Hadrons in hot and dense matter I

Hadrons in hot and dense matter I Hendrik van Hees Goethe University Frankfurt and FIAS July 17-21, 2017 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 1 / 38

Outline 1 Plan of the lectures and motivation 2 The phase diagram of strongly interacting matter 3 QCD and chiral symmetry Particles and forces in the Standard Model Quantum Electrodynamics (QED) Quantum Chromodynamics (QCD) and chiral symmetry Chiral effective models for hadrons 4 Strongly interacting matter: QCD/hadronic models at finite T,µ 5 References 6 Quiz Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 2 / 38

Plan of the lectures Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 3 / 38

Plan of the Lectures Lecture I: Fundamentals symmetries and conservation laws in (quantum) field theory the Standard Model in a nutshell QCD, chiral symmetry, and effective models for hadrons Lecure II: electrodynamic probes in heavy-ion collisions (fundamentals) transport and hydrodynamics collective flow radiation of electromagnetic probes from a thermal transparent medium (McLerran-Toimela formula) effective hadronic models for vector mesons Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 4 / 38

Plan of the Lectures Lecture III: Dileptons in heavy-ion collisions (SIS@GSI) hadronic models for transport models: baryon resonances Gießen Boltzmann-Uehling-Uhlenbeck (GiBUU) Ultrarelativistic Quantum Molecular Dynamics (UrQMD) medium modifications: transport-hydro hybrid and coarse-graining approach Lecture IV: Electromagnetic probes in heavy-ion collisions (SPS@CERN, RHIC@BNL, LHC@CERN) hard-thermal-loop approved dilepton rates (emission from QGP) hadronic many-body theories (emission from hadron gas) dileptons at SPS and RHIC access to the phase diagram? Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 5 / 38

Phase diagram, HICs, dileptons Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 6 / 38

The phase diagram of strongly interacting matter hot and dense matter: quarks and gluons inside hadrons compressed highly energetic scatterings deconfinement/chiral phase transition quarks and gluons relevant d.o.f. Quark-Gluon Plasma still strongly couples: fast thermalization! in the vacuum quark condensate q q 0 at high T,µ B : chiral phase transition q q = 0 T [GeV] 0.25 Quark-Gluon Plasma 0.20 RHIC qq = qq = 0 SPS 0.15 crit. point AGS 0.10 0.05 hadron gas qq 0 SIS muclei 0.0 µ N 1st order qq 0 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 7 / 38

Ultra-relativistic heavy-ion collisions highly energetic collisions of (heavy) nuclei many collisions of partons inside the nucleons creation of many particles hot and dense fireball generation of the Quark-Gluon Plasma (QGP)? properties of QGP and/or compressed baryonic matter? signatures of 1st-order phase transition (critical end point)? Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 8 / 38

Hydrodynamical behavior particle spectra compatible with collective motion of an ideal fluid small viscosity Medium in local thermal equilibrium (after short formation time 1 fm/c ) y Au Au x v 2 = p 2 x p 2 y p 2 x +p2 y p t (GeV) Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 9 / 38

Why Electromagnetic Probes? γ,l ± : only e. m. interactions reflect whole history of collision chance to see chiral symm. rest. directly? dn ee / dydm π o,η Dalitz-decays ρ,ω Φ DD J/Ψ π,... ρ/ω γ e Ψ l e + Drell-Yan a 1 Low- Intermediate- High-Mass Region > 10 fm > 1 fm < 0.1 fm 0 1 2 3 4 5 Fig. by A. Drees (from [RW00]) mass [GeV/c 2 ] Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 10 / 38

Vacuum Baseline: e + e hadrons 10 2 10 u, d, s 3 loop pqcd Naive quark model R 1 10 1 Sum of exclusive Inclusive measurements measurements s(gev) 0.5 1 1.5 2 2.5 3 R := σ e + e hadrons σ e +e µ + µ probes all hadrons with quantum numbers of γ R QM = N c f =u,d,s Q 2 f = 3 [(2/3) 2 + ( 1/3) 2 + ( 1/3) 2 ] = 2 Our aim pp l + l, pa l + l, AA l + l (l = e,µ) Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 11 / 38

The CERES findings: Dilepton enhancement pp (pbe): elementary reactions ; baseline (mandatory to understand first!) pa: cold nuclear matter effects ; next step (important as baseline for other observables like J /ψ suppression ) AA: medium effects ; hope to learn something about in-medium properties of vector mesons, fundamental QCD properties Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 12 / 38

The CERES findings: Dilepton enhancement Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 13 / 38

QCD and chiral symmetry Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 14 / 38

The standard model in a nutshell: particles and forces [graphics from http://www.isgtw.org/spotlight/go-particle-quest-first-cern-hackfest] Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 15 / 38

Quantum Electrodynamics (QED) Literature: [Nac90, DGH92, B + 12], conventions as in [Nac90] electrons+positrons: massive spin-1/2 Dirac field ψ 4 describes electron (charge q e = 1) and antielectron (=positron) photon: massless vector field A µ antisymmmetric field-strength tensor ( E, B ) 0 E 1 E 2 E 3 E F µν = µ A ν ν A µ = 1 0 B 3 B 2 E 2 B 3 0 B 1 E 3 B 2 B 1 0 Lagrangian (e > 0: em. coupling constant e 2 /(4π) = α em 1/137 = 1 4 F µνf µν + ψ[i( / + iq e e /A) m]ψ, q e = 1 Dirac matrices: γ µ 4 4, µ {0, 1, 2, 3}, γ µ,γ ν = 2η µν = diag(1, 1, 1, 1), ψ = ψ γ 0 Feynman slash /A = A µ γ µ, / = γ µ µ = γ µ Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 16 / 38 x µ

Symmetries of QED as a classical field theory: Least-action principle equations of motion action (Lorentz invariant!) S[A,ψ] = d 4 x symmetries lead to conservation laws (Noether s Theorem) space-time symmetries time translations: energy conservation space translations: momentum conservation rotations: angular-momentum conservation intrinsic symmetry: invariant under change of phase factor ψ exp( iq e e α)ψ, α electric-charge conservation here even local gauge symmetry: j (e)µ em = q ee ψγ µ ψ, µ j (e)µ em = 0 ψ exp[ iq e e χ(x )]ψ, A µ A µ + q e µ χ local symmetry gauge boson Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 17 / 38

Quantization fields operators physical quantities S-matrix elements: T f i 2 transition probabilities for scattering from asymptotic free initial to asymptotic free final state local, microcausal quantum field theory with stable ground state spin-statistics relation: half-integer spin fermions, integer spin bosons can only evaluate in perturbation theory Feynman rules Internal lines: Propagators External lines: Initial and final states µ µ p ν = ig γ µν ε µ (p) p (ε µ ) = ieγ µ = ig e (p) G µν γ = η µν /(p 2 + i0 + ), G e = (/p m)/(p 2 m 2 + i0 + ) e + in final state or e in initial state e + in initial state or e in final state Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 18 / 38

Quantum Chromodynamics: QCD Theory for strong interactions: QCD QCD = 1 4 F µν a F a µν + ψ(i /D ˆM )ψ non-abelian gauge group SU(3) color each quark: color triplet covariant derivative: D µ = µ + ig Tˆ a A a (a {1,..., 8}) field-strength tensor F a = µν µa a ν νa a g f a b c A b µ µ Ac ν group structure constants: [ Tˆ a, Tˆ b ] = if a b c Tˆ c, Tˆ a = ( Tˆ a ) 3 3 Particle content: ψ: Quarks with flavor (u,d ; c, s ; t, b ) (mass eigenstates!) ˆM = diag(m u,m d,m s,...) = current quark masses A a µ : gluons, gauge bosons of SU(3) color Symmetries fundamental building block: local SU(3) color symmetry in light-quark sector: approximate chiral symmetry ( ˆM 0) dilation symmetry (scale invariance for ˆM 0) Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 19 / 38

Features of QCD asymptotically free: at large momentum transfers α s = 4πg 2 s 0 running from renormalization group (due to self-interactions of gluons!): Nobel prize 2004 for Gross, Wilczek, Politzer 0.3 α s 0.2 0.1 0 1 10 100 µ(gev) quarks and gluons confined in hadrons theoretically not fully understood (nonperturbative phenomenon!) need of effective hadronic models at low energies: (Chiral) symmetry! Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 20 / 38

Chiral Symmetry of (massless) QCD Consider only light u, d quarks u iso-spin 1/2 doublet: ψ = = d ψ1 ψ 2 NB: ψ has three indices : Dirac spinor, color, flavor iso-spin! γ matrices: γ µ,γ ν = 2ηµν 1, γ 5 := iγ 0 γ 1 γ 2 γ 3, γ 5 γ µ = γ µ γ 5, γ 5 = γ 5, γ 2 5 = 1 Diracology of left and right-handed components ψ L = 1 γ 5 ψ = P L ψ, ψ R = 1 + γ 5 ψ = P R ψ, 2 2 P 2 L/R = P L/R, P R P L = P L P R = 0, P L/R γ 5 = γ 5 P L/R = P L/R P L/R γ µ = γ µ P R /L, P L ψ = ψp R, P R ψ = ψp L ψγ µ ψ = ψ L γ µ ψ L + ψ R γ µ ψ R, ψψ = ψ L ψ R + ψ R ψ L ψ := ψ γ 0, γ 5 ψ = ψ γ 5 γ 0 = ψγ 5 in the massless limit (m u = m d = 0) u,d = ψi /D ψ = ψ L i /D ψ L + ψ R i /D ψ R Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 21 / 38

Chiral Symmetry of (massless) QCD in the massless limit (m u = m d = 0) a lot of global chiral symmetries: change of independent phases for left and right components: ψ L (x ) exp( iφ L )ψ L (x ), ψ R (x ) exp( iφ R )ψ R (x ) symmetry group U(1) L U(1) R independent iso-spin rotations ψ L (x ) exp( i α L T )ψ L (x ), ψ R (x ) exp( i α R T )ψ R (x ) T = τ/2, τ: Pauli matrices; symmetry group SU(2) L SU(2) R alternative notation scalar-pseudoscalar phases/iso-spin rotations ψ exp( iφ s )ψ, ψ exp( iγ 5 φ a )ψ ψ exp( i α V T )ψ, ψ exp( iγ 5 α A T )ψ U(1) s and SU(2) V are subgroups that are symmetries even if m u = m d 0 Heisenberg s iso-spin symmetry! Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 22 / 38

Currents: relation to mesons based on [Koc97, Sch03, Din11] Noether: each global symmetry leads to a conserved quantity from chiral symmetries j µ s = ψγµ ψ, j µ a = ψγµ γ 5 ψ j µ V = ψγµ T ψ, j µ A = ψγµ γ 5 T ψ Link to mesons: Build Lorentz-invariant objects with corresponding quantum numbers σ: ψψ (scalar and iso-scalar) π s: iψ T γ 5 ψ (pseudoscalar and iso-vector) ρ s: ψγ µ T ψ (vector and iso-vector) a 1 s: ψγ µ γ 5 T ψ (axialvector and iso-axialvector) in nature: σ and π s; ρ s and a 1 s do not have same mass! reason: QCD ground state not symmetric under pseudoscalar and pseudovector trafos since vac ψψ vac 0 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 23 / 38

Spontaneous symmetry breaking spontaneously broken symmetry: ground state not symmetric vacuum necessarily degenerate vacuum invariant under scalar and vector transformations: U(1) L U(1) R broken to U(1) s ; SU(2) L SU(2) R broken to SU(2) V for each broken symmetry massless scalar Goldstone boson there are three pions which are very light compared to other hadrons (finite masses due to explicit breaking through m u, m d!) but no pseudoscalar isoscalar light particle! (m η 548 MeV) reason: U(1) a anomaly axialscalar symmetry does not survive quantization! good for explanation of correct decay rate for π 0 γγ axialscalar current not conserved µ j µ a = 3/8α s ε µνρσ G a µν G a ρσ explicit breaking due to quark masses can be treated perturbatively chiral perturbation theory axial-vector current only approximately conserved PCAC a lot of low-energy properties of hadrons derivable Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 24 / 38

Chiral transformations σ meson and pions (chiral partners) meson = q -q bound state infinitesimal chiral transformations for quarks ( T = τ/2) in SU(2) L SU(2) R model ψ (1 iδ α V τ/2)ψ ψ (1 iγ 5 δ α A τ/2)ψ (iso-vector transformation) (axial-vector transformation) transformation properties under χ transformations as σ ψψ, π iψ τγ 5 ψ σ σ δ α A π, π π + δ α V π + δ α A σ σ 2 + π 2 invariant χ symmetry realized as SO(4) rotations Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 25 / 38

The minimal linear σ model chiral symmetry realized by SO(4): meson fields φ 4 describes σ and pions (π ±, π 0 ) χlimit = 1 2 ( µφ)( µ φ) V (φ) = 1 2 ( µφ)( µ φ) λ 4 (φ2 f 2 π )2 spontaneous symmetry breaking: mexican-hat potential V φ0 φ2 φ1 doesn t cost energy to excite field in direction of the rim massless Nambu-Goldstone bosons (pions) vacuum expectation value φ 0 = f π 0 symmetry spontaneously broken from SO(4) to SO(3) V particle spectrum: 4 field-degrees of freedom vacuum inv. 3-dim SO(3) 3 massless pions 4 3 = 1 massive σ Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 26 / 38

Pion decay and PCAC weak decay π + µ + + ν µ weak interaction J µ V J µ A pion pseudoscalar decay must be due to axial current 0 J a µ A (x ) π b (p ) = ip µ δ a b f π exp( ip x ) decay rate from Fermi model f π 93 MeV 0 µ J a µ A (x ) π b (p ) = f π p 2 δ a b exp( ip x ) = f π m 2 π δa b exp( ip x ) exact chiral symmetry m π = 0 (Goldstone theorem!) µ J a µ A = 0 Noether m π 0 but small partial conservation of axial current (PCAC) in effective model J a µ A,π = f π µ φ a, a {1, 2, 3} Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 27 / 38

Explicit symmetry breaking explicit breaking due to finite quark masses symmetry-breaking term in QCD: χsb = m ψψ m = (m u + m d )/2; from ψψ σ now write for σ-pion potential χsb = εσ V (σ, π) = λ (σ 2 + π 2 ) v 2 2 0 εσ 4 tilted in σ-direction selects direction of vacuum expectation value minimum at f π v 0 = f π ε, m 2 2λfπ 2 σ = 2λf 2 π + ε, m 2 π f = ε π f π Noether with symmetry breaking + PCAC (consistency!): µ J a µ A = επ a PCAC = f π m 2 π πa ε = f π m 2 π χsb in QCD as in effective model Gell-Mann-Oaks-Renner Relation 0 εσ 0 = f π ε = m 2 π f 2 π = m 0 ψψ 0 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 28 / 38

Adding nucleons axial current of nucleon J µ A,nucl = g a Ψγ µ γ 5 τ 2 Ψ from neutron-β decay g a = 1.25 total axial current J µ A = J µ A,π + J µ A,nucl should fulfill PCAC µ J µ A = f πm 2 π π Goldberger-Treiman relation extension of linear σ model ( + m 2 π )πa = g a i M f π Ψγ 5 τψ g πn N = g a M f π 12.6 vs. g exp πn N = 13.4 nucl = Ψi / Ψ g πn N iψγ5 τψ π + ΨΨσ + µψψ GT relation: 0 σ 0 = f π M = g A M µ µ = (g A 1)M M 4 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 29 / 38

QFT@finite T,µ Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 30 / 38

Finite Temperature/Density: Idealized theory picture partition sum: Z (V,T,µ q,φ) = Tr{exp[ (H[Φ] µ q N)/T ]} Real Time Z[V,T, µ,φ] Imag. Time Dynamical quantities off equilibrium: derivation of BUU,... T, µ 0 vacuum analytic continuation Thermodyn. potentials bulk properties lattice QCD [CSHY85, Lv87, LeB96, KG06] Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 31 / 38

Finite Temperature Asymptotic freedom quark condensate melts at high enough temperatures/densities all bulk properties from partition sum: Free energy: Ω = T V lnz = P Quark condensate: ψ q ψ q T,µ q = V T Z (V,T,µ q ) = Tr{exp[ (H µ q N)/T ]} P m q Lattice QCD (at µ q = 0) chiral symmetry ψψ deconfinement transition Polyakov Loop tr P exp(i β 0 dτa0 ) Chiral symmetry restoration and deconfinement transition at same T c Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 32 / 38

What can we learn from em. probes in heavy-ion collisions? only penetrating probe leptons and photons leave hot and dense fireball unaffected they are produced during the entire fireball evolution dileptons provide information on in-medium spectral properties of hadrons theoretical challenge need an understanding of QCD medium at all stages of its evolution transport models, hydrodynamics need to identify all sources of dileptons and photons perturbative QCD not applicable non-perturbative QCD, effective hadronic models evaluate dilepton and photon rates QFT at finite T and µ B Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 33 / 38

Bibliography I [B + 12] J. Beringer, et al., Review of Particle Physics (RPP), Phys. Rev. D 86 (2012) 010001. http://dx.doi.org/10.1103/physrevd.86.010001 [CSHY85] K. Chou, Z. Su, B. Hao, L. Yu, Equilibrium and Nonequilibrium Formalisms made unified, Phys. Rept. 118 (1985) 1. http://dx.doi.org/10.1016/0370-1573(85)90136-x [DGH92] [Din11] [KG06] J. F. Donoghue, E. Golowich, B. R. Holstein, Dynamics of the Standard Model, Cambridge University press (1992). M. Dine, Goldstone Bosons and Chiral Symmetry Breaking in QCD (2011), lecture notes. http: //scipp.ucsc.edu/~dine/ph222/goldstone_lecture.pdf J. I. Kapusta, C. Gale, Finite-Temperature Field Theory; Principles and Applications, Cambridge University Press, 2 ed. (2006). Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 34 / 38

Bibliography II [Koc97] V. Koch, Aspects of chiral symmetry, Int. J. Mod. Phys. E 6 (1997) 203. http://dx.doi.org/10.1142/s0218301397000147 [LeB96] [Lv87] [Nac90] [RW00] M. LeBellac, Thermal Field Theory, Cambridge University Press, Cambridge, New York, Melbourne (1996). N. P. Landsmann, C. G. van Weert, Real- and Imaginary-time Field Theory at Finite Temperature and Density, Physics Reports 145 (1987) 141. http://dx.doi.org/10.1016/0370-1573(87)90121-9 O. Nachtmann, Elementary Particle Physics - Concepts and Phenomenology, Springer-Verlag, Berlin, Heidelberg, New York, London, Paris, Tokyo (1990). R. Rapp, J. Wambach, Chiral symmetry restoration and dileptons in relativistic heavy-ion collisions, Adv. Nucl. Phys. 25 (2000) 1. http://arxiv.org/abs/hep-ph/9909229 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 35 / 38

Bibliography III [Sch03] S. Scherer, Introduction to chiral perturbation theory, Adv. Nucl. Phys. 27 (2003) 277. http://arxiv.org/abs/hep-ph/0210398 Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 36 / 38

Quiz 1 What are the peaks in the following figure of R e +e hadrons? 2 Can you explain the horizontal lines (values: 2, 3.333, 3.667)? Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 37 / 38

Quiz 4 Why is chiral symmetry (intuitively) only true for massless quarks? 5 What s the main consequence of spontaneous symmetry breaking? 6 Why can one measure the vector and axial-vector current-current correlators from τ even/odd number of pions + ν? Hendrik van Hees (GU Frankfurt/FIAS) Hadrons in hot and dense matter I July 17-21, 2017 38 / 38