QCD Thermodynamics from Effective Field Theories. Chihiro Sasaki Frankfurt Institute for Advanced Studies, Germany ITP, University of Wroclaw, Poland

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1 QCD Thermodynamics from Effective Field Theories Chihiro Sasaki Frankfurt Institute for Advanced Studies, Germany ITP, University of Wroclaw, Poland Selected Issues: interplay between chiral symmetry breaking and confinement hadrons near chiral symmetry restoration chiral thermodynamics of heavy-light hadrons

2 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 ingredients Where does the nucleon mass ( 1 GeV) come from? 10 % from EW/Higgs, and 90 % from QCD!

3 scale and confinement? from elementary particle to composite states? fundamental issues in non-abelian gauge theories! effective field theories: particles contained: either fundamental or composite! based on fundamental/emergent symmetries broad spectrum of applications to QCD, QCD-like, SM, BSM

4 Quantum Chromodynamics (QCD) underlies Hadron Physics theory of strong interaction among quarks (q) and gluons (F µν ) SU c (3) : L = q a f ( iγµ D µ ab δ abm f ) q b f 1 4 F α µνf µν α colors = (r, g, b) and flavors = (u, d, s, c, b, t) asymptotic freedom color confinement α s q q q q- q Baryons (Fermions) Mesons (Bosons) Hadrons (proton, neutron, pion...) 1 Λ QCD 200 MeV energy hadrons are colorless : qqq, qq SU(3) c singlet exotic hadrons: tetraquarks ( q qqq), pentaquarks ( qqqqq), dibaryons (qqqqqq), glueballs

5 origin of hadron masses? proton u u d u u d 15 MeV 940 MeV current quark masses (cf. QCD scale 200 MeV) m u m d < m s m c m b m t 4 MeV 7 MeV 100 MeV 1.2 GeV 4 GeV 180 GeV } {{ } light quarks } {{ } heavy quarks chiral symmetry in u, d-quark sector (m u,d /Λ QCD 1) L = L(q L ) + L(q R ) + m u,d ( q L q R + q R q L ) SU L (2) SU R (2) Is it manifest in hadron spectra?... NO

6 Is it manifest in hadron spectra?... NO helicity eigenstates parity eigenstates degenerate parity partners: m S m P, m V m A Energy (MeV) f 0( ) π (140) qq-excitations of the QCD vacuum a ρ (1260) (1285) ω (770) (782) f 1 1 f 1 φ (1420) (1020) P-S, V-A splitting in the physical vacuum Dynamical breaking of chiral symmetry SU L (N f ) SU R (N f ) SU L+R (N f ) transition between q L and q R : qq = 0 Nambu-Goldstone theorem: pions as NG bosons gluon -q q q

7 Questions How are hadrons formed from quarks and gluons? how are color-singlet states formed? exotic hadrons? dynamical origin of hadronic interactions and masses? composition of QCD matter at finite temperature/density? Dynamical χsb & Color Confinement L = F F µν + q(id - - m)q 4 color confinement trace anomaly pion as NG boson µν / instantons U(1) A anomaly dynamical chiral symmetry breaking nuclear force

8 Why QCD matter under extreme conditions? the origin of hadron masses disentangle the mechanism of the mass generation by reversing the process! dynamical disappearance of the mass heated: going backward in time in the evolution of the Universe compressed: several times of the nuclear matter density interior of compact stars interplay between chiral dynamics and color confinement Lattice QCD at small µ/t : no true phase transition but crossover which takes place first? at µ 0 LQCD is not accessible. accessible in current and future experiments hot matter: RHIC (BNL), LHC (CERN) dense matter: FAIR (GSI), NICA (Dubna), J-PARC (Tokai)

9 I. QCD Phase Transition

10 How to characterize a phase transition? order parameter: 0 φ 0 = 0 vs. 0, susceptibility χ φ χ Z[J] = e iw [J] = Dφ e [ i d 4 x(l[φ]+jφ)] δ 2 W [J], χ = = φ(x)φ(y) φ(x) φ(y). δj(x)δj(y) J=0 chiral oder parameter and its susceptibility from Lattice QCD inflection point of qq peak of χ [HotQCD Collaboration ( 11)] T l,s asqtad: N τ =8 N τ =12 HISQ/tree: N τ =6 N τ =8 N τ =12 N τ =8, m l =0.037m s stout cont. T [MeV] fk scale χ R /T 4 HISQ/tree: N τ =6 N τ =8 asqtad: N τ =8 N τ =12 stout: N τ =8 N τ =10 10 f K scale 5 T [MeV]

11 Chiral crossover at finite temperature chiral restoration in N f = 2 QCD (massless quarks): O(4) universality class [Pisarski and Wilczek ( 84)] M b /h 1/δ O(2) all masses t/h 1/βδ m l /m s =2/5 1/5 1/10 1/20 1/40 1/ T c [MeV] N τ = 8 12 scaling fctn. scaling fctn.+regular n f = m l /m s confirmed by Lattice QCD simulations: also true in QCD with physical m π! [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

12 (De)confinement in Yang-Mills theory center of SU(N c ) [Note: quarks break Z(N c ) explicitly.] Ω c = e iφ 1, and det Ω c = 1 φ = 2πj, j = 0, 1,, (N c 1) : Z(N c ) symmetry N c Polyakov loop (A 4 = ia 0 ) [ ] ˆL( x) = P exp i Φ = Tr ˆL/N c, 1/T 0 dτa 4 ( x, τ) Φ = Tr ˆL /N c Φ e i2πj Nc Φ: its VEV is an order parameter of Z(N c ) breaking heavy-quark potential [McLerran-Svetitsky (81)] { Φ( x) e F q( x)/t = 0 confined phase 0 deconfined phase effective potential U = a ΦΦ + b( Φ 3 + Φ 3 ) + c( ΦΦ) 2 + [Pisarski ( 00)]

13 (De)confinement in QCD NJL/QM models under a constant background A 0 [Meisinger-Ogilvie (96), Fukushima (03), Mocsy-Sannino-Tuominen (04), Ratti-Thaler-Weise (06)] L kin = q (i/ /A 0 ) q d 3 p [ ] Ω q = d q T (2π) 3 ln 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 NJL PNJL chiral limit ψψ pure ψψ Φ 0.4 gauge nd order 1 st order correct dof at low/high T limit similar inflection point in T quarks at any T! how good is Z(3)? T/T c

14 Probing deconfinement with baryon number fluctuations chemical potential for a particle: µ = Bµ B + Sµ S + Qµ Q B: baryon number, S: strangeness, Q: electric charge kurtosis of baryon number fluctuations κ = χ B 4 /χb 2, χb n = n (P/T 4 )/ (µ B /T ) n asymptotic values: { κ = B 2 1, T T c = 1/9, T T c <B 4 >-3<B 2 > 2 <B 2 > n f =2+1, m π =220 MeV n f =2, m π =770 MeV Resonance gas 1 Lattice QCD 0.5 [Schmidt [for RBC-Bielefeld Collaboration] ( 08)] 0 SB T [MeV]

15 Probing deconfinement with Polyakov-loop fluctuations R T n f =0 (48 3 x4) n f =0 (48 3 x6) n f =0 (64 3 x8) n f =2+1 (32 3 x8) T/T c Polyakov loop L: quite broaden in QCD [HotQCD Collaboration] susceptibilities of L and their ratios: [Lo, Friman, Kaczmarek, Redlich, CS ( 13)] modulus (A), real (R) and imaginary (I) part of L: χ A, χ R, χ I ratios: R A = χ A /χ R, R T = χ I /χ R vs. T ambiguity of renor. prescription can be avoided to large extent! pure YM vs. N f = QCD (HISQ) on lattice (T chiral = 155 MeV): broad R T due to exp. Z(3) symmetry

16 1.2 R A n f =0 (48 3 x4) n f =0 (48 3 x6) n f =0 (64 3 x8) n f =2+1 (32 3 x8) T/T c [Lo, Friman, Kaczmarek, Redlich, CS ( 13)] [Lo, Friman, Kaczmarek, Karsch, Redlich, CS ( 14)] a clear remnant of Z(3) in R A! pseudo-critical temperatures T ch T dec! quark number fluctuations: kurtosis measures fermion number B 2 Lattice QCD: T ch T dec at zero density Polyakov-loop model with quarks (a la PNJL/PQM) does not reproduce R A, R T dynamics of confinement missed!

17 II. Hadrons with Unbroken Chiral Symmetry

18 Confinement vs. dynamical chiral symmetry breaking (DχSB) to which extent does DχSB contain information on confinement? Banks-Casher relation: low-lying Dirac eigenmodes generate qq. qq = lim m q lim πρ(0), ρ(λ) = 1 δ(λ λ n ) V V removal of low-lying Dirac modes NO DχSB Q. does confinement disappear simultaneously? linking Polyakov loop to spectral function of lattice Dirac operator [Gattringer ( 06); Bruckmann, Gattringer, Hagen ( 07); Synatschke, Wipf, Langfeld ( 08)] manifestly gauge invariant formalism [Gongyo, Iritani, Suganuma ( 12); Doi, Iritani, Suganuma ( 13,14)] NO particular Dirac-modes that crucially affect confinement! disappearance of DχSB DOES NOT mean deconfinement? n

19 Fate of hadron masses toward chiral symmetry restoration mass [1/m ρ ] k N(+) 0th state N(-) 0th state N(+) 1st state N(-) 1st state (+) 0th state (-) 0th state (+) 1st state (-) 1st state mass [1/m ρ ] k a 1, 0th state b 1, 0th state ρ, 1st state σ [MeV] σ [MeV] hadron masses vs. truncation level on lattice [Glozman, Lang, Schrock ( 12)] removal of lowest Dirac-eigenmodes NO qq parity partners degenerate and stay quite massive! no universal scaling (m meson 2m, m baryon 3m) found the system remains confined! origin of a scale in χ-sym. phase? trace anomaly T µ µ α s π G2 in-medium gluon condensate: melting toward T ch, but non-vanishing

20 Dilaton as a conformal compensator [Paeng, Lee, Rho, CS ( 12-13)] m * / m chiral Lagrangian with dilaton: N N Fπ 2 cχ 2 from nonlinear to linear: Σ = Uχ = s + i π τ IR fixed point: g V g A U(1) vs. SU(2) at finite density U(2) L U(2) R SU(2) L SU(2) R U(1) V SU(2) V U(1) V 1-loop RGE analysis: ρnn-int. runs whereas ωnn-int. walks with n B! m N const.: emergence of a chiral-invariant mass!

21 III. Hadrons near Chiral Symmetry Restoration

22 Chiral symmetry restoration and in-medium axial-vector spectrum 0.08 Vacuum T 100 MeV T 140 MeV ΡV, A s Π s Vector Axial vector Vector Axial vector Vector Axial vector T 150 MeV T 160 MeV T 170 MeV ΡV, A s Π s Vector Axial vector Vector Axial vector Vector Axial vector s GeV s GeV s GeV 2 [Hohler, Rapp ( 14)] Weinberg sum rules: f π ρ V,A (s) [Weinberg ( 67); Kapusta, Shuryak ( 94)] strategy: phenomenologically accepted ρ V (T ) & ansatz for m a1 (T ), Γ a1 (T ) WSRs in-medium ρ A (T ) smooth reduction of m a1 m ρ and broader width importance of higher-lying states: ρ, a 1,...

23 V-A mixing at finite density longitudinal transverse average L z R ImG V 10-1 L R s 1/2 [GeV] no charge-conjugation inv. at finite density ρ-a 1 mixing at tree level L mix = 2Cɛ 0νλσ tr [ ν V λ A σ + ν A λ V σ ] transv: p 2 0 p2 = 1 [ m 2 ρ + m 2 a 2 1 ± (m 2 a 1 m 2 ρ) C 2 p 2] mixing strength: χeft [Harada, CS ( 09)] vs. AdS/QCD [Domokos, Harvey ( 07)] C = 0.1 GeV vs. 1 GeV at n B = n 0! C = 1 GeV vector-meson condensation at n 0!? L V R

24 why C(n 0 ) = 1 GeV in AdS/QCD? C ω C = C ω + C ω + C ω +? role of infinite KK modes: current correlator at large Q 2 from open moose model [Son, Stephanov ( 03)] Q G 2 V 1 8π correct high-energy behavior! 2 ln Q2 nuclear potential from Sakai-Sugimoto model [Hashimoto, Sakai, Sugimoto ( 09)] repulsive core, tensor force, spin/isospin dep., V (r) 1/r 2 truncation vs. integrating-out different r dependence! V (r) = 1 r + e r r V N (r) = 1 r + e r r + e 2r r + e 2r r + + e Nr r + small r 1 r 2 exp[ mr] 1 N r

25 Role of higher-lying hadronic states near phase transitions increasing T and n B toward QCD phase transition: More and more hadronic states activated!... How to handle them? holographic QCD models: 1/N c corrections? 4d effective field theories: unknown interactions among higher-lying states? if renormalized/integrated out, T, n B -dep. interactions? L (π, ρ, ω, ) L low lying (π, ρ, ω, ; g(t, n B )) might be bridged via DS eq. and FRG? fixed along with comparison to lattice observables?

26 IV. Chiral Thermodynamics with Charm

27 Symmetries of QCD in the heavy quark mass limit flavor symmetries chiral symmetry : m u,d /Λ QCD 1, m s /Λ QCD < 1. heavy quark symmetry : Λ QCD /m c,b 1. heavy-light (Q q) mesons Q : heavy quark and q : light quark e.g. D mesons: Q = c, q = u, d, s physical picture (m Q ) flavor symmetry (c b): cloud does not feel the flavor of Q. spin symmetry: cloud does not feel the spin of Q. Spin and flavor symmetries of heavy quarks are entangled!

28 SU(2N Qf ) spin-flavor symmetry: [Shuryak ( 81), Isgur-Wise ( 89)] light d.o.f. (q) do not feel the flavor and spin of the heavy quark (Q). B b flavor c D spin spin B * b flavor c D * spin partners: D(0 ) and D(1 ), B(0 ) and B(1 ) real world: m D m D = 142 MeV, m B m B = 46 MeV Λ QCD 1/m Q corrections m Ds m Dd = 100 MeV, m Bs m Bd = 90 MeV Λ QCD m q corrections

29 Role of light flavor (chiral) symmetry observation: 2nd lowest spin doublets D u,d (0 + ) : 2308 MeV [Belle (03)] D u,d (1 + ) : 2427 MeV [Belle (03)] D s (0 + ) : 2317 MeV [Babar (03)] D s (1 + ) : 2460 MeV [CLEO (03)] mass difference of parity doublets: δm = MeV Λ QCD NOTE: potential model for D mesons (cf. hydrogen atom) does not work! chiral doubling [Nowak-Rho-Zahed (92); Bardeen-Hill (93)] chiral sym heavy quark sym D(0 + ) D(1 + ) D(0 - ) D(1 - ) heavy quark sym chiral sym effective theory for heavy-light system based on the two relevant symmetries

30 In-medium charmed-meson masses g π s (T)/gπ s (T=0) M Ds [GeV] T/T pc T/T pc chiral splitting at T pc : δm D δm Ds insensitive to light flavors! heavy quark symmetry light mesons at T pc : δm π-σ δm K-κ SU(2+1) SU(3) cf. chiral SU(4): [Roder-Ruppert-Rischke ( 03)] δm D δm Ds D( 0 ) D( 0 ) D s D s ( 0 ) ( 0 )

31 Correlations between light and heavy-light mesons [CS-Redlich ( 14)] σ q,s vs. D q,s D q,s vs. D q,s χ(t)/χ(t=0) σ q D q σ s D q σ q D s σ s D s χ(t)/χ(t=0) D q D q D q D s D s D s T/T pc T/T pc qualitative changes set in at T T pc : ˆχ σd = ˆχ ch Ĉ HL ˆχ D, ˆχ Dσ = ˆχ D Ĉ HL ˆχ ch, ˆχ DD = ĈD ĈHL ˆχ ch Ĉ HL ˆχ D. in-medium D s as a probe of O(4)!

32 Summary Polyakov-loop ratio & quark number fluc. suggest T ch T dec at µ = 0 survival hadrons in confined phase at µ 0? dual Meissner effect monopole condensation: confinement and DχSB induced by monopole condensation how monopoles and Dirac eigenmodes are related? importance of higher-lying states near phase transitions effective theory of HL mesons: D, D s to measure a remnant of O(4) More to come from Exp, LQCD, EFTs! Exciting moments in hadron physics!