Fedora Core 1 GNU/Linux; LATEX 2; xfig

Color : accomodating the strange Mark Alford Washington University Saint Louis, USA

1 Outline

I The phases of quark Confined, quark- plasma, color superconducting II High density QCD Color-superconducting quark matter, Color-flavor locking (CFL) III Quark matter in compact stars

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

2 I. The phases of quark matter Low temperatures and densities : confined, broken chiral symmetry.

High temperatures (T & 150 MeV): quark-gluon plasma (QGP) • chiral symmetry restored • deconfinement • signatures sought at heavy-ion colliders

−3 High densities (nquark & 1 fm ): color superconductivity • pair in color non-singlets. • Rich variety of phases depending on which colors and flavors participate.

3 Conjectured QCD phase diagram Light strange quark Heavy strange quark or large pairing gap or small pairing gap T T

RHIC RHIC

QGP QGP

non−CFL confined

gas liq CFL gas liq CFL

nuclear µ nuclear compact star µ heavy ion collisions: chiral critical point and first-order line compact stars: color superconducting quark matter core

4 T II. High density QCD QGP At sufficiently high density and low tempera- ture, there is a Fermi sea of almost free quarks. crystal?

gas liq CFL

nuclear µ µ = EF E F = E − µN p But quarks have attractive QCD interactions.

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

5 High-density QCD calculations Guess a color-flavor-spin pairing pattern P ; mini- mize free energy wrt ∆: gap equation for ∆. α β αβ = hqiaqjbi1P I = Pij ab∆

1. Weak-coupling methods. First-principles calculations direct from QCD Lagrangian, valid in the asymptotic regime, currently µ & 106 MeV. 2. Nambu–Jona-Lasinio models, ie quarks with four- fermion coupling based on instanton vertex, single gluon exchange, etc. This is a semi-quantitative guide to physics in the compact star regime µ ∼ 400 MeV, not a systematic approximation to QCD. NJL gives ∆ ∼ 10−100 MeV at µ ∼ 400 MeV. Both methods agree on the favored pairing pattern.

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

α β α β α β αβA hqi qj i ∼ (κ+1)δi δj + (κ−1)δj δi =  ijA + κ(· · · ) color α, β This is invariant under equal and opposite flavor i, j rotations of color and (vector) flavor SU(3) × SU(3) × SU(3) ×U(1) → SU(3) ×Z color | L {z R} B | {zC+L+R} 2

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

7 III. 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) > ρnuclear ∼ 30 MeV

The star cools by neutrino emission for the first 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 infinite 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 flavor is not conserved. So quark matter in a compact star might be CFL, or something else: gapless CFL; kaon-condensed CFL, 2SC, 1SC, crystalline,. . .

9 Neutrality and weak equilibrium

• Weak equilibrium + 2 u → d e ν¯ µu = µ − 3 µe + 1 u → s e ν¯ µd = µs = µ + 3 µe • Electromagnetic neutrality [NB: But there may be a globally neutral ∂Ω Q = = 0 mixture of positive nuclear matter with ∂µ e negative quark matter.] • Color neutrality. In unitary gauge, number of red, green, blue quarks must be the same. The cost of projecting to a color singlet

is then negligible. T3 = diag(1, −1, 0), T8 = diag(1, 1, −2) ∂Ω ∂Ω hQ3i = = 0 hQ8i = = 0 ∂µ3 ∂µ8

10 So a general color-superconductivity calculation takes the form

1. Write down free energy Ω(µ, Ms, µ3, µ8, µe, ∆) 2. Solve ∂Ω ∂Ω ∂Ω Neutrality conditions: = = = 0 ∂µe ∂µ3 ∂µ8 ∂Ω Gap equations: = 0 ∂∆i For a complicated pairing ansatz there may be many gap parameters

∆i each with its own gap equation. Now we know how to impose weak equilibrium and gauge neutrality, we just have to include the strange quark mass and see what happens to CFL pairing.

11 Color superconductivity with realistic strange quark mass

With Ms ∼ 120−500 MeV, the pure CFL pairing pattern is distorted and perhaps broken. Approaches to studying this are: 1. Use an effective theory of the low-energy degrees of freedom, including flavor rotations of the CFL condensate (“pions” and “kaons”). 2. Allow condensates with momentum: crystalline color superconductivity. 3. Use a more general pairing ansatz, eg allowing d-u pairing to differ from d-s and u-s. This is how we get gapless CFL (gCFL). 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 µb − µa > 2∆. For CFL, unpairing starts when µ > 2∆CF L.

12 Distortions of CFL with rising Ms

Keep only the κ = 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. d ∆ud ∆ds Plug it into an NJL model and minimize the s ∆us ∆ds free energy, imposing ∆ud d − neutrality and weak equi- u −∆ud librium. s −∆us In each 2 × 2 block, u −∆us chemical potentials and Ms try to pull the Fermi s −∆ds momenta apart; pairing ∆ds d − holds them together.

13 Unpairing, blocking regions, gaplessness, etc Quasiparticle dispersion relations E−µ

blocking δµ =0 for 2 species of massless fermion, region with chem pots µ¯  δµ, that pair |δµ| >∆ with gap parameter ∆. ˛ 2 2˛ E(p) = ˛δµ  p(p − µ¯) + ∆ ˛ ∆ ˛ ˛ p ∆ µ gapless modes When |δµ| = ∆, unpairing begins: a “blocking” or “breached pairing” region opens up, bordered by gap- less modes. We can apply this to the 2 × 2 blocks of the CFL pairing. Strange 2 quarks effectively have a contribution −Ms /2µ to their chem pot near the Fermi surface (p = µ). 2 In the ds sector, unpairing begins when Ms /µ > 2∆ds.

14 Gapless CFL (gCFL)

10 ]

4 g2SC unpaired NJL model (gluon-like interaction) 0 MeV

6 with fixed Ms. -10 2SC • CFL → gCFL (continuously) -20 gCFL when sd and us pairs start break- -30 2 ing, at Ms /µ ≈ 2∆CF L(µ). -40 CFL • gCFL → g2SC → unpaired (first- Energy Difference [10 Difference Energy -50 2 0 25 50 75 100 125 150 s 2 µ order) at higher M /µ. MS/ [MeV] 30 ∆ ud 40 25 ud unpairing line gCFL 20 ∆ 20 us µ e neutrality 15 CFL line 0 10

unpairing line ds unpairing line Gap Parameters [MeV] ∆ -20 us 5 ds

0 25 50 75 100 125 150 0 2 0 25 50 75 100 125 150 M /µ [MeV] 2 µ S (MS ) / [MeV]

15 Gapless CFL (gCFL) summary • gCFL is a form of CFL in which some unpairing is occurring in

response to Ms and neutrality.

• CFL smoothly turns into gCFL with increasing Ms (decreasing density).

• gCFL contains electrons (µe > 0) and is metallic, not insulating. • Charge of electrons is cancelled by unpaired u quarks. • gCFL contains gapless quasiquarks, including ones with an (almost) quadratic dispersion relation. Interesting effects on transport properties expected.

16 gCFL at T > 0

∆ 3=0

15 dSC ∆ 2=0 2SC ∆ ∆ 1> 2 10 g2SC [MeV] T uSC ∆ ∆ 5 2> 1

CFL ∆ 1=0 gCFL

0 0 50 100 150 2 / µ ms [MeV]

(Fukushima, Kouvaris, Rajagopal, in preparation.)

17 IV. Signatures of color superconductivity in compact stars Gaps in spectra affect Transport properties. Pairing energy affects 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) 2. Gravitational waves: r-mode instability (Madsen, astro-ph/9912418) 3. Glitches and crystalline (“LOFF”) pairing (Alford, Bowers, Rajagopal, hep-ph/0008208 )

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

4 2 2 2 2 pressure p = (1 − c) (·) µ − (·) Ms µ + (·) ∆ µ − B

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

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

19 Mass-radius relationship We fix the bag const B, which is not well known, by requiring the nuclear→quark phase transition to occur at ρ = 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, nucl-th/0312007; Alford, Braby, Paris, and Reddy, to be published

20 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? Difficult. 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 difficult. 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.

21 V. Looking to the future

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

– Response of CFL to Ms: gapless CFL, kaon condensation,. . .? – Better weak-coupling calculations, include vertex corrections – Go beyond mean-field, include fluctuations.

22