QCD at Finite Density (Nuclear/Quark Matter) · 2008. 6. 30. · Dense Baryonic Matter Low Density...
Transcript of QCD at Finite Density (Nuclear/Quark Matter) · 2008. 6. 30. · Dense Baryonic Matter Low Density...
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QCD at Finite Density
(Nuclear/Quark Matter)
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Schematic Phase Diagram
T
µ
µeneutron
matter
nuclearmatter
quark
matter?
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Dense Baryonic Matter
Low Density
Equation of state of nuclear/neutron matter
Neutron/proton superfluidity, pairing gaps
Moderate Density
Pion/kaon condensation, hyperon matter
Pairing, equation of state at high density
High Density
Quark matter
Color superconductivity, Color-flavor-locking
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Dense Baryonic Matter
Low Density (constrained by NN interaction, phenomenology)
Equation of state of nuclear/neutron matter
Neutron/proton superfluidity, pairing gaps
Moderate Density (very poorly known)
Pion/kaon condensation, hyperon matter
Pairing, equation of state at high density
High Density (weak coupling methods apply)
Quark matter
Color superconductivity, Color-flavor-locking
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Low Density: Nuclear Effective Field Theory
Low Energy Nucleons:
Nucleons are point particles
Interactions are local
Long range part: pions
+ : : : = ++ : : :
Advantages:
Systematically improvable
Symmetries manifest (Chiral, gauge, . . .)
Connection to lattice QCD
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Effective Field Theory
Effective field theory for point-like, non-relativistic neutrons
Leff = ψ†(
i∂0 +∇2
2M
)
ψ− C0
2(ψ†ψ)2 +
C2
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[
(ψψ)†(ψ↔
∇2
ψ)+h.c.]
+ . . .
Simplifications: neutrons only, no pions (very low energy)
Effective range expansion
p cot δ0 = −1
a+
1
2
∑
n
rnp2n
Coupling constants
C0 =4πa
M, C2 =
4πa2
M
r
2, . . . a = −18 fm, r = 2.8 fm
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Neutron Matter: Universal Limit
Consider limiting case (“Bertsch” problem)
(kFa) → ∞ (kF r) → 0
Why is this limit interesting? (Close to real world!)
Scale (and conformal) invariance
Universal equation of state E/A = ξ(E/A)0
Cross section saturates QM unitarity bound
Perfect fluid?
Connection to cold atoms
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Cold Fermi Gases
Universal equation of state
(E/A) = 0.42(E/A)0
Experiment, Quantum MC, ǫ expansion
Transport: η/s from damping of collective modes
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0.2 0.4 0.6 0.8
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
η/s
T/TF
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Epsilon Expansion
Bound state wave function ψ ∼ 1/rd−2. Nussinov & Nussinov
d ≥ 4: Non-interacting bosons ξ(d = 4) = 0
d ≤ 4: Effective lagrangian for atoms Ψ = (ψ↑, ψ†↓) and dimers φ
L = Ψ†
(
i∂0 +σ3∇2
2m
)
Ψ + µΨ†σ3Ψ + Ψ†σ+Ψφ+ h.c.
Nishida & Son (2006)
Perturbative expansion: φ = φ0 + gϕ (g2 ∼ ǫ)
+ +
O(1) O(1) O(ǫ)
ξ =1
2ǫ3/2 +
1
16ǫ5/2 ln ǫ
− 0.0246 ǫ5/2 + . . .
ξ = 0.475 ∆ = 0.62EF
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Nuclear Matter
isospin symmetric matter: first order onset transition
ρ0 ≃ 0.14 fm−3 (kF ≃ 250 MeV) B/A = 15 MeV
can be reproduced using accurate VNN (V3N crucial, V4N ≈ 0)
EFT methods: explain need for V3N if Nf > 1 (and V4N ≪ V3N )
+ +systematic calculations difficult since kFa≫ 1, kF r ∼ 1
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Nuclear Matter at large Nc
Nucleon nucleon interaction is O(Nc)
mN = O(Nc) rN = O(1)
VNN = O(Nc)
Get SU(2Nf ) (Wigner symmetry) relations
C0(ψ†ψ)2 ≫ CT (ψ†~σψ)2
Dense matter: kF = O(1) (EF ∼ 1/Nc)
crystallization
Note: E ∼ Nc (no phase transition?)
B1 B2
B3 B4
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Very Dense Matter
Consider baryon density nB ≫ 1 fm−3
quarks expected to move freely
Ground state: cold quark matter (quark fermi liquid)
PF
PF
n(p)
p
only quarks with p ∼ pF scatter
pF ≫ ΛQCD → coupling is weak
No chiral symmetry breaking, confinement, or dynamically generated
masses
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High Density: Pairing in Quark Matter
QQ scattering in perturbative QCD
(~T )ac(~T )bd = −1
3(δabδcd−δadδbc)+
1
6(δabδcd+δadδbc)
[3] × [3] = [3] + [6]
Fermi surface: pairing instability in weak coupling
Φab,αβij = 〈ψa,α
i Cψb,βj 〉
Phase structure in perturbation theory
Minimize Ω(Φab,αβij )
In practice: consider Φab,αβij with residual symmetries
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Superconductivity
Thermodynamic potential
Ω = + + +. . .
Variational principle δΩ/δΦ gives gap equation
=∆(p0) =g2
18π2
∫
dq0 log
(
ΛBCS
|p0 − q0|
)
∆(q0)√
q20 + ∆(q0)2
ΛBCS = ci256π4µg−5
(ci depends on phase)∆i = 2ΛBCS exp
(
− 3π2
√2g
)
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Nf = 2: 2SC Phase
Nf = 2, color-anti-symmetric: spin-0 BCS condensate
〈ψbiCγ5ψ
cj〉 = Φaǫabcǫij
Order parameter φa ∼ δa3 breaks SU(3)c → SU(2)
SU(2)L × SU(2)R unbroken
4 gapped, 2 (almost) gapless fermions
light U(1)A Goldstone boson
SU(2) confined (Λconf ≪ ∆)
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Nf = 3: CFL Phase
Consider Nf = 3 (mi = 0)
〈qai q
bj〉 = φ ǫabIǫijI
〈ud〉 = 〈us〉 = 〈ds〉〈rb〉 = 〈rg〉 = 〈bg〉
Symmetry breaking pattern:
SU(3)L × SU(3)R × [SU(3)]C
× U(1) → SU(3)C+F
All quarks and gluons acquire a gap
[8] + [1] fermions, Q integer
FLL C C F
R R
<q q > <q q >L L R R
... have to rotate right
flavor also !
Rotate left flavor Compensate by rotating
color
〈ψLψL〉 = −〈ψRψR〉
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CFL Phase: What does it look like?
CFL phase is fully gapped
transparent insulator
CFL is a superfluid
rotational vortices
CFL is not an electric superconductor
magnetic flux only partially expelled
CFL is “confined”
excitations: mesons and baryons
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Towards the real world: Non-zero strange quark mass
Have ms > mu,md: Unequal Fermi surfaces
δpFδpF ≃ m2
s
2pF
Also: If psF < pu,d
F have unequal densities
Charge neutrality not automatic
Strategy
Consider Nf = 3 at µ≫ ΛQCD (CFL phase)
Study response to ms 6= 0
Constrained by chiral symmetry
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Effective theory: (CFL) baryon chiral perturbation theory
L =f2
π
4
Tr(
∇0Σ∇0Σ†)
− v2πTr
(
∇iΣ∇iΣ†)
+A
[Tr (MΣ)]2 − Tr (MΣMΣ) + h.c.
+ Tr(
N†ivµDµN)
−DTr(
N†vµγ5 Aµ, N)
− FTr(
N†vµγ5 [Aµ, N ])
+∆
2
Tr (NN) − [Tr (N)]2
∇0Σ = ∂0Σ + iµLΣ − iΣµR
DµN = ∂µN + i[Vµ, N ]
f2π =
21 − 8 log 2
18
µ2
2π2v2
π =1
3A =
3∆2
4π2D = F =
1
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Phase Structure and Spectrum
Phase structure determined by effective potential
V (Σ) =f2
π
2Tr
(
µLΣµRΣ†)
−ATr(MΣ†) −B1
[
Tr(Mӆ)]2
+ . . .
V (Σ0) ≡ min
Fermion spectrum determined by
L = Tr(
N†ivµDµN)
+ Tr(
N†γ5ρAN)
+∆
2
Tr (NN) − [Tr (N)]2
,
ρV,A =1
2
ξµLξ† ± Rξ
ξ =√
Σ0
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Phase Structure of CFL Phase
ms*
m*
CFL K0
η η+K0
ms*
m*
mcrits ∼ 3.03m
1/3
d ∆2/3
m∗ ∼ 0.017α4/3s ∆
QCD realization of s-wave meson condensation
Driven by strangeness over-saturation of CFL state
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Fermion Spectrum
20 40 60 80 100
20
40
60
80
m2s=(2pF ) [MeV
!i [MeV
p + n 0 0 p +n 0 0 n 0 0
mcrits ∼ (8µ∆/3)1/2
gapless fermion modes (gCFLK)
(chromomagnetic) instabilities ?
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Phase Diagram: ms 6= 0
T
Hadrons
Nuclear
Plasma
Matter
µ
E
E
2SC
CFL−K
CFL
exoticsnuclearsuperfluid
p−CFLK, LOFF
Phase structure at moderate µ (and ms, µe 6= 0) complicated and
poorly understood.Systematic calculations
m2s ≪ µ2,ms∆ ≪ µ2, g ≪ 1
Use neutron stars to rule out certain phases
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Composition of Neutron Stars
Hydrogen/Heatmosphere
R ~ 10 km
n,p,e, µ
neutron star withpion condensate
quark−hybridstar
hyperon star
g/cm3
1011
g/cm3
106
g/cm3
1014
Fe
−π
K−
s ue r c n d c t
gp
oni
u
p
r ot o
ns
color−superconductingstrange quark matter(u,d,s quarks)
CFL−K +
CFL−K 0
CFL−0π
n,p,e, µquarks
u,d,s
2SCCSLgCFLLOFF
crust
N+e
H
traditional neutron star
strange star
N+e+n
Σ,Λ,
Ξ,∆
n superfluid
nucleon star
CFL
CFL
2SC
F. Weber (2005)
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Observational Constraints
Mass-radius relationship, maximum mass
Equation of state
Cooling behavior
Phase structure, low energy degrees of freedom
Rotation
Equation of state, Viscosity
Spin-down, glitches
Superfluidity
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Resources
Star, Phenix, Phobos, Brahms, Discoveries at RHIC, Nuclear Physics
A750 (2005).
E. Shuryak, The QCD Vacuum, Hadrons, and Superdense Matter,
World Scientific, Singapore.
T. Schaefer, Phases of QCD, hep-ph/0509068.
T. Schaefer, Effective Theories of Dense and Very Dense Matter,
nucl-th/0609075.
J. Kogut, M. Stephanov, The Phases of QCD, Cambridge University
Press (2004).
M. Alford, K. Rajagopal, T. Schaefer, A. Schmitt, Color Supercon-
ductivity, arXiv:0709.4635.
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J. Lattimer and M. Prakash, The Physics of Neutron Stars,
astro-ph/0405262.
U. Heinz, Concepts of Heavy-Ion Physics, hep-ph/0407360.
D. Son, A. Starinets, Viscosity, Black Holes, and Quantum Field
Theory, arXiv:0704.0240.
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