Transport properties of quark matter

Prof. Mark AlfordWashington University in St. Louis

Schematic QCD phase diagram

superconductingquark matter

= color−

liq

T

µ

gas

QGP

CFL

nuclear

/supercondsuperfluid

compact star

non−CFL

heavy ioncollider

hadronic

M. Alford, K. Rajagopal, T. Schafer, A. Schmitt, arXiv:0709.4635 (RMP review)

A. Schmitt, arXiv:1001.3294 (Lecture notes)

Low-temperature critical pointMarch 10 talk, T. Hatsuda

Phase transition lines associated with spontaneous breaking of exactsymmetries cannot end.If the symmetry is explicitly broken, the line can end at a critical point.

3 massless quarksno instantons

µ

qLqR

qLqL

qLqL

qLqR

U(1)AChiralT (1)U B

2LZ Z2R

Z2L : qL → −qL, qR → qR

Z2R : qL → qL, qR → −qR

3 massless quarks

µ

qLqR

qLqL

qLqR

qLqL

T (1)U B

Chiral

approx Z2R2LZ

2 light + 1 massive

µ

qLqL

qLqR

CFL

qLqL

qLqR

fluid

hadronsuper−

hadronfluid

2LZ Z2R

T

QGP

(1)U Bapprox

approx Chiral

Color superconducting phases

Attractive QCD interaction ⇒ Cooper pairing of quarks.

We expect pairing between different flavors .

Quark Cooper pair: 〈qαiaqβ

jb〉color α, β = r , g , bflavor i , j = u, d , sspin a, b =↑, ↓

Each possible BCS pairing pattern P is an 18× 18 color-flavor-spinmatrix

〈qαiaqβ

jb〉1PI = ∆P Pαβij ab

The attractive channel is:

space symmetric [s-wave pairing]color antisymmetric [most attractive]spin antisymmetric [isotropic]

⇒ flavor antisymmetric

Quark matter in the real world

In the real world there are three factors that combine to oppose pairingbetween different flavors. March 10 talk, A. Schmitt

1. Strange quark mass is not infinite nor zero, but intermediate. Itdepends on density, and ranges between about 500 MeV in thevacuum and about 100 MeV at high density.

2. Neutrality requirement. Bulk quark matter must be neutralwith respect to all gauge charges: color and electromagnetism.

3. Weak interaction equilibration. In a compact star there is timefor weak interactions to proceed: neutrinos escape and flavor is notconserved.

Cooper pairing vs. the strange quark mass

Unpaired

blue

s

p

red green

F

u

dMs

24µ

2SC pairing

blue

s

p

red green

F

u

d

CFL-K 0

δm<0

K0E

p

CFL pairing

blue

p

red green

F

CFL: Color-flavor-locked phase, favored at the highest densities.

〈qαi qβ

j 〉 ∼ δαi δ

βj − δα

j δβi = εαβNεijN

breaks chiral symmetry by a new mechanism: 〈qq〉 instead of 〈qq〉.2SC: Two-flavor pairing phase. May occur at intermediate densities.

〈qαi qβ

j 〉 ∼ εαβ3εij3 ∼ (rg − gr)(ud − du)

The CFL-K 0 phaseCFL breaks chiral symmetry ⇒ octet of light pseudoscalar mesons.Lightest ones are K 0 and K + (because they have s quarks).

EK0(p) = −M2s

2µ+√

13p2 + m2

K0 ≈ mK0 − M2s

2µ︸ ︷︷ ︸δm

+13p2

2mK0

+ · · ·

CFL: δm > 0

K0

δm>0K0m

E

p

CFL-K 0: δm < 0

δm<0

K0E

p

K 0 can condense ifits energy gap δmgoes negative(Bedaque, Schafer,

hep-ph/0105150).

Phases of quark matter, again

liq

T

µ

gas

QGP

CFL

nuclearsuperfluid/supercond

compact star

non−CFL

heavy ioncollider

hadronic

NJL model, uniform phases only

µB/3 (MeV)

T(M

eV)

550500450400350

60

50

40

30

20

10

0

g2SC

NQ

NQ

2SC

uSCguSC →CFL

← gCFL

CFL-K0

p2SC−→χSB

t

t tt

Warringa, hep-ph/0606063

But there are also non-uniform phases, such as the crystalline(“LOFF”/“FFLO”) phase. (Alford, Bowers, Rajagopal, hep-ph/0008208)

Low-energy excitations of uniform quarkmatter phases

For 2 light and 1 massive flavor, approx symmetry group is

SU(3)c × SU(2)L × SU(2)R × U(1)B × U(1)S∗

Phase unbroken sym gapless modes light modesunpaired no breaking 9 quarks

2SC SU(2)c× global 5 quarks

CFL-K 0 SU(2)Vsf phonon “H”sf kaon∗

CFL SU(2)V × U(1)S∗ sf phonon “H” K 0, K +

∗ weak interactions break strangeness slightly.

Signatures of color superconductivity incompact stars

Pairing energy

affects Equation of state . Hard to detect.(Alford, Braby, Paris, Reddy, nucl-th/0411016)

Gaps in quark spectraand Goldstone bosons

affect Transport properties :emissivity, heat capacity, viscosity (shear, bulk),conductivity (electrical, thermal). . .

1. Gravitational waves: r-mode instability, shear and bulk viscosity2. Glitches and crystalline (“LOFF”) pairing3. Cooling by neutrino emission, neutrino pulse at birth

r-modes and gravitational spin-down

An r-mode is a quadrupole flowthat emits gravitational radiation. Itbecomes unstable (i.e. arises spon-taneously) when a star spins fastenough, and if the shear and bulkviscosity are low enough.

Side viewPolar view

mode pattern

star

The unstable r -mode can spin the star down very quickly, in days toyears (Andersson gr-qc/9706075; Friedman and Morsink gr-qc/9706073; Lindblom

astro-ph/0101136).

So if we see a star spinning quickly, we can infer that the interiorviscosity must be high enough to damp the r -modes.

Constraints from r-modesRegions above curves are forbidden because viscosity is too low to hold backthe r -modes.

Nuclear matter

610 710 810 910 1010 11100

0.2

0.4

0.6

0.8

1

T (K)

KΩ/

Ω

LMXBs

n=1n=2

)APR (M=1.4 M)APR (M=1.8 M

Quark matter

610 710 810 910 1010 11100

0.2

0.4

0.6

0.8

1

LMXBs

T (K)

KΩ/

Ω

BagCFLAPR+BagAPR+CFL

(Jaikumar, Rupak, Steiner arXiv:0806.1005)

This motivates the calculation of bulk and shear viscosity for all phasesthat might occur in a neutron star.

What is viscosity?

(L. viscum; It: vischio, Eng: mistletoe, Ger: Mistelzweig, Jp:Bp ', Fr: gui,

Sp: muerdago, Po: jemio la . . .)

A sticky glue was made from mistletoe berries and coated onto small tree

branches to catch birds.

What is viscosity?

(L. viscum; It: vischio, Eng: mistletoe, Ger: Mistelzweig, Jp:Bp ', Fr: gui,

Sp: muerdago, Po: jemio la . . .)

A sticky glue was made from mistletoe berries and coated onto small tree

branches to catch birds.

What is bulk viscosity?

Energy consumed in acompression cycle:

V (t) = V + Re[δV exp(iωt)]p(t) = p + Re[δp exp(iωt)]⟨

dE

dt

⟩= −ζ

τ

∫ τ

0

(div~v)2dt =ζ

2ω2 δV 2

V 2= − 1

τ V

∫ τ

0

p(t)dV

dtdt

⇒ ζ(ω,T ) = − V

δV

Im(δp)

ω

Physically, bulk viscosity arises from re-equilibration processes. If some

quantity goes out of equilibrium on compression, and re-equilibrates on a

timescale comparable to τ , then pressure gets out of phase with volume and

energy is consumed. (Just like V and Q in an R-C circuit.)

Flavor re-equilibration processes

Bulk visc gets a separate contribution from each equilibrating quantity:K ≡ S − D (can equilibrate nonleptonically).

U − S − D (needs leptons to carry charge) (Sa’d, Shovkovy, Rischkeastro-ph/0703016).

phase unpaired/2SC CFL/CFL-K 0

lightestmodes

unpaired quarks(in 2SC: blue only )

H , K 0

flavorequilibrationprocesses

u + d ↔ s + u K 0 ↔ H HK 0 H ↔ H

u s

d uW

d

s

uW

H

H

Leptonic processes:small phase space

How bulk viscosity depends on flavorequilibration rate

ζ(ω,T ) = C (T )γeff(T )

γeff(T )2 + ω2

12

γω

ωCζ

I ω is angular frequency of applied compression cycle.I C measures the sensitivity of nK and nq to changes in µK and µq.I γeff is the effective rate/particle of the re-equilibration process.

I As γeff → 0, ζ → 0. No equilibration.I As γeff →∞, ζ → 0. Infinitely fast equilibration.

I In unp/2SC, Fermi surface modes dominate, so C is constant forT µq, and bulk viscosity peaks when ω = γeff(T ).

I In CFL/CFL-K 0, C (T ) washes out the peak.

How bulk viscosity depends on flavorequilibration rate

ζ(ω,T ) = C (T )γeff(T )

γeff(T )2 + ω2

12

γω

ωCζ

I ω is angular frequency of applied compression cycle.I C measures the sensitivity of nK and nq to changes in µK and µq.I γeff is the effective rate/particle of the re-equilibration process.

I As γeff → 0, ζ → 0. No equilibration.I As γeff →∞, ζ → 0. Infinitely fast equilibration.

I In unp/2SC, Fermi surface modes dominate, so C is constant forT µq, and bulk viscosity peaks when ω = γeff(T ).

I In CFL/CFL-K 0, C (T ) washes out the peak.

How bulk viscosity depends on flavorequilibration rate

ζ(ω,T ) = C (T )γeff(T )

γeff(T )2 + ω2

12

γω

ωCζ

I ω is angular frequency of applied compression cycle.I C measures the sensitivity of nK and nq to changes in µK and µq.I γeff is the effective rate/particle of the re-equilibration process.

I As γeff → 0, ζ → 0. No equilibration.I As γeff →∞, ζ → 0. Infinitely fast equilibration.

I In unp/2SC, Fermi surface modes dominate, so C is constant forT µq, and bulk viscosity peaks when ω = γeff(T ).

I In CFL/CFL-K 0, C (T ) washes out the peak.

Bulk viscosity of quark matter phases

0.01 0.1 1 10T (MeV)

100_

105_

1010

_

1015

_

1020

_

1025

_

1030

_µ = 400 MeV Ms = 90 MeV τ = 1 ms

bulk

vis

c ζ

(g

cm-1

s-1

)

ϕ m.f.p. greater than 10 km

unpaired2SC

δm =

4 M

eV

CFL-K0, δm = -0.5 MeV

CFL,

ϕ

CFL, δm = 0.5 MeV

CFL-K0, δm = -4 MeV

ϕ + ϕ

δm = kaon mass gapδm > 0: CFLδm < 0: CFL-K 0

ζCFL and ζCFL−K0

would peak aroundT ∼ 10 MeV butstrong C (T ) depen-dence masks the peak.

Alford, Schmitt, nucl-th/0608019; Manuel, Llanes-Estrada, arXiv:0705.3909;Alford, Braby, Reddy, Schafer, nucl-th/0701067; Alford, Braby, Schmitt, arXiv:0806.0285

Features of the bulk viscosity

1. Why are unpaired and 2SC more dissipative than CFL/CFL-K 0?• unp/2SC: K = D − S equilibration is dominated by quark modesnear the Fermi surface, so rate is high (lots of phase space).• CFL/CFL-K 0: equilibration is dominated by kaons, so rate is slower(less phase space).

2. Why do unpaired and 2SC have the same peak structure?Bulk viscosity generically has a resonant peak.In 2SC, fewer unpaired flavors just reduces γeff by a factor of 9.

3. Why does CFL drop fast at T . δm?In CFL, δm > 0, so thermal kaon density ∼ exp(−δm/T ), dropsrapidly for T δm.

4. Why is CFL more dissipative than CFL-K 0 at T ∼ 1 MeV?Massive K 0 can then go to on-shell H .

Features of the bulk viscosity

1. Why are unpaired and 2SC more dissipative than CFL/CFL-K 0?• unp/2SC: K = D − S equilibration is dominated by quark modesnear the Fermi surface, so rate is high (lots of phase space).• CFL/CFL-K 0: equilibration is dominated by kaons, so rate is slower(less phase space).

2. Why do unpaired and 2SC have the same peak structure?Bulk viscosity generically has a resonant peak.In 2SC, fewer unpaired flavors just reduces γeff by a factor of 9.

3. Why does CFL drop fast at T . δm?In CFL, δm > 0, so thermal kaon density ∼ exp(−δm/T ), dropsrapidly for T δm.

4. Why is CFL more dissipative than CFL-K 0 at T ∼ 1 MeV?Massive K 0 can then go to on-shell H .

Features of the bulk viscosity

1. Why are unpaired and 2SC more dissipative than CFL/CFL-K 0?• unp/2SC: K = D − S equilibration is dominated by quark modesnear the Fermi surface, so rate is high (lots of phase space).• CFL/CFL-K 0: equilibration is dominated by kaons, so rate is slower(less phase space).

2. Why do unpaired and 2SC have the same peak structure?Bulk viscosity generically has a resonant peak.In 2SC, fewer unpaired flavors just reduces γeff by a factor of 9.

3. Why does CFL drop fast at T . δm?In CFL, δm > 0, so thermal kaon density ∼ exp(−δm/T ), dropsrapidly for T δm.

4. Why is CFL more dissipative than CFL-K 0 at T ∼ 1 MeV?Massive K 0 can then go to on-shell H .

Features of the bulk viscosity

1. Why are unpaired and 2SC more dissipative than CFL/CFL-K 0?• unp/2SC: K = D − S equilibration is dominated by quark modesnear the Fermi surface, so rate is high (lots of phase space).• CFL/CFL-K 0: equilibration is dominated by kaons, so rate is slower(less phase space).

2. Why do unpaired and 2SC have the same peak structure?Bulk viscosity generically has a resonant peak.In 2SC, fewer unpaired flavors just reduces γeff by a factor of 9.

3. Why does CFL drop fast at T . δm?In CFL, δm > 0, so thermal kaon density ∼ exp(−δm/T ), dropsrapidly for T δm.

4. Why is CFL more dissipative than CFL-K 0 at T ∼ 1 MeV?Massive K 0 can then go to on-shell H .

Bulk viscosity of nuclear matterIn npeµ matter, there is bulk viscosity from “Modified Urca” processes

N n N p e− νe

` µ− ` e−νe νµ

The leptonic contribution has recently been calculated.

0.001 0.01 0.1 1 10T [MeV]

1e-08

0.0001

1

10000

1e+08

1e+12

1e+16

1e+20

1e+24

1e+28

1e+32

ζ [g

cm

-1s-1

]

µl = 110 MeV

µl = 200 MeV

µl = 300 MeV

Nucleonic

Alford, Good, 1003.1093

• Oscillation freq f = 1 kHz.• At T Tc for nucleon su-perfluidity (1 MeV) leptoniccontribution dominates.• For r -modes, shear viscos-ity is more important at thesetemperatures.

Suprathermal bulk viscosityIn the large amplitude regime, which is inevitably reached by unstablemodes like r-modes, bulk viscosity depends on amplitude as well astemperature:

Unpaired quark matter

Alford, Mahmoodifar, andSchwenzer, unpublished.

March 9 talk, K. Schwenzer

Future directionsPhenomenology of dense matter in compact stars:

I extending to the high-amplitude (“suprathermal”) regimeI shear viscosity (Manuel, Dobado, Llanes-Estrada, hep-ph/0406058;

Alford, Braby, Mahmoodifar, arXiv:0910.2180)

I recalculation of r-mode damping timesI detailed analysis of r -mode profiles in hybrid starI crystalline phase (glitches) (gravitational waves?)I other transport properties (conductivities, heat capacity, emissivity

etc) of various phases (2SC etc)I CFL: vortices but no flux tubes (March 9 talk by K. Itakura)

More basic issues:I magnetic instability of gapless phasesI better weak-coupling calculationsI better models of quark matter: PNJL, Schwinger-Dyson

(K. Schwenzer)I solve the sign problem and do lattice QCD at high density

Enhancement of CFL bulk visc at T ∼ 1 MeVImaginary part of kaon self-energy

Σ =~

G 2ds f 2

π f 2H(p2

0 − v 4ds |~p|2)2Im Π(p)

(p20 − v 2|~p|2)2 + Im Π2(p)

Peaked when K can go to on-shell H ,i.e. when εK (p) = v p.

pp k

εH

(CFL)K

(CFL−K0)K

Plot of kaon decay rate

∆m=+2 MeV

∆m=-

2 MeV

CFL-K0

CFL

0.01 0.1 1 1010-37

10-32

10-27

10-22

10-17

10-12

TMeV

ΛM

eV3

• At T ∼ p, CFL is en-hanced.• At very low T , massivekaons in CFL are absent.