The Phases of QCD

Thomas Schaefer

North Carolina State University

1 Motivation

Diﬀerent phases of QCD occur in the universe Neutron Stars, Big Bang Exploring the phase diagram is important to understanding the phase that we happen to live in Structure of hadrons is determined by the structure of the vacuum

Need to understand how vacuum can be modiﬁed QCD simpliﬁes in extreme environments Study QCD matter in a regime where quarks and gluons

are the correct degrees of freedom

2 QCD Phase Diagram

Big Bang T Quark Gluon Plasma

RHIC

Hadron Gas CERN Supernova

Nuclear Liquid

Color Superconductivity

Neutron Stars µ

3 Plan

1. QCD and Symmetries 2. The High Temperature Phase: Theory 3. Exploring QCD at High Temperature: Experiment 4. QCD at Low Density: Nuclear Matter 5. QCD at High Density: Quark Matter 6. Matter at Finite Density: From the Lab to the Stars

4 Quantumchromodynamics Elementary ﬁelds: Quarks Gluons

color a = 1, . . . , 3 a  a color a = 1, . . . , 8 (qα) spin α = 1, 2 A  f  µ §   spin ²µ ﬂavor f = u, d, s, c, b, t    Dynamics: non-abelian gauge theory

1 L = q¯ (iD/ − m )q − Ga Ga f f f 4 µν µν

a a a abc b c Gµν = ∂µAν − ∂ν Aµ + gf AµAν

µ a a iD/ q = γ i∂µ + gAµt q ¡ ¢

5 “Seeing” Quarks and Gluons

L LLLL 3 LLLLL333

e+ q e+ q g e− q e− q

6 Running Coupling Constant

7 Asymptotic Freedom

Study modification of classical field by quantum fluctuations

1 1 k2 A = Acl + δA F 2 → + c log F 2 µ µ µ g2 µg2 µµ2 ¶¶

¤ ¤ £ ¡ £ ¢¡

¥ ¤

¦¨§

dielectric ² > 1 paramagnetic µ > 1 dielectric ² > 1 µ² = 1 ⇒ ² < 1

g3 1 2 β(g) = 2 − 4 Nc + Nf (4π) nh3 i 3 o

8 About Units

QCD Lite∗ is a parameter free theory The lagrangian has a coupling constant, g, but no scale. After renormalization g becomes scale dependent g is traded for a scale parameter Λ Λ is the only scale, the QCD “standard kilogram”

−1 ΛQCD ' 200 MeV ' 1 fm

∗ QCD Lite is QCD in the limit mq → 0, mQ → ∞

9 Phases of Gauge Theories

Coulomb Higgs Conﬁnement ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

e2 e−mr V (r) ∼ V (r) ∼ V (r) ∼ kr r r

Standard Model: U(1) × SU(2) × SU(3)

10 Phases of Matter phase order broken rigidity Goldstone param symmetry phenomenon boson crystal ρk translations rigid phonon magnet M~ rotations magnetization magnon superﬂuid hΦi particle number supercurrent phonon supercond. hψψi gauge symmetry supercurrent none (Higgs)

11 Gauge Symmetry

Local gauge symmetry U(x) ∈ SU(3)c

ψ → Uψ Dµψ → UDµψ † † † Aµ → UAµU + iU∂µU Fµν → UFµν U

Gauge “symmetries” cannot be broken Gauge “symmetries” can be realized in different modes Coulomb Higgs confined d.o.f: 2 (massless) 3 (massive) 3 (massive) Distinction between Higgs and confinement phase not always sharp

12 Chiral Symmetry Deﬁne left and right handed ﬁelds L 1 ψ = (1 § γ )ψ R L,R 2 5

Fermionic lagrangian, M = diag(mu, md, ms)

¯ ¯ L = ψL(iD/ )ψL + ψR(iD/ )ψR L L R R

M M ¯ ¯ + ψLMψR + ψRMψL L R R L

M = 0: Chiral symmetry (L, R) ∈ SU(3)L × SU(3)R

ψL → LψL, ψR → RψR

13 Chiral Symmetry Breaking

Chiral symmetry implies massless, degenerate fermions

(1/2)+ ∗ − mN = 935 MeV mN (1/2) = 1535 MeV Chiral symmetry is spontaneously broken

¯f g ¯f g 3 fg hψLψR + ψLψRi ' −(230 MeV) δ

SU(3)L × SU(3)R → SU(3)V (G → H)

Consequences: dynamical mass generation mQ = 300 MeV À mq

mN = 890 MeV + 45 MeV (QCD, 95%) + (Higgs, 5%)

14 a Goldstone Bosons: Consider broken generator Q5

a a a a a a [H, Q5] = 0 Q5|0i = |π i H|π i = HQ5|0i = Q5H|0i = 0

Low energy eﬀective theory for the Goldstone modes Step 1: Parameterize G/H =pseudoscalar GB’s

† U(x) : U → LUR (L, R) ∈ SU(3)L × SU(3)R

Vacuum U fg = δfg. Massless ﬂuctuations (G/H)

a a a § 0 § 0 0 U(x) = exp(iφ λ /fπ) φ = (π , π , K , K , K¯ , η)

Step 2: Write most general G invariant eﬀective lagrangian f 2 L = π Tr[∂ U∂µU †] + . . . 4 µ Non-linear sigma model

15 Expand lagrangian (SU(2) sector)

1 1 ∂4 L = (∂ φa)2 + (φa∂ φa)2 − (φa)2(∂ φb)2 + O 2 µ 6f 2 µ µ µf 4 ¶ π £ ¤ π Step 3: Low energy expansion (power counting)

2 2 4 4 Tππ ∼ k1,2/fπ + k1,2/fπ + . . .

Relation to fπ: Couple weak gauge ﬁelds

§ ∓ π W ∂µU → (∂µ + igWµ τ )U e f § µ ∓ π ν L = gfπWµ ∂ π

16 Quark Masses † Non-zero quark masses: L = ψ¯LMψR + ψ¯RM ψR M → LMR† spurion ﬁeld M Chiral lagrangian at leading order in M L = BTr[MU] + h.c.

Mass matrix M = diag(mu, mdms). Minimize eﬀective potential

Uvac = 1, Evac = −BTr[M] hψ¯ψi = −B

Expand around Uvac: pion mass 2 2 ¯ mπfπ = (mu + md)hψψi Chiral expansion m n 4 ∂U mπ L = fπ Λχ = 4πfπ µ Λχ ¶ µ Λχ ¶ 17 Symmetries of the QCD Vacuum: Summary

Local SU(3) gauge symmetry conﬁned: V (r) ∼ kr

Chiral SU(3)L × SU(3)R symmetry

spontaneously broken to SU(3)V

Axial U(1)A symmetry

N anomalous : ∂ A0 = f Ga G˜a µ µ 16π2 µν µν

Vectorial U(1)B symmetry unbroken: B = d3x ψ†ψ conserved R

18 Notes

QCD with general Nf , Nc (with or without SUSY) ﬁnd theories without conﬁnement and/or chiral symmetry breaking

QCD with Nf = Nc = 3 conﬁnement implies chiral symmetry breaking

∗ symmetry breaking pattern SU(3)L × SU(3)R → SU(3)V unique order parameter hψ¯ψi 6= 0

19 QCD Phase Diagram: Nc and Nf

QCD SUSY QCD

14 14

b=0 b=2 IR free N f conformal 12 12

10 10 b'=0

8 8 conformal

6 6

'weak' chiral symmetry breaking Coulomb 4 IILM, chisb 4 lattice, conf

chiral symmetry breaking, confinement ¡ ¡ no ¡ chisb, ¡ ¡ confinement ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡

¡ ¡ chiral ¡ ¡ symmetry ¡ ¡ ¡ breaking, ¡ ¡ no ¡ confinementground ¡ ¡ state ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ ¡ 2 3 4 2 3 4 N c N c

20 Approaching the Phase Diagram: Symmetries and Weak Coupling Arguments

T QGP critical

critical

critical color pion superconductor µ gas e µ

neutron gas

21 Approaching the Phase Diagram: Strongly Correlated Phases

T QGP critical sQGP

Hagedorn critical

critical color BEC− superconductor pion nuclear liquid BCS µ gas e µ

neutron gas

22 Approaching the Phase Diagram: Experiments and Numerical Simulations

RHIC's lattice

HIC's

neutron stars µ e µ