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: 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 

❑ 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 3   Density in coordinate space: ( r ,t ) = d p f ( r , p,t ) ( 2 )3     Density in momentum space: g ( p , t ) =  d 3 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 3rd 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 =  + rr + 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 (NON-relativistic formulation!) - free propagation of particles in the self-generated HF mean-field potential with an on-shell collision term:  d      p          f     f ( r , p,t )  f ( r , p,t )+ r f ( r , p,t )− rU( r ,t )  p f ( r , p,t ) =   dt t m  t coll

Collision term for 1+2→3+4 (let‘s consider fermions) : Probability including Pauli blocking of fermions  f  1 3 3 3 Icoll     d p2 d p3 d p4  w( 1 + 2 → 3 + 4 )   t  (( 2 )3 )3  coll     3 3     p1 p2 p3 p4 ( 2 )  ( p1 + p2 − p3 − p4 ) ( 2 ) ( + − − ) 2m1 2m2 2m3 2m4 d 3 Transition probability for 1+2→3+4: w( 1+ 2 → 3 + 4 )   12 d 3q    where  = | p − p | - relative velocity of the colliding nucleons 12 m 1 2 d 3    - differential cross section, q – momentum transfer q = p − p d 3q 1 3 19 BUU: Collision term

4     d I = d 3 p d 3 p d | | 3 ( p + p − p − p ) ( 1 + 2 → 3 + 4 ) P coll ( 2 )3  2 3  12 1 2 3 4 d

Probability including Pauli blocking of fermions:         P = f ( r , p3 ,t ) f ( r , p4 ,t ) 1 − f ( r , p1 ,t ) 1 − f ( r , p2 ,t )         − f ( r , p1 ,t ) f ( r , p2 ,t ) 1 − f ( r , p3 ,t ) 1 − f ( r , p4 ,t )

 f3 f4 ( 1 − f1 )( 1 − f2 ) − f1 f2 ( 1 − f3 )( 1 − f4 ) Pauli blocking factors for fermions * Gain term Loss term 3+4→1+2 1+2→3+4

For particle 1 and 2: I = G − L Collision term = Gain term – Loss term coll

*Note: for bosons – enhancement factor 1+f (where f<<1); often one neglects bose enhancement for HIC, i.e. 1+f →1 20 Dynamical transport model: collision terms

❑ BUU eq. for different particles of type i=1,…n d Df  f = I  f , f ,..., f  (20) i dt i coll 1 2 n Drift term=Vlasov eq. collision term

* i : Baryons : p,n,( 1232 ), N( 1440 ), N( 1535 ),...,, , , , ; C

* Mesons :  ,h ,K ,K , , ,K ,h, ,a1 ,...,D,D ,J / , ,...

➔ coupled set of BUU equations for different particles of type i=1,…n

DfN = Icoll  f N , f , f N ( 1440) ,..., f , f  ,...

Df = Icoll  f N , f , f N ( 1440) ,..., f , f  ,... ...

Df = Icoll  f N , f , f N ( 1440) ,..., f , f  ,... ...

21 (only 1→2, 2→2 E.g., Nucleon transport in N,, system : DfN=Icoll reactions indicated here)

Full collision term consists of >10000 different particle combinations

22 ➔ set of transport equations coupled via Icoll and mean field Dynamical transport model: collision terms

Collision terms for (N,, ) system:    N * Relativistic formulation

g d 3 p d 3 p Df =  N | M |2  4 ( p + p − p )  f ( p ) f ( p )( 1 − f ( p ))   3  N  N    N N    ,N ( 2 ) E EN  g d 3 p d 3 p −  N | M |2  4 ( p + p − p )  f ( p ) ( 1 − f ( p ))( 1 + f ( p ))  3  N  N    N N    ,N ( 2 ) E EN

Eq. Eq. for = Gain (N →  ) − Loss (  → N )  production  decay

g d 3 p d 3 p Df =  N | M |2  4 ( p + p − p )  f ( p ) ( 1 + f ( p ))( 1 − f ( p ))   3  N  N      N N N , ( 2 ) E EN

 g d 3 p d 3 p −  N | M |2  4 ( p + p − p )  f ( p ) f ( p )( 1 − f ( p ))  3  N  N    N N    ,N ( 2 ) E EN

Eq. for = Gain (  → N ) − Loss (N →  )  production  absorbtion by  decay by nucleon

23 Dynamical transport model: possible interactions

Consider possible interactions for the sytem of (N,R,m), where N-nucleons, R- resonances, m-mesons ❑ elastic collisions: Baryon-baryon (BB): meson-Baryon (mB) meson-meson (mm) NN → NN mN → mN m m → m m NR → NR mR → mR   RR → RR Detailed balance: a + b  c a + b  c + d ❑ inelastic collisions: Baryon-baryon (BB): meson-Baryon (mB) meson-meson (mm) ~ NN  NR mN  R m m  m NR  NR mR  R m m  mm   NN  RR mB  m B ...... m m → X BB → X mB → X X - multi-particle state

24 Elementary hadronic interactions

Consider all possible interactions – elastic and inelastic collisions - for the sytem of (N,R,m), where N-nucleons, R- resonances, m-mesons, and resonance decays

Low energy collisions: High energy collisions: binary 2→2 and ▪ (above s1/2~2.5 GeV) 2→3(4) reactions +p ▪ 1→2 : formation and decay of baryonic and Inclusive particle mesonic resonances production: BB→X , mB→X, mm→X BB → B´B´ X =many particles BB → B´B´m described by mB → m´B´ mB → B´ string formation and decay mm → m´m´ pp (string = excited color mm → m´ . . . singlet states q-qq, q-qbar) using LUND string model Baryons: B = p, n, (1232), N(1440), N(1535), ... Mesons: M = , h, , , , ...

25 Covariant transport equation

From non-relativistic to relativistic formulation of transport equations:

Non-relativistic Schrödinger equation → relativistic Dirac equation

Non-relativistic dispersion relation: Relativistic dispersion relation:

 2 p  * 2 * 2 * 2 E = + U ( r ) E = m + p 2m * US – scalar potential m = m + U S (attractive)     U(r) – density dependent potential p* = p + U V U m = ( U 0 ,U V ) (with attractive and repulsive parts) E * = E − U vector 4-potential (repulsive) 0 m = 0 ,1 ,2 ,3 ! Not Lorentz invariant, i.e. ! Lorentz invariant, i.e. dependent on the frame independent on the frame

➔ Consider the Dirac equation with local and non-local mean fields:

MD ( i m  − m ) ( x ) − U MF ( x ) ( x ) − d 4 y U ( x , y ) ( x ) = 0 m  here   x x x  ( t ,r ) y  ( t ,r  ) y 26 Covariant transport equation

❑ Covariant relativistic on-shell BUU equation : from many-body theory by connected Green functions in phase-space + mean-field limit for the propagation part (VUU)

p  * p  m x  * x  m  ( m −   (  mUV ) − m (  mU S ))  x + (  (  mUV ) + m (  mU S ))  p  f ( x, p ) = Icoll

I  d 2 d 3 d 4 [G +G ]  4 (  +  −  −  ) coll  1+ 2→3+4 2 3 4 3  d p 2 ,3 ,4 d 2  2 E2  { f ( x, p3 ) f ( x, p4 ) ( 1 − f ( x, p )) ( 1 − f ( x, p2 )) Gain term Loss term 3+4→1+2 1+2→3+4 − f ( x, p ) f ( x, p2 ) ( 1 − f ( x, p3 )) ( 1 − f ( x, p4 )) }  x where  m  (  t , r ) - effective mass m*(x,p) = m + Us (x,p) m (x,p) = pm – Um (x,p) - effective momentum

Us (x,p), Um (x,p) are scalar and vector part of particle self-energies

m 2 (m  −m* ) – mass-shell constraint W. Ehehalt, W. Cassing, Nucl. Phys. A 602 (1996) 449 27 Brueckner theory

+ 4 Transition rate for the process 1+2→3+4 [ G G ]1+ 2→ 3+ 4  (  +  2 −  3 −  4 ) follows from many-body Brueckner theory:

1) 2-body scattering in vacuum: 1 Scattering amplitude: T ( E ) = V + V T ( E ) E − t( 1 ) − t( 2 ) + ih A 1 with the hamiltonian: H =  t( i ) + V ( ij ) i=1 2 i j

p p   1 2 p1 p2 p p 1 2 V ( 12 ) V ( 12 ) p p T ( E )  + 3 3 + ... V ( 12 )

p1 p2 p1 p2 p1 p2 ‚ladder‘ resummation

28 Brueckner theory

2) 2-body scattering in the medium:

Scattering amplitude → from Brueckner theory: 1 G( E ) = V + V ( 1 − n − n ) G( E ) E − h( 1 ) − h( 2 ) + ih 3 3 Pauli-blocking MF n3 – occupation number with single-particle hamiltonian: h( 1 ) = t( 1 ) + U ( 1 ) Note: vacuum case : h( 1 ) = t( 1 ) and n3 = n3 =0  G − matrix → T − matrix p p   1 2 p1 p2 p p 1 2 V ( 12 ) V ( 12 ) p p G ( E )  + 3 3 + ... V ( 12 )

p1 p2 p1 p2 p1 p2

Propagation between scattering V(12) with mean field hamiltonian h(1), h(2) ! only allowed if intermediate states 3,3‘ are not accupied ! 29 Hadron-String-Dynamics – a microscopic transport model for heavy-ion reactions

• very good description of particle production in pp, pA, pA, AA reactions • unique description of nuclear dynamics from low (~100 MeV) to ultrarelativistic (>20 TeV) energies

104 Au+Au (central) 102

0 10 +   10-2 h D(c) J/

Multiplicity __ -4 10 K+ D(c) AGS − SPS K HSD ' 99 RHIC 10-6 10-1 100 101 102 103 104 Energy [A GeV]

HSD predictions from 1999; data from the new millenium W. Cassing, E.B., Phys.Rept. 308 (1999) 65 30