The Phase Structure of Quantum Chromodynamics

— thermodynamics from effective field theories —

Chihiro Sasaki

Institute of Theoretical Physics, University of Wroclaw, Poland

Selected Issues:

• interplay between chiral symmetry breaking and confinement

• phases of hot/dense QCD

• fate of hadrons near the phase transition

What is the origin of matter?

• after EW phase transition: Standard Model (EM & Weak plus Strong int.)

• QCD phase transition is the last transition in the evolution of the Universe!

up, down quarks (∼ 5 MeV): SM ingredientsWhere does the nucleon mass (∼ 1 GeV) come from?

10 % from EW/Higgs, and 90 % from QCD!

• scale and confinement? from elementary particle to composite states?· · · fundamental issues in non-abelian gauge theories!

∼ 150

∼ 200

• effective field theories:

– “particles” contained: either fundamental or composite

– based on fundamental/emergent symmetries

– broad spectrum of applications to QCD(-like), SM, BSM

Quantum Chromodynamics (QCD) underlies Hadron Physics

• theory of strong interaction among quarks (q) and gluons (Fµν)

SUc(3) : L = qaf

(iγµD

µab − δabmf

)qbf −

1

4Fα

µνFµνα

colors = (r , g , b) and flavors = (u, d, s, c, b, t)

asymptotic freedom⇒ color confinement

αs

Λ 200 MeV energyQCD

1

qq q q q-

Baryons (Fermions) Mesons (Bosons)

Hadrons (proton, neutron, pion ...)

• hadrons are “colorless”: qqq , qq ∼SU(3)c singlet

• exotic hadrons: tetraquarks (qqqq),pentaquarks (qqqqq), dibaryons(qqqqqq), glueballs

• origin of hadron masses?

u u du

u d15 MeV

940 MeV

proton

• current quark masses (cf. QCD scale ∼ 200 MeV)

mu . md < ms ¿ mc ¿ mb ¿ mt4 MeV 7 MeV 100 MeV 1.2 GeV 4 GeV 180 GeV

︸ ︷︷ ︸light quarks

︸ ︷︷ ︸heavy quarks

• chiral symmetry in u, d-quark sector (mu,d/ΛQCD ¿ 1)

L = L(qL) + L(qR) + mu,d (qLqR + qRqL) ∼ SUL(2)⊗ SUR(2)

Is it manifest in hadron spectra? ... NO

Is it manifest in hadron spectra? ... NOhelicity eigenstates ' parity eigenstates⇒ degenerate parity partners: mS ' mP ,mV ' mA

qq-excitations of the QCD vacuum

π

ρ ω

φ

(140)

(770) (782)

(1260)

(1020)

(1285)

(400-

1200)

a f

f

f

1 1

1

0

Energy (MeV)

P-S, V-A splitting

in the physical vacuum

(1420)

Dynamical breaking of chiral symme-try

• SUL(Nf )⊗ SUR(Nf ) → SUL+R(Nf )

• transition between qL and qR: 〈qq〉 6= 0

• Nambu-Goldstone theorem: pions as NGbosons

q

-q qgluon

QCD ground state in low energy/temperature/density

• color confinement: only SU(3)c singlet observed

• dynamical breaking of (approximate) SU(2)f chiral symmetry:pions as NG bosons, no degenerate parity partners

pion as NG boson

color confinement instantons

nuclear force

trace anomaly U(1)A anomaly

dynamical chiral symmetry breaking

L = - F F + q(iD - m)q-41 µν

µν /-

changing external parameter(s): different ground state!

∗ how are color-singlet states formed? exotic hadrons?

∗ dynamical origin of hadronic interactions and masses?

∗ composition of QCD matter at finite temperature/density?

I. Confinement vs. Dynamical Chiral symmetry Breaking

How to characterize a phase transition?

• order parameter: 〈0|φ|0〉 = 0 vs. 6= 0, susceptibility χ

T

χφ

Z[J ] = e−iW [J ] =

∫Dφ e[i

∫d4x(L[φ]+Jφ)] , χ =

δ2W [J ]

δJ(x)δJ(y)

∣∣∣J=0

= 〈φ(x)φ(y)〉 − 〈φ(x)〉〈φ(y)〉 .

• chiral oder parameter and its susceptibility from Lattice QCDinflection point of 〈qq〉 ' peak of χ [HotQCD Collaboration (’11)]

0

0.2

0.4

0.6

0.8

1

120 140 160 180 200

T [MeV]

∆l,sfK scale

asqtad: Nτ=8Nτ=12

HISQ/tree: Nτ=6Nτ=8

Nτ=12Nτ=8, ml=0.037ms

stout cont. 0

5

10

15

20

25

30

35

40

45

120 140 160 180 200 220 240

T [MeV]

χR/T4

fK scale

HISQ/tree: Nτ=6Nτ=8

asqtad: Nτ=8Nτ=12

stout: Nτ=8 Nτ=10

Chiral crossover at finite temperature

• chiral restoration in Nf = 2 QCD (massless quarks): O(4) universalityclass [Pisarski and Wilczek (’84)]

0.00

0.50

1.00

1.50

2.00

-5 -4 -3 -2 -1 0 1 2 3 4 5

t/h1/βδ

Mb/h1/δ

O(2)

all masses

ml/ms=2/51/5

1/101/201/401/80

140

150

160

170

180

190

200

210

0.00 0.05 0.10 0.15 0.20 0.25 0.30

ml/ms

Tc [MeV]

nf=2+1Nτ= 8

12scaling fctn.

scaling fctn.+regular

• confirmed by Lattice QCD simulations:also true in QCD with physical mπ! [Karsch (’11), Bazavov et al., (’12)]

• chiral crossover in QCD governed by O(4)

– important guide in building models

– critical behaviors independent of models! — universality

Confinement in Yang-Mills theory

• Polyakov loop (A4 = iA0)

L(~x) = P exp

[i

∫ 1/T

0dτA4(~x, τ )

]

Φ = TrL/3 , Φ = TrL†/3

– Φ → ei2πj3 Φ: its VEV is an order parameter of Z(Nc = 3) breaking

– heavy-quark potential [McLerran-Svetitsky (81)]

〈Φ(~x)〉 ∼ e−Fq(~x)/T

= 0 confined phase

6= 0 deconfined phase

• effective potential U = aΦΦ + b(Φ3 + Φ3) + c(ΦΦ)2 + · · · [Pisarski (’00)]

– T-dep. coefficients unconstrained by Z(3)⇒ putting T-dep. by hand & fitting Lattice EoS not unique!

– where T-dep. comes from? · · · thermal gluon excitations ∼ LYM⇒ closer contact with the underlying theory

• full thermodynamics potential: Ω = Ωg + ΩHaar [CS, Redlich (’12)]

Ωg = 2T

∫d3p

(2π)3ln

1 +

8∑

n=1

Cn(Φ, Φ) e−n|~p|/T ,

ΩHaar = −a0T ln[1− 6ΦΦ + 4

(Φ3 + Φ3

)− 3

(ΦΦ

)2]

,

C1 = C7 = 1− 9ΦΦ , C2 = C6 = 1− 27ΦΦ + 27(Φ3 + Φ3

),

C3 = C5 = −2 + 27ΦΦ− 81(ΦΦ

)2,

C4 = 2[−1 + 9ΦΦ− 27

(Φ3 + Φ3

)+ 81

(ΦΦ

)2]

, C8 = 1

⇒ energy distributions solely determined by group characters of SU(3)

one parameter in ΩHaar: a0 ⇔ T latc = 270 MeV

• any finite temperature in confined phase: Φ = 0

Ωg(Φ = Φ = 0) ∼ 2T

∫d3p

(2π)3ln

(1+e−|~p|/T

)

wrong sign! ⇒ unphysical EoS p, s, ε < 0

Gluons are NOT correct dynamical variables below Tc!

Confinement in QCD

• NJL/QM models under a constant background A0[Meisinger-Ogilvie (96), Fukushima (03), Mocsy-Sannino-Tuominen (04), Ratti-Thaler-Weise (06)]

Lkin = q (i/∂ − /A0) q

⇒ Ωq = dq T

∫d3p

(2π)3ln

[1 + 3Φe−E/T + 3Φe−2E/T + e−3E/T

]

〈Φ〉 ' 0 at low T: 1- and 2-quark states thermodynamically irrelevant⇒ mimicking confinement

• partial success, and missing pieces

NJLHchiral limitL

PNJL

puregauge

1st order2nd order

0.0 0.5 1.0 1.50.0

0.2

0.4

0.6

0.8

1.0

0.0

0.2

0.4

0.6

0.8

1.0

|〈ψψ〉||〈ψψ〉|

0

〈Φ〉

T/Tc

– correct dof at low/high T limit

– similar inflection point in T

– quarks at any T!

– how good is Z(3)?

QCD thermodynamics at µq ' 0 from lattice simulations

• deconfinement and SU(2) chiral restoration set in at Tpc ∼ 155 MeV.

→ conserved charges → melting chiral condensate

• fluct. of conserved charges with:µ = BµB + SµS + QµQ

e.g. kurtosis of baryon number fluctuations

κ = χB4 /χB

2 = B2 →

1 , T ¿ Tc

1/9 , T À Tc,

χBn = ∂n(P/T 4)/∂(µB/T )n

• Polyakov-loop sus. and their ratios:a clear remnant of Z(3)in RA = χabsolute/χreal! [Lo et al. (’13)]

cf. pure YM: exact Z(3), deconf p.t.

0.2

0.4

0.6

0.8

1

1.2

0.5 1 1.5 2 2.5 3 3.5 4

RA

T/Tc

nf=0 (483x4)nf=0 (483x6)nf=0 (643x8)

nf=2+1 (323x8)

To which extent does DχSB contain information on confinement?

• Banks-Casher relation: low-lying Dirac eigenmodes generate 〈qq〉.removal of low-lying Dirac modes ⇒ NO DχSBQ. does confinement disappear simultaneously?

[Gattringer (’06); Bruckmann et al. (’07); Synatschke et al. (’08); Gongyo et al.(’12); Doi et al. (’13,14)]

unmodified string tension, Polyakov loop and its fluctuations ... A. NO!

• hadron masses vs. lowest Dirac-eigenmodes [Glozman, Lang, Schrock (’12)]

– parity partners degenerate and stay quite massive!

– cf. LQCD at µ = 0: mN−T→Tch→ mN+ ∼ m

(vac)N+ [Aarts et al., (’15) ]

– no universal scaling (mmeson ∼ 2m, mbaryon ∼ 3m) found⇒ the system remains confined! χ-restored phase with conf.

• topology and confinement [’t Hooft (’75); Mandelstam (’76)]

how monopoles and Dirac zero modes are related? [e.g. Larsen, Shuryak (’14,’15)]

• is it odd if the two transitions happen separately?cf. adjoint QCD, Tch ∼ 8Tdec

II. Hadronic Matter at High Density

How to suppress unphysical d.o.f. at T = 0? [Benic, Mishustin, CS (’15)]

• IR cutoff b: NJL, SD eq., AdS/QCD [Ebert, Feldmann, Reinhardt (’96); · · · ]1/b ∼ typical size of a hadron ⇒ modified FD distribution functions

nq = θ(~p2−b2)fq , nN± = θ(α2b2−~p2)fN± , fX = 1/1+eβ(EX∓µX) .

• asymptotic behavior at high density:non-int. quark gas, restored chiral symmetry⇒ density dep. IR cutoff, upper limit αmax = 2−1/3 ∼ 0.8

• model setup: Ω =∑

X=N±,q ΩX + Vσ + Vω + Vχ + Vb

Vb = −κ2b

2b2 +

λb

4b4 .

parameter fixing: (κb, λb) ← (εvac, Tch) from lattice QCD

• deconfinement criteria

YN++ YN− = Yq ; YN± =

ρN±ρB

, Yq =1

3

ρq

ρB.

Phase diagram from effective theories

800 1000 1200 1400 1600 1800 2000µ

B (MeV)

0

0.2

0.4

0.6

0.8

1

YX

αb0=300 MeV

1000 1500 2000 2500 3000 3500 4000µ

B (MeV)

0

0.2

0.4

0.6

0.8

1

YX N

+N

-qαb0=440 MeV

0 5 10 15 20n

B (n

0)

250

500

750

1000

1250

1500

p (M

eV/f

m3 )

αb0=300 MeV

αb0=440 MeV

deconfinement

chiral

• quark-meson-nucleon hybrid model at T = 0 and large µB⇒ ρch separated from ρdec α-dep.: ∆ρcr/ρ0 ∼ 4-12

• decreasing ∆ρcr(T )due to bosonic thermal fluctuations

• holographic QCD approach:

– large Nc limit

– less parameters

temp

LQCD

liquid-gasch.pot.

?

Role of a tetraquark: more exotic phases [Harada-CS-Takemoto (09)]

µ

T chiral sym. restored

<σ> = 0

chiral sym. broken

<σ> = 0

I

II

<χ> = 0& deconfined

III

<χ> = 0 <χ> = 0

<σ> = 0

µ

σχ χB

0

0

µz2 µchiral

• symmetry breaking: SU(Nf)L × SU(Nf)R → SU(Nf)V × ZNf→ SU(Nf)V

2-quark state σ ∼ qq and 4-quark state χ ∼ (qq)2 + qq-qq

• 3 phases: baryon number flutc. χB becomes max. when 〈σ〉 → 0 ,whereas ch.sym. remains broken by 〈χ〉 6= 0

• multiple critical points, the same universality class as the “ordinary” CP⇔ different universality from anomaly induced CP [Hatsuda et al. (06-07)]

∵ U(1)B is broken in CFL phase.

• hadron mass spectra

phase I: σ0 6= 0 , χ0 6= 0 phase II: σ0 = 0 , χ0 6= 0

SU(2)V SU(2)V × (Z2)AmS 6= 0 ,mP = 0 mS 6= mP 6= 0 ,mP ′ = 0mV 6= mA mV 6= mAmN+ 6= mN− mN+ = mN− 6= 0

phase I: σ0 6= 0 , χ0 6= 0 phase II: σ0 = 0 , χ0 6= 0

SU(3)V SU(3)V × (Z3)AmS 6= 0 ,mP = 0 mS = mP 6= 0 ,mP ′ = 0mV 6= mA mV 6= mAmN+ 6= mN− mN+ = mN− 6= 0

• Nf = 2 + 1:(non-strange) mS 6= mP mN+ ' mN−(strange) mS ' mP mN+ ' mN−⇒ early onset of χSR for the strange mesons and non-/strange baryons!

Toward QCD p.t.: more and more hadrons activated!

• role of higher-lying hadrons

– V/A spectra: importance of heavier states towards Tch [Hohler, Rapp (’14)]

– role of higher KK modes in open moose model [Son, Stephanov (’03)]

correct high-energy behavior for current correlator

– nuclear matter saturation in Walecka modeldensity-dep. parameters: many-body effects integrated out

How to handle them?

• holographic QCD models: 1/Nc corrections?

• microscopic approach: lattice QCD, DS/FRGlattice ch. condensates at µ = 0: T -dep. int. of charmed mesons⇒ Ds more sensitive to O(4) than kaons [CS (’14); CS, Redlich (’14)]

• 4d effective theories: higher resonances, careful treatment of broad reso-nances [Broniowski et al. (’15); Friman et al. (’15)]

Summary

•Dynamical chiral symmetry breaking vs. confinement

– various fluctuations: remnant of underlying symmetry

– why Tch ' Tdec at µ = 0?: PNJL/QM models, instanton-dyon picture

– not clear at high density: Dirac eigenmode expansion, model w/ IRcutoff, cf. adjoint QCD: Tdec 6= Tch

•Nature of hadrons in vacuum and in a medium

– exotic phases at high density, new CPs, χB

– role of higher-lying hadrons

– careful treatment of broad resonances required

– exotic hadrons

•Our goal: from hadrons to quarks and gluonsmultifaceted studies of gauge dynamics guided by symmetry, topology,ideal limits AND available data from LQCD simulations and HIC