Lecture Models for heavy-ion collisions (Part III: transport models SS06: Dynamical models for relativistic heavy-ion collisions
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: 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 -body potential
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 (classical Fock term: non-local mean-field exchange potential (quantum statistics TDHF approximation describes only the interactions of particles with the time-dependent mean-field U HF (r,t! EoM: propagation of particles in the self-generated mean-field In order to describe the collisions between the individual(! particles, one has to go beyond the mean-field level!
Density-matrix formalism In order to go beyond the one-body TDHF limit one has to include N-body operators (or at least -body operators density-matrix formalism A density matrix is a matrix that describes a quantum system in a mixed state, a statistical ensemble of several pure quantum states ψ k > Introduce the density operator: where P k is a probability to find a quantum system in a pure quantum state ψ k > (in Hilbert space By choosing the ortogonal basis { ϕ k >} we can resolve the density operator to the density matrix, those elements are The expectation value of any operator O is given as
Density-matrix formalism Schrödinger equation for a system of N fermions: Hamiltonian operator: Schrödinger eq. in density operator representation von Neumann (or Liouville eq.: -body potential kinetic energy operator ( Density operator for N-body system summed over all possible quantum state k with amplitude P k (P k meaning of probability: (for any possible quantum state k of N-body system Notation: j particle index of many body system (j=[,n] in different representations : discrete state 5
Density-matrix formalism von Neumann (or Liouville eq. in matrix representation describes an N-particle system in or out-off equilibrium ( E.g.: Introduce a reduced density matrices ρ n ( n, n ;t by taking the trace (integrate over particles n+, N: Recurrence ( (tensor of rank n: n<n Normalization: Trρ N =N! such that 6
Density matrix formalism: BBGKY-Hierarchy Taking corresponding traces (i.e. Tr (n+, N of the von-neumann equation we obtain BBGKY-Hierarchy (Bogolyubov, Born, Green, Kirkwood and Yvon ( This set of equations is equivalent to von-neumann equation. The approximations or truncations of this set will reduce the information about the system The explicit equations for n=, n= read: (5 (6 Eqs. ( are not closed since the eq. For ρ requires information from ρ. Its equation reads: (7 7
Density matrix formalism: BBGKY-Hierarchy Introduce the cluster expansion Correlation dynamics: -body density matrix: -body density matrix (consider fermions: initial state of particle final state of the same particle (8 (9 PI= -particle-irreducible approach -body antisymmetrization operator: PI = -particle-irreducible approach + (TDHF approximation -body correlations Permutation operator By neglecting c in (9 we get the limit of independent particles (Time-Dependent Hartree-Fock. This implies that all effects from collisions or correlations are incorporated in c and higher orders in c etc. -body density matrix: (0 8
Correlation dynamics From eq. (5 for ρ (by substitution of eq. (8 for ρ, we obtain equation-of-motion (EoM for the one-body density matrix: ( From eq. (6 for ρ (by substitution of eq. (0 for ρ and discarding explicit -body correlations c, we obtain EoM for the two-body correlation matrix c : ( 9
To reduce the complexity we introduce: a one-body Hamiltonian by Correlation dynamics ( kinetic term + interaction in the self-generated time dependent mean field Pauli-blocking operator is uniquely defined by ( Effective interaction in the medium: (5 Resummed interaction G-matrix approach 0
Correlation dynamics * EoM for the one-body density matrix: (6 TDHF -body correlations EoM (6 describes the propagation of a particle in the self-generated mean field U s (i with additional -body correlations that are further specified in EoM (7 for c : * EoM for the -body correlation matrix: Propagation of two particles and in the mean field U s Born term: bare -body scattering (7 G-matrix theory: resummation of the in-medium interaction with intermediate Pauli blocking Note: Time evolution of c depends on the distribution of a third particle, which is integrated out in the trace! The third particle is interacting as well! *: EoM is obtained after the cluster expansion and neglecting the explicit -body correlations c Particle-hole interaction (important for graundstate correlations and damping low energy modes
BBGKY-Hierarchie - PI: 0 Vlasov equation perform Wigner transformation of one-body density distribution function ρ(r,r,t r r r r i rr r s r s f (, p,t = d s exp ps ρ +,r, t h f is the single particle phase-space distribution function After the st order gradient expansion Vlasov equation of motion - free propagation of particles in the self-generated HF mean-field potential: t r r f (, p,t + r U(,t r p r r r r r r r r r f (, p,t r U(,t r p f (, p,t = 0 m ( πh = β occ d r d r r r r pv (,t f (, p,t (8 (9
Uehling-Uhlenbeck equation: collision term ( TDHF Vlasov equation -body correlations Collision term: I(,t= ( perform Wigner transformation Formally solve the EoM for c (with some approximations in momentum space: and insert c in the expression for I(,t :
Boltzmann (Vlasov-Uehling-Uhlenbeck (B(VUU equation : Collision term d dt r r f (, p,t t r r f (, p,t + r p r m r r r f (, p,t r r r U(,t r r p r r f (, p,t = f t coll Collision term for + + (let s consider fermions : r r r r Icoll = d p d p d ( p + p p p ( Ω υ δ π dσ ( + dω + P Probability including Pauli blocking of fermions: P = f f( f ( f f f( f ( f Gain term + + Loss term + + For particle and : Collision term = Gain term Loss term I coll = G L The VUU equations describes the propagation in the self-generated mean-field U(r,t as well as mutual two-body interactions respecting the Pauliprinciple (also denoted as BUU ect..
Numerical solution of the BUU equation The Vlasov part is solved in the testparticle approximation r r f (, p,t = N N i= r r r r δ ( i ( t δ ( p pi ( t with N denoting the number of testparticles per nucleon (N infinity. The trajectories r i (t, p i (t result from the solution of classical equations of motion. The collision term is solved by a Monte Carlo treatment of collisions: an interaction takes place at impact parameter b if πb < σ! the final state is selected by Monte Carlo according to the angular distribution dσ/dω the final state is accepted again by Monte Carlo according to the probability (-f (-f for + + 5
Covariant transport equation From non-relativistic to relativistic formulation of transport equations: Non-relativistic Schrödinger equation relativistic Dirac equation Non-relativistic dispersion relation: E r p r = + U( m U(r density dependent potential (with attractive and repulsive parts! Not Lorentz invariant, i.e. dependent on the frame Relativistic dispersion relation: E * * * * = m + U U S scalar potential S (attractive * r r r = p + UV U = (U0,UV - vector * -potential (repulsive = E U0 = 0,,,! Lorentz invariant, i.e. independent on the frame m r p E = m r + p Consider the Dirac equation with local and non-local mean fields: ( i γ MF m ψ ( x U ( x ψ ( x d y U ( x, y ψ ( x = 0 (9 here r r x ( t,r y ( t,r 6
Local mean field potential: Covariant transport equation U MF MF MF ( x = U S V ( x + γ U ( x scalar potential + vector potential (0 Non-local mean field potential (non-local interaction: U ( x, y = U S ( x, y + γ UV ( x, y ( scalar potential + vector potential General form: U U S ( x, y = C D ( x, y = C D ( x V S V S V ( x y ψ ( y ψ ( x y ψ ( y γ ψ ( x ( Here C S,C V are the coupling constants strength of interactions D(x-y D-function: in the perturbative limit (weak coupling the D-function has the meaning of a Green function or meson propagator when the interaction between the nucleons occurs by meson exchange 7
Covariant transport equation Perform a Wigner transformation of eq. ( in phase-space: U U S ( x, p = C ( x, p = C S ( π ( π d d D D S ( p m ( p V V V Π * ( x, ( x, f ( x, f ( x, ( where f(x,p is the single particle phase-space distribution function m * ( x, p = Π ( x, p = m + U p MF S U MF V ( x + U S ( x, p U ( x, p V ( x, p - effective mass - effective momentum ( where Ansatz for D-function: D D S,V S(V ( p ( p = d z D S,V ( z e iz( p Fourier transform Λ S(V Λ S(V ( p (5 (6 8
Covariant transport equation Covariant relativistic Vlasov equation : { ( ( } p ν * p ν x ν * x ν Π Π U m ( U + ( U + m ( U f ( x, p = 0 ν ( V S x Π ν V S p where x r ( t, r (7 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 ν * p ν x ν * x ν Π Π ν ( U m ( U + Π ν ( U + m ( U S p f ( x, p = Icoll V S x V Gain term + + I coll + d d d [G G ] + + {,, f ( x, p f ( x, p f ( x, p f ( x, p ( f ( x, p ( ( f ( x, p δ ( Π + Π ( Π f ( x, p f ( x, p Π } d p d E Loss term + + 9 (8
Brueckner theory Transition rate for the process + + follows from many-body Brueckner theory: + [ G G ] + + δ ( Π + Π Π Π -body scattering in vacuum: Scattering amplitude: T( E = V + V E T( E t( t( + iη (8 with the hamiltonian: H = A i= t( i + i< j V ( ij T( E p p V ( p p V ( + p +... ladder resummation V ( p p 0
Brueckner theory -body scattering in the medium: Scattering amplitude from Brueckner theory: with single-particle hamiltonian: Note: vacuum case ( : G( E p p G( E = V + V ( n n E h( h( + iη h( = t( + U V ( MF p p ( V ( V ( p p G( E Pauli-blocking n occupation number h( = t( and n = n = 0 G matrix T matrix + p +... (9 Propagation between scattering V( with mean field hamiltonian h(, h(! only allowed if intermediate states, are not accupied!
Elementary reactions with resonances Consider the reaction a + b R c + d a R c Cross section: intermediate resonance b d dσ ab cd ( π = δ ( pa+ pb pc pd M ab cd p a s (( π d p E c c d p E d d (6 Matrix element: M ab cd = M ab R P R M R cd (7 Propagator: P R s M = R + Π (8 where self-energy Π = i s Γ tot ( s, Γtot( s = total width j Γ j (9
Elementary reactions with resonances The spin averaged/sumed matrix element squared is Mab cd Mab R PR M R cd = (0 R a b Partial width: Γ p = 8 π s a R ab( s M R ab ( p a momentum of a in the rest frame of R or cms a+b σ ab R cd J R + π sγ R ab( s Γ R cd ( s = ( J + ( J + p ( s M + sγ a b a R tot ( s ( s (
Spectral function Production of resonance with effective mass spectral function = Breight-Wigner distribution: ( Γtot( A( = π ( M + Γ R tot ( Normalization condition: 0 Γtot ( = Γ j ( j d A( = The total width = the sum over all partial channels: Life time of resonance with mass : τ ( = hc Γ tot ( ( Note: Experimental life time with pole mass =M R τ R = Γ tot hc ( = M R
Decay rate Decay rate: dn dt ~ τ N N( t ~ e t τ Total probability to decay P decay = e Γ tot t Total probability to survive: P = e Γ tot t pole mass Branching ratio= probability to decay to channel j: Br P j = Γ ( Γ j tot ( 5
Detailed balance: a + b c + d Detailed balance Note: DB is important to get the correct equilibrium properties σ ( Ja + ( Jb + pab( s ( s = σ a+ b c d ( J + ( J + p ( s c + d a+ b + c d cd Momentum of particle a (or b in cms: p ab ( s = [( s ( m + m ( s ( m m ] Momentum of particle c (or d in cms: p cd ( s = a b s [( s ( m + m ( s ( m m ] c d s a c b d / / ( s J- spin ( ( (5 6