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Lecture: Part 1 Introduction to the dynamical models of heavy-ion collisions Elena Bratkovskaya (GSI Darmstadt & Uni. Frankfurt) The CRC-TR211 Lecture Week “Lattice QCD and Dynamical models for Heavy Ion Physics” Limburg, Germany, 10-13 December, 2019 1 The ‚holy grail‘ of heavy-ion physics: The phase diagram of QCD • Search for the critical point • Study of the phase transition from hadronic to partonic matter – Quark-Gluon-Plasma • Search for signatures of ? chiral symmetry restoration • Search for the critical point • Study of the in-medium properties of hadrons at high baryon density and temperature 2 Theory: Information from lattice QCD I. deconfinement phase transition II. chiral symmetry restoration with increasing temperature + with increasing temperature lQCD BMW collaboration: mq=0 qq Δ ~ T l,s qq 0 ❑ Crossover: hadron gas → QGP ❑ Scalar quark condensate 〈풒풒ഥ〉 is viewed as an order parameter for the restoration of chiral symmetry: ➔ both transitions occur at about the same temperature TC for low chemical potentials 3 Experiment: Heavy-ion collisions ❑ Heavy-ion collision experiment ➔ ‚re-creation‘ of the Big Bang conditions in laboratory: matter at high pressure and temperature ❑ Heavy-ion accelerators: Large Hadron Collider - Relativistic-Heavy-Ion-Collider - Facility for Antiproton and Ion Nuclotron-based Ion Collider LHC (CERN): RHIC (Brookhaven): Research – FAIR (Darmstadt) fAcility – NICA (Dubna) Pb+Pb up to 574 A TeV Au+Au up to 21.3 A TeV (Under construction) (Under construction) Au+Au up to 10 (30) A GeV Au+Au up to 60 A GeV 4 Signals of the phase transition: • Multi-strange particle enhancement in A+A • Charm suppression • Collective flow (v1, v2) • Thermal dileptons • Jet quenching and angular correlations • High pT suppression of hadrons • Nonstatistical event by event fluctuations and correlations • ... Experiment: measures final hadrons and leptons How to learn about physics from data? Compare with theory! 5 Basic models for heavy-ion collisions • Statistical models: basic assumption: system is described by a (grand) canonical ensemble of non-interacting fermions and bosons in thermal and chemical equilibrium = thermal hadron gas at freeze-out with common T and mB [ - : no dynamical information] • Hydrodynamical models: basic assumption: conservation laws + equation of state (EoS); assumption of local thermal and chemical equilibrium - Interactions are ‚hidden‘ in properties of the fluid described by transport coefficients (shear and bulk viscosity h, z, ..), which is ‘input’ for the hydro models [ - : simplified dynamics] • Microscopic transport models: based on transport theory of relativistic quantum many-body systems - Explicitly account for the interactions of all degrees of freedom (hadrons and partons) in terms of cross sections and potentials - Provide a unique dynamical description of strongly interaction matter in- and out-off equilibrium: - In-equilibrium: transport coefficients are calculated in a box – controled by lQCD - Nonequilibrium dynamics – controled by HIC Actual solutions: Monte Carlo simulations [+ : full dynamics | - : very complicated] 7 Models of heavy-ion collisions thermal model thermal+expansion final initial hydro transport 7 Dynamical description of heavy-ion collisions The goal: to study the properties of strongly interacting matter under extreme conditions from a microscopic point of view Realization: dynamical many-body transport approaches This lecture: 1) Dynamical transport models (nonrelativistic formulation): from the Schrödinger equation to Vlasov equation of motion ➔ BUU EoM 2) Density-matrix formalism: Correlation dynamics 3) Quantum field theory ➔ Kadanoff-Baym dynamics ➔ generalized off-shell transport equations 4) Transport models for HIC 8 1. From the Schrödinger equation to the Vlasov equation of motion 9 Quantum mechanical description of the many-body system Dynamics of heavy-ion collisions is a many-body problem! Schrödinger equation for the system of N particles in three dimensions: nonrelativistic formulation Hartree-Fock approximation: •many-body wave function → antisym. product of single-particle wave functions •many-body Hamiltonian → single-particle Hartree-Fock Hamiltonian kinetic term N-body potential kinetic term 2-body potential 10 Hartree-Fock equation Time-dependent Hartree-Fock equation for a single particle i: Single-particle Hartree-Fock Hamiltonian operator: •Hartree term: self-generated local mean-field potential •Fock term: non-local mean-field exchange potential (quantum statistics) ➔ Equation-of-motion (EoM): propagation of particles in the self-generated mean-field: local potential We‘ll neglected the exchange (Fock) term Note: TDHF approximation describes only the interactions of particles with the time-dependent mean-field ! In order to describe the collisions between the individual(!) particles, one has to go beyond the mean-field level ! (see Part 2: Correlation dynamics) 11 Single particle density matrix ❑ Introduce the single particle density matrix: * ( r ,r ,t ) ( r ,t ) ( r ,t ) occ Thus, the single-particle Hartree-Fock Hamiltonian operator can be written as h( r ,t ) = T( r )+ d 3r V ( r − r,t )( r,r,t ) = T( r )+ U( r ,t ) occ local potential ❑ Consider equation: i ( r ,t ) = h( r , t ) ( r ,t ) (1) t 휓∗ 푟Ԧ′, 푡 ∗(1): 훼 * ( r , t )i ( r , t ) = * ( r , t )h( r , t ) ( r , t ) (2) t + (1) |푓표푟 푟Ԧ′ ∗ 휓훼 푟, 푡 : * * (3) − i ( r ,t ) ( r ,t ) = h( r' ,t ) ( r ,t ) ( r ,t ) t ෍ 2 − 3 : 훼 12 Wigner transform of the density matrix i ( r , r , t ) = h( r , t ) − h( r , t ) ( r , r , t ) (4) t * ( r , r , t ) ( r , t ) ( r , t ) The single-particle Hartree-Fock Hamiltonian: o cc h(r , t) = T (r ) + d 3r V (r − r , t)(r , r , t) = T (r ) + U (r , t) kinetic term + potential (local) term ➔EoM: 2 2 i 2 2 (5) ( r ,r,t )+ r + U( r ,t )− r −U( r,t )( r ,r,t ) = 0 t 2m 2m Rewrite (5) using x instead of r 2 2 i 2 2 ( x , x , t ) + x + U ( x , t ) − x − U ( x , t ) ( x , x , t ) = 0 t 2 m 2 m 13 Wigner transform of the density matrix ➔EoM: ❑ Instead of considering the density matrix , let‘s find the equation of motion for its Fourier transform, i.e. the Wigner transform of the density matrix: i s s f ( r , p,t ) = d 3s exp− ps r + ,r − ,t 2 2 old : x x New variables: x + x r = , s = x − x 2 f ( r , p,t ) is the single-particle phase-space distribution function 1 ( r ,t ) = d 3 p f ( r , p,t ) Density in coordinate space: 3 i 2 (22 ) 2 2 Density ( inx ,momentumx , t ) + space: x + U ( x ,t ) − 3 x − U ( x , t ) ( x , x , t ) = 0 t 2 m g ( p , t ) =2 m d r f ( r , p , t ) 14 Wigner transformation + Taylor expansion 2 2 s s i 2 s 2 s s s (r + , r − , t) + s + U (r + , t) − s − U (r − , t) (r + , r − , t) = 0 (1) r + r − t 2 2 2m 2 2 2m 2 2 2 2 ❑ Make Wigner transformation of eq.(2) i s s d 3 s exp − ps r + ,r − ,t t 2 2 (2) 2 i 3 i 2 2 s s + d s exp − ps s − s r + ,r − ,t r + r − 2m 2 2 2 2 i 3 i s s s s + d s exp − ps U ( r + ,t ) − U ( r − ,t ) r + ,r − ,t = 0 2 2 2 2 2 2 (3) ❑ Use that s − s = 2 r s r + r − 2 2 s s ❑ Consider U ( r + ,t ) − U ( r − ,t ) 2 2 Make Taylor expansion around r; s→0 n n n 1 1 1 1 1 (4) = s r U |s=0 − − s r U |s=0 = 2 s r U |s=0 n=0 n! 2 n=0 n! 2 odd 2 terms even in n cancel s r U ( r ,t ) Classical limit: keep only the first term n=1 15 Vlasov equation-of-motion From (2) and (3),(4) obtain 2 i 3 i s s f ( r , p,t ) + d s exp − ps 2 r s r + ,r − ,t t 2m 2 2 (5) i 3 i s s + d s exp − ps s U ( r ,t ) r + ,r − ,t = 0 r 2 2 3 i s s (*): f ( r , p , t ) = d s exp − ps r + , r − , t 2 2 using (*) using Vlasov equation - free propagation of particles in the self-generated HF mean-field potential: p (6) f ( r , p , t ) + f ( r , p , t ) − U ( r , t ) f ( r , p , t ) = 0 t m r r p Eq.(6) is entirely classical (lowest order in s expansion). Here U is a self-consistent potential associated with f phase-space distribution: 1 (7) U (r , t) = d 3rd 3 pV (r − r , t) f (r , p, t) (2)3 16 Vlasov EoM Vlasov equation of motion - free propagation of particles in the self-generated HF mean-field potential: p f ( r , p,t )+ f ( r , p,t )− U( r ,t ) f ( r , p,t ) = 0 t m r r p Vlasov EoM is equivalent to: d f ( r , p,t ) = 0 = + rr + p p f ( r , p,t ) = 0 dt t ➔ Classical equations of motion : dr p r = = trajectoty : r( t ) dt m 1 dp p = = − U( r ,t ) dt r 2 Note: the quantum physics plays a role in the initial conditions for f: the initial f in case of fermions must respect the Pauli principle 17 Dynamical transport models with collisions ➔ In order to describe the collisions between the individual(!) particles, one has to go beyond the mean-field level ! (See Part 2: Correlation dynamics) add 2-body collisions: 3 t d 1 t d - average distance 5 2 Interaction 1+2→3+4 6 34 Vcoll 12 4 * * * * In cms: p1 + p2 = p3 + p4 = 0 ( r , p ) ( r , p ) → ( r , p ) ( r , p ) p* 1 1 2 2 3 3 4 4 3 * * ❑ If the phase-space around ( r 3 , p 3 ) and( r4 , p4 ) p * p 1 2 is essentially empty then the scattering is allowed, ❑ if the states are filled → Pauli suppression * = Pauli principle p4 18 BUU (VUU) equation Boltzmann (Vlasov)-Uehling-Uhlenbeck equation
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