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