Color Superconductivity

Simon Hands Swansea University

Color Superconductivity in Dense Matter?

A warm-up: the NJL Model

Overview of QC2D:Thermodynamic observables & deconfinementHadron spectrumGluodynamics

Collaborators: Phil Kenny, Seyong Kim, Peter Sitch,Jon-Ivar Skullerud, David Walters

IPPP Durham, 25th October 2007. – p.1/30

The QCD Phase Diagram

µ

T(MeV)

(MeV)

T

quarkmatter?

quark−gluon plasma

nuclearmatter

c

µ

100

200

500 1500

colorsuperconductor

hadronic fluid

RHIC/A

LICE

compact stars

critical endpoint

crossover(µ Ε ,Τ )Ε

FAIR ?

onset

crystalline

. – p.2/30

χSB vs. Cooper Pairing

pairing instability

FE

k

E

2Σ

E

k

E

k

E

k

instabilitypairing

Dirac Sea

Fermi Sea

2∆

. – p.3/30

Color Superconductivity

In the asymptotic limit µ→ ∞, g(µ) → 0, the ground stateof QCD is the color-flavor locked (CFL) state characterisedby a BCS instability, [D. Bailin and A. Love, Phys.Rep. 107(1984)325]

ie. diquark pairs at the Fermi surface condense via

〈qαi (p)Cγ5q

βj (−p)〉 ∼ εAαβεAij × const.

SU(3)c⊗SU(3)L⊗SU(3)R ⊗U(1)B ⊗U(1)Q →SU(3)∆⊗U(1)QGround state is simultaneously superconducting (8 gappedgluons with mass O(gµ)), superfluid (1 exact Goldstone),chirally broken (8 pseudo-Goldstones), and transparent(quasiparticles with Q 6= 0 gapped).

[M.G. Alford, K. Rajagopal and F. Wilczek, Nucl.Phys.B537(1999)443]

Asymptotically gap scales as ∆ ∝ µ exp(

− 3π2

√2g

)

[D.T. Son, Phys.Rev.D59(1999)094019]. – p.4/30

At smaller densities such that µ/3 ∼ kF<∼ ms, expect

pairing between u and d only ⇒ “2SC” phase

〈qαi (p)Cγ5q

βj (−p)〉 ∼ εαβ3εij × const.

SU(3)c −→ SU(2)c ⇒ 5/8 gluons get gappedGlobal SU(2)L⊗SU(2)R⊗U(1)B unbroken

Another possibility in isopsin asymmetric matter is theso-called “LOFF” phase:

〈u(kuF ; ↑)d(−kd

F ; ↓)〉 6= 0

In the electrically-neutral matter expected in compact stars,kd

F − kuF = µe ∼ 100MeV ⇒ 〈ψψ〉 condensate has ~k 6= 0

breaking translational invariance ⇒ crystallisationOther ideas:a 2SC/normal mixed phase (plates? rods?)

or a gapless 2SC where 〈qq〉 6= 0 but ∆ = 0. – p.5/30

A Gentle Reminder

QCD condensed matter physics could be very rich![M.G. Alford, A. Schmitt, K. Rajagopal & T. Schäfer, arXiv:0709.4635]

BUT

. – p.6/30

A Gentle Reminder

QCD condensed matter physics could be very rich![M.G. Alford, A. Schmitt, K. Rajagopal & T. Schäfer, arXiv:0709.4635]

BUTThe most urgent issue of all – whether quark matter existsin a stable form in the cores of compact stars – requiresquantitative knowledge of the Equation of Statenq(µ), p(µ), ε(µ) for all µ > µo

the oldest problem in strong interaction physics?[L.D. Landau, Phys.Z.Sowjetunion 1(1932)285]

. – p.6/30

What can we say at smaller densites µ ∼ O(1 GeV) whereweak coupling methods can’t be trusted? Lattice QCDsimulations can’t help because the Euclidean path integralmeasure detM(µ) is not positive definite.

The “Sign Problem”

In condensed matter theory there are two tractable limits:

Weak Coupling Strong Coupling

physical d.o.f.’s weakly interacting fermions tightly-bound bosons

superfluiditymechanism BCS condensation Bose-Einstein Condensation

QFT example NJL model QC2D

Both model QFT’s can be studied with µ 6= 0 using latticesimulations which evade the Sign Problem.

High-Tc superconducting compounds, cold atoms near aFeshbach resonance, and perhaps QCD, are difficultproblems because they belong to neither limit

. – p.7/30

The NJL Model SJH, D.N. Walters, PRD69:076011 (2004)

Effective description of soft pions interacting withnucleons/constituent quarks

LNJL = ψ(∂/ +m+ µγ0)ψ − g2

2[(ψψ)2 − (ψγ5~τψ)2]

∼ ψ(∂/ +m+ µγ0 + σ + iγ5~π.~τ)ψ +2

g2(σ2 + ~π.~π)

Full global symmetry is SU(2)L⊗SU(2)R⊗U(1)B

Dynamical χSB for g2 > g2c ⇒ isotriplet Goldstone ~π

Scalar isoscalar diquark ψtrCγ5 ⊗ τ2 ⊗Acolorψ breaks U(1)B

⇒ diquark condensation signals high density ground stateis superfluid

. – p.8/30

In 3+1d, NJL is non-renormalisable, so an explicit cutoff isrequired. We follow the large-Nf (Hartree) approach ofKlevansky (1992) and match lattice parameters to lowenergy phenomenology:

Phenomenological Lattice ParametersObservables fitted extractedΣ0 = 400MeV ma = 0.006

fπ = 93MeV 1/g2 = 0.495

mπ = 138MeV a−1 = 720MeV

Lattice regularisation preserves SU(2)L⊗SU(2)R⊗U(1)B. . .

. . . but phenomenological NJL is barely a field theory

. – p.9/30

Equation of State and Diquark Condensation

! " # !

$ "

% #

&

. – p.10/30

Equation of State and Diquark Condensation

' () *+ , - . /0 1 /20 1 - 3 -4 , / 5 + 4 678 -9:

;< < =

>

?@ A?B@ CB@ DB@ EB@ ABB@ C

B@ FB@ D

B@ GB@ E

B@ HB@ A

B@ ?B

I JLK MN O PQ MK R SUT VK PNW M

XZY Y [\] ^ _X Y Y [

`

ab cadb edb fdb gdb cddb e

db hdb f

db idb g

db jdb c

db ad

kml lon p qrqs t uvw xzy vw y |vw y vw y ~vw y

xz | z ~ z

z z

z z x

z xz

Add source j[ψtrψ + ψψtr]

Diquark condensate esti-mated by taking j → 0

Our fits exclude j ≤ 0.2

. – p.10/30

The Superfluid Gap

Quasiparticlepropagator:

〈ψu(0)ψu(t)〉 = Ae−Et +Be−E(Lt−t)

〈ψu(0)ψd(t)〉 = C(e−Et − e−E(Lt−t))

Results from 96 × 122 × Lt, µa = 0.8 extrapolated toLt → ∞ (ie. T → 0) then j → 0

z

¡¢¤£ ¥£

¦§ ¨ ¨ © ¡¢¤£ ¥£ª «

¬

­ ®¯ °

±²³ ±´±µ± ´¶¶£ ·

¶¸ ¶£ ·

¸ ¶£ ª¸ ¶£ ¹

¸ ¶£ º¸ ¶£ »

¸ ¶£ ¼¸ ¶£ ½

¸ ¶£ ¾¸ ¶£ ¿

The gap at the Fermi surface signals superfluidity

. – p.11/30

ÀÁ Â ÃÄ ÅÇÆ È ÉÊË ÌÍ Î Ì ÏÀÁ Â ÃÄ ÅÇÆ È ÉÊ Î ÌÍÐZÑ Ñ Ò ÓÔ

ÍÕ ÖÍÕ ×ÍÕ ØÍÕ ÊÍÕ ÙÍÕ ÏÍÕ ÚÍÕ ÌÍÕ ÉÍÍÕ Ì

ÍÕ É ÙÍÕ É

ÍÕ Í ÙÍ

ÛÍÕ Í Ù

• Near transition, ∆ ∼ const, 〈ψψ〉 ∼ ∆µ2

• ∆/Σ0 ' 0.15 ⇒ ∆ ' 60MeVin agreement with self-consistent approaches

• ∆/Tc = 1.764 (BCS) ⇒ Ltc ∼ 35explains why j → 0 limit is problematic

• Currently studying µI = (µu − µd) 6= 0,which “re”introduces a sign problem!

. – p.12/30

Two Color QCD - the large-N−1c limit

• τ2D/ (µ)τ2 = D/ ∗(µ) ⇒ det(D/ +m) is real• Spectrum contains degenerate qq mesons and qqbaryons so that chiral group at µ = 0 is enlarged fromSU(Nf )L⊗SU(Nf )R⊗U(1)B to SU(2Nf ) −→ Sp(2Nf )

Can analyse response to µ 6= 0 using chiral effectivetheory:LχPT =

f2π

2ReTr

Ü

∂νΣ∂νΣ† − 2m2

π

1

−1Σ + 4µ

1

−1Σ

†∂tΣ

− 2µ2Σ

1

−1Σ

† 1

−1+

1

1

Ý

The µ-dependent terms in LχPT are completely determinedby the global symmetries

. – p.13/30

Quantitatively, for µ >∼ µo χPT predicts (µo ≡ 1

2mπ)

〈ψψ〉〈ψψ〉0

=

(

µo

µ

)2

; nq = 8Nff2πµ

(

1 − µ4o

µ4

)

;〈qq〉〈ψψ〉0

=

√

1 −(

µo

µ

)4

[Kogut, Stephanov, Toublan, Verbaarschot & Zhitnitsky, Nucl.Phys.B582(2000)477]

confirmed by QC2D simulations with staggered fermions

0.0 1.0 2.0 3.0 4.02µ/mπ

0.0

0.2

0.4

0.6

0.8

1.0

1.2

m=0.10m=0.05m=0.01

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

µ

0.0

1.0

2.0

3.0

4.0

<ψψ>

<ψtrψ>

(<ψψ>2+<ψtrψ>2)0.5

nB

[SJH, I. Montvay, S.E. Morrison, M. Oevers, L. Scorzato J.I. Skullerud,

Eur.Phys.J.C17(2000)285, ibid C22(2001)451]

. – p.14/30

Thermodynamics at T = 0 from χPT

quark number density nχPT = 8Nff2πµ(

1 − µ4o

µ4

)

[KSTVZ]

pressure pχPT = −Ω

V=∫ µ

µonqdµ = 4Nff

2π

(

µ2 + µ4o

µ2 − 2µ2o

)

energy density εχPT = −p+ µnq = 4Nff2π

(

µ2 − 3µ4o

µ2 + 2µ2o

)

interaction measure (aka conformal anomaly)

δχPT = Θµµ = ε− 3p = 8Nff2π

(

−µ2 − 3µ4o

µ2 + 4µ2o

)

NB δχPT < 0 for µ >√

3µo

speed of sound vχPT =√

∂p

∂ε=

(

1−µ4o

µ4

1+3µ4

oµ4

)1

2

. – p.15/30

This is to be contrasted with our other paradigm for colddense matter, namely a degenerate system of weaklyinteracting (deconfined) quarks populating a Fermi sphereup to some maximum momentum kF ≈ EF = µ

⇒ nSB =NfNc

3π2µ3; εSB = 3pSB =

NfNc

4π2µ4;

δSB = 0; vSB =1√3

Superfluidity arises from condensation of diquark Cooperpairs from within a layer of thickness ∆ centred on theFermi surface:

⇒ 〈qq〉 ∝ ∆µ2

. – p.16/30

1 1.5 2 2.5

µ/µo

0

0.5

1

1.5

2

nχPT/n

SB

εχPT/ε

SB

pχPT/p

SB

vχPT/v

SBµ

d /µ

o

c

By equating free energies, we naively predict a first orderdeconfining transition from BEC to quark matter;eg. for f 2

π = Nc/6π2, µd ≈ 2.3µo.

. – p.17/30

Simulation Details (Nf = 2 Wilson flavors)

Initial runs used a 83 × 16 lattice with parametersβ = 1.7, κ = 0.1780 (Wilson gauge action)⇒ a = 0.220 fm, mπa = 0.79(1), mπ/mρ = 0.779(4)

Now have preliminary data from a matched 123 × 24 latticewith β = 1.9, κ = 0.1680

⇒ a = 0.15 fm, mπa = 0.68(1), mπ/mρ = 0.80(1)

⇒ T ≈ 60MeV in both cases

To counter IR fluctuations and to maintain ergodicity, weintroduce a diquark source jκ(−ψ1Cγ5τ2ψ

tr2 + ψtr

2 Cγ5τ2ψ1)

So far have accumulated roughly 300 trajectories of meanlength 0.5 on 83 × 16 and 100 trajectories on 123 × 24

SJH, Kim, Skullerud, Eur.Phys. J. C48 (2006)193

. – p.18/30

Computer effort

0 0.2 0.4 0.6 0.8 1µ

0

1000

2000

3000

Ncg

j=0j=0.02j=0.04j=0.06

The number of congrad iterations required forconvergence during HMC guidance rises as µ increases⇔ accumulation of small eigenvalues of M .

. – p.19/30

Equation of State (83 × 16)

0.2 0.4 0.6 0.8 1

µ

0

2

4

6

8 nq/n

SB

εq/ε

SB

p/pSB

Open symbols denote j → 0 extrapolation

. – p.20/30

Evidence for Deconfinement at µa ' 0.65

0.2 0.4 0.6 0.8 1

µ

0

0.1

0.2

0.3

0.4

0.5 <qq+>/µ2

Polyakov Loop x4j → 0

. – p.21/30

We conclude there is a transition from confined bosonic“nuclear matter” to deconfined fermionic “quark matter” atµd ≈ 0.65. Both phases are superfluid, but for µ > µd thescaling is that expected of a degenerate system.

In condensed matter parlance we are observing aBEC/BCS crossover.

What is the nature of the transition between these tworégimes?

. – p.22/30

Towards the continuum limit. . .

0.2 0.3 0.4 0.5 0.6 0.7 0.8

µa

0

0.02

0.04

0.06

0.08 nqa

3

<qq+>a

3

εga

4

L

mπa/2

0.2 0.3 0.4 0.5 0.6 0.7

µ

0

4

8

12

16

20

nq/n

SB

εq/ε

SB

p/pSB

123 × 24 results at β = 1.9 κ = 0.168 ja = 0.04

Identify onset transition at µa ≈ 0.32 and (very tentatively)a deconfining transition at µa ≈ 0.5

i.e. with µq ≈ 670 MeV, nq ≈ 5 fm−3, ∆εg<∼ 2 GeVfm−3

On Nτ = 24 the unrenormalised Polyakov loop L has very poor signal:noise

. – p.23/30

Mesons on 83 × 16 SJH, P. Sitch, J.I. Skullerud arXiv:0710.1966

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

µa

0

0.5

1

1.5

pionrhoI=0 0

-

I=0 0+

I=1 1+

Meson spectrum roughly constant up to onset. Thenmπ ≈ 2µ in accordance with χPT, while mρ decreases oncenq > 0, in accordance with effective spin-1 action

[Lenaghan, Sannino & Splittorff PRD65:054002(2002)]

Cf. Hiroshima group [Muroya, Nakamura & Nonaka PLB551(2003)305]. – p.24/30

Diquark Spectrum on 83 × 16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

µa

0

0.5

1

1.5

2 isoscalar 0+ anti-diquark

isoscalar 0+ diquark

isovector 1+ anti-diquark

isovector 1+ diquark

isoscalar 0- anti-diquark

isoscalar 1- anti-diquark

Diquark spectrum modelled by mπ,ρ ± 2µ up to onset, whilepost-onset:

Splitting of “Higgs/Goldstone” degeneracy in I = 0 0+ channel

Meson/Baryon degeneracy in I = 0 0+ and I = 1 1+ channels. – p.25/30

Kaon Spectrum on 83 × 16

0 0.2 0.4 0.6 0.8

µa

0.5

0.75

1

1.25 K +

K−

K*+

K*−

“Kaons” have one quark propagating with µs ≡ 0.Kaon spectrum modelled by mπ,ρ ± µ up to onset, whilepost-onset mK−

>∼ µ, as if it were a weakly-bound state ofan s-quark and a u-hole.Suggestive of bound kaonic states in nuclear medium?

. – p.26/30

Electric gluon propagator (Landau gauge)

1 1.5 2 2.5

|q|a

0

0.5

1

1.5

2

µ=0.35µ=0.50µ=0.90µ=1.00

Plot DE(q0, ~q) for fixed q0 as a function of |~q|. The electricgluon in the static limit q0 = 0 shows some evidence ofDebye screening as |~q| → 0 for µ >∼ 0.9.

. – p.27/30

Magnetic gluon

0 0.5 1 1.5 2 2.5

|q|a

0

1

2

3

4

µ=0.35µ=0.50µ=0.90µ=1.00

The effect of µ 6= 0 is much more dramatic in the magneticsector.This is significant because in perturbation theory magneticgluons are not screened in the static limit.

. – p.28/30

A Star is Born?

0.2 0.4 0.6 0.8 1

µ

0

0.5

1

1.5

2

2.5

3

3.5

εq/n

q raw data

εq/n

qcorrected data

shallow minimum here

Remarkably, εq/nq exhibits a robust minimum for µ >∼ µd,implying that macroscopic objects such as Two Color Starsare largely made of quark matter. . .

. – p.29/30

A Star is Born?

0.2 0.4 0.6 0.8 1

µ

0

0.5

1

1.5

2

2.5

3

3.5

εq/n

q raw data

εq/n

qcorrected data

shallow minimum here

Remarkably, εq/nq exhibits a robust minimum for µ >∼ µd,implying that macroscopic objects such as Two Color Starsare largely made of quark matter. . .

0.2 0.4 0.6 0.8 1

µ

0

0.5

1

1.5

2

2.5

3

3.5

εq/n

q raw data

εq/n

qcorrected data

mantle

(stiff)

atmosphere

interior of

star (soft)

(soft)

. – p.29/30

Summary & Outlook

Thermodynamic results support a BEC at intermediateµ, and deconfined BCS superfluid at large µ

A non-vanishing gluon energy density may be a morereliable indicator of deconfinement than the Polyakovline as T → 0

In-medium decrease of ρ mass, meson/baryondegeneracy, and kaonic-nuclear bound states

Non-perturbative screening of gluon propagator inmagnetic sector

Bulk quark matter may be more stable energeticallythan predicted by χPT

Future analysis to include: nature of deconfinement,topological excitations. . .

. – p.30/30