: c=0.3 1000_ QM: c=0 APR 50_

Energy Density (MeV/fm Pressure (MeV/fm Pressure 0_ QM: c=0.3 100_ -50_ 300 350 400 450 500 1 10 100 1000 µ (MeV) Pressure (MeV/fm3)

For c 0.3, the quark matter EoS is very similar to that of APR nuclear matter.

16 Mass-radius relationship We x the bag const B, which is not well known, by requiring the nuclear→quark phase transition to occur at = 1.5nsat.

ρ =1.5 n0 ms=150 2.5_

c=0, ∆=0 c=0, ∆=50 2_ c=0.3, ∆=0 c=0.3, ∆=50 APR98 only Hybrid star can be 1.5_ M indistinguishable (Mo.) 1_ from pure NM star up to M 1.9 M. 0.5_

0_ 8 10 12 14 R (km)

(Alford, Braby, Paris, and Reddy, nucl-th/0411016)

17 M(R) measurements and quark matter

What would rule out quark matter? Observed mass M & 2 M (Based on bag model with pert corrections and color SC, ignoring mixed phases).

What would indicate the presence of quark matter? Dicult. Regions of M-R space that cannot be reached by any nuclear matter EoS also cannot be reached by hybrid NM-QM EoS.

What would indicate the presence of color superconducting quark matter? More dicult. Even if we found an M(R) characteristic of quark matter, we would need an independent determination of the bag constant to claim that it was color-superconducting.

18 V. Looking to the future

Neutron-star phenomenology of color superconducting quark matter: – Structure: nuclear-quark interface – Crystalline phase and glitches – Vortices but no ux tubes – Eects of gaps in quark spectrum conductivity and emissivity (neutrino cooling) shear and bulk viscosity (r-mode spin-down) Particle theoretic questions:

– Response of CFL to Ms: gapless CFL, kaon condensation,. . .? – Magnetic instability of gapless phases – Better weak-coupling calculations, include vertex corrections – Go beyond mean-eld, include uctuations.

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