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11
Lecture
Models for heavy-ion collisions
(Part II):
transport models
SS2019: ‚Dynamical models for relativistic heavy-ion collisions‘
2
kinetic term 2-body potential
Hartree-Fock approximation:
•many-body wave function →
antisym. product of single-particle wave functions
•many-body Hamiltonian → single-particle Hartree-Fock Hamiltonian
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:
kinetic term N-body potential
3
TDHF approximation describes only the interactions of particles with
the time-dependent mean-field UHF(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 !
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)
4
Density-matrix formalism
➢ Introduce the density operator for N-body system:
In order to go beyond the one-body TDHF limit one has to include N-body operators
(or at least 2-body operators)
→ density-matrix formalism
❑ A density matrix is a matrix that describes a quantum N-body system in
a mixed state, as statistical ensemble of several ‚pure‘ quantum states |yk>
where Pk is a probability to find a quantum N-body system in a ‚pure‘ quantum
‚k‘ state described by |yk> (in Hilbert space)
❑ By choosing the ortogonal N-body basis {|jn>} (Slater determinants) we can
resolve the density operator to the density matrix, those elements are
❑ The expectation value of any operator O is given as
5
Density-matrix formalism
❑ Schrödinger equation for a system of N fermions:
❑ Schrödinger eq. in density operator representation
→ von Neumann (or Liouville) eq.:
Hamiltonian operator:
Density operator for N-body system summed over all possible quantum
state k with Pk probability:
Notation: j – particle index of many body system
(j=[1,N]) in different representations :
discrete state
(1)
2-body potentialkinetic energy operator
(for any possible quantum state k of
N-body system)
6
❑ Introduce a reduced density matrices rn(1…n,1´…n´;t) by taking the trace
(integrate) over particles n+1,…N:
Density-matrix formalism
Normalization: TrrN =N! such that
❑ von Neumann (or Liouville) eq. in matrix representation describes an N-particle
system in- or out-off equilibrium
E.g.:
(tensor of rank 2n): n<N
(2)
(3)Recurrence
1=1‘;2=2‘
7
Density matrix formalism: BBGKY-Hierarchy
Taking corresponding traces (i.e. Tr(n+1,…N)) of the von-Neumann equation we obtain
BBGKY-Hierarchy (Bogolyubov, Born, Green, Kirkwood and Yvon)
❑ The explicit equations for n=1, n=2 read:
▪ This set of equations is equivalent to von-Neumann equation.
▪ The approximations or truncations of this set will reduce the information about the system
Eqs. (5,6) are not closed since the eq. for r2 requires information from r3. Its equation reads:
(4)
(5)
(6)
(7)
8
Density matrix formalism: BBGKY-Hierarchy
❑ 1-body density matrix:
❑ 2-body density matrix (consider fermions):
2-body correlations1PI = 1-particle-irreducible approach +
(TDHF approximation)
2PI= 2-particle-irreducible approach
➢ Introduce the cluster expansion ➔ Correlation dynamics:
2-body antisymmetrization
operator:
By neglecting c2 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 c2 and higher
orders in c2 etc.
Permutation
operator
(9)
❑ 3-body density matrix:
(8)
(10)
9
Correlation dynamics
❑ From BBGKY-Hierarchy eq. (5) for r1 (by substitution of eq. (8) for r2) , we
obtain equation-of-motion (EoM) for the one-body density matrix:
❑ From eq. (6) for r2 (by substitution of eq. (10) for r3) and discarding explicit 3-body
correlations c3, we obtain EoM for the two-body correlation matrix c2 :
(11)
(12)
10
Correlation dynamics
To reduce the complexity we introduce:
❑ a one-body Hamiltonian by
(13)
(14)❑ Pauli-blocking operator is uniquely defined by
❑ Effective interaction in the medium:
(15)
kinetic term + interaction in the self-generated time dependent mean field
! Resummed interaction ➔ G-matrix approach
11
Correlation dynamics
*: EoM is obtained after the ‚cluster‘ expansion and neglecting the explicit 3-body correlations c3
TDHF2-body correlations
❑* EoM for the one-body density matrix:
❑* EoM for the 2-body correlation matrix:
(16)
EoM (16) describes the propagation of a particle in the self-generated mean field Us(i) with
additional 2-body correlations that are further specified in EoM (17) for c2 :
Note: Time evolution of c2 depends on the distribution of a third particle, which is integrated out in the trace!
The third particle is interacting as well!
Propagation of two particles
1 and 2 in the mean field Us
Born term: bare 2-body scattering
G-matrix theory: resummation of the in-medium
interaction with intermediate Pauli blocking
Particle-hole interaction (important
for graundstate correlations) and
damping low energy modes
(17)
12
BBGKY-Hierarchie - 1PI:
0
Vlasov equation
−+
−= t,
2
sr,
2
srsp
iexpsd)t,p,r(f
3
r
f is the single particle phase-space distribution function
After the 1st order gradient expansion➔ Vlasov equation of motion
- free propagation of particles in the self-generated HF mean-field potential:
0)t,p,r(f)t,r(U)t,p,r(fm
p)t,p,r(f
tprr =−+
(18)
(19)
➢ perform Wigner transformation of one-body density distribution function
r(x,x‘,t)➔
),,(),()2(
1),(
33
3tprftrrpVdrdtrU
−=
13
Uehling-Uhlenbeck equation: collision term
Collision term:
❑ perform Wigner transformation
❑ Formally solve the EoM for c2 (with some approximations in momentum space):
and insert c2 in the expression (22) for I(11´,t) :→
TDHF – Vlasov equation (1PI)2-body correlations
(22)I(11´,t)=
(21)
(2PI)
14
Boltzmann (Vlasov)-Uehling-Uhlenbeck (B(V)UU) equation :
Collision term
P)4321(d
d)pppp(||dpdpd
)2(
4I 4321
3
123
3
2
3
3coll +→+−−+=
)f1)(f1(ff)f1)(f1(ffP 43212143 −−−−−=
Probability including Pauli blocking of fermions:
Gain term
3+4→1+2Loss term
1+2→3+4
For particle 1 and 2:
Collision term = Gain term – Loss termLGI coll −=
The VUU equations describes the propagation in the self-generated
mean-field U(r,t) as well as mutual two-body interactions respecting the Pauli-
principle (also denoted as BUU ect.).
coll
prrt
f)t,p,r(f)t,r(U)t,p,r(f
m
p)t,p,r(f
t)t,p,r(f
dt
d
=−+
Collision term for 1+2→3+4 (let‘s consider fermions) :
15
Numerical solution of the BUU equation
1) The Vlasov part is solved in the testparticle approximation
with N denoting the number of testparticles per nucleon (N→infinity).
The trajectories ri(t), pi(t) result from the solution of classical
equations of motion.
2) The collision term is solved by a Monte Carlo treatment of collisions:
➢ an interaction takes place at impact parameter b if b2 < !
➢ the final state is selected by Monte Carlo according to the
angular distribution d/d
➢ the final state is accepted again by Monte Carlo according to the
probability (1-f3)(1-f4) for 1+2→3+4 (Pauli blocking)
=
−−=N
1i
ii ))t(pp())t(rr(N
1)t,p,r(f
16
US – scalar potential
(attractive)
- vector
4-potential (repulsive)
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:
)r(Um2
pE
2
+=
V
*
S
*
2*2*2*
Upp
Umm
pmE
+=
+=
+=
! Not Lorentz invariant, i.e.
dependent on the frame
)U,U(U V0
=
0
*UEE −=
! Lorentz invariant, i.e.
independent on the frame
➔ Consider the Dirac equation with local and non-local mean fields:
here
(9)
)r,t(y)r,t(x
3,2,1,0=
0)x()y,x(Uyd)x()x(U)x()mi(MD
4MF =−−− yyy
U(r) – density dependent potential
(with attractive and repulsive parts)
17
Covariant transport equation
❑ Local mean field potential:
❑ Non-local mean field potential (non-local interaction):
)x(U)x(U)x(UMF
V
MF
S
MF
+=
scalar potential + vector potential
(10)
)y,x(U)y,x(U)y,x(UMD
V
MD
S
MD
+= (11)
)x()y()yx(DC)y,x(U
)x()y()yx(DC)y,x(U
MD
VV
MD
V
MD
SS
MD
S
yy
yy
−=
−=
scalar potential + vector potential
(12)
Here CS ,CV 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
➢General form:
x
x y
18
Covariant transport equation
Perform a Wigner transformation of eq. (12) ➔ in phase-space:
)p,x(f)p,x()pp(Dpd)2(
4C)p,x(U
)p,x(f)p,x(m)pp(Dpd)2(
4C)p,x(U
MD
V
4
3V
MD
V
*MD
S
4
3S
MD
S
−=
−=
(13)
where
)p,x(U)p,x(Up)p,x(
)p,x(U)x(Um)p,x(m
MD
V
MF
V
MD
S
MF
S
*
−−=
++= (14)- effective mass
- effective momentum
)pp(izMD
V,S
4MD
V,S e)z(Dzd)pp(D−
−
Fourier transform
‚Ansatz‘ for D-function:
22
)V(S
2
)V(SMD
)V(S)pp(
)pp(D−−
=−
(15)
(16)
where f(x,p) is the single particle phase-space distribution function
19
Covariant transport equation
( ) ( ) 0)p,x(f)U(m)U()U(m)U( pS
x*
V
x
xS
p*
V
p =++−−
❑ Covariant relativistic Vlasov equation :
),( rt
x
(17)where
❑ Covariant relativistic on-shell BUU equation :
( ) ( ) collpS
x*
V
x
xS
p*
V
pI)p,x(f)U(m)U()U(m)U( =++−−
}))p,x(f1())p,x(f1()p,x(f)p,x(f
))p,x(f1())p,x(f1()p,x(f)p,x(f{
)(]GG[4d3d2dI
432
243
432
4
4321
4,3,2
coll
−−−
−−
−−+ +→+
+
from many-body theory by connected Green functions in phase-space + mean-field
limit for the propagation part (VUU)
Gain term
3+4→1+2Loss term
1+2→3+4
(18)
2
2
3
E
pd2d
20
Brueckner theory
)(]GG[ 432
4
4321 −−++→+
+Transition rate for the process 1+2→3+4
follows from many-body Brueckner theory:
1) 2-body scattering in vacuum:
Scattering amplitude:
with the hamiltonian:
)E(Ti)2(t)1(tE
1VV)E(T
+−−+=
=
+=ji
A
1i
)ij(V2
1)i(tH
1p2p
1p 2p
)E(T
1p2p
1p 2p
)12(V
1p2p
1p 2p
)12(V
)12(V
3p3p + ...+
‚ladder‘ resummation
(18)
21
Brueckner theory
2) 2-body scattering in the medium:
Scattering amplitude → from Brueckner theory:
with single-particle hamiltonian:
)E(G)nn1(i)2(h)1(hE
1VV)E(G 33
−−+−−
+=
)1(U)1(t)1(hMF+=
1p2p
1p 2p
)E(G
1p2p
1p 2p
)12(V
1p2p
1p 2p
)12(V
)12(V
3p3p + ...+
Pauli-blocking
n3 – occupation number
Note: vacuum case (1) : matrixTmatrixG0nnand)1(t)1(h 33 −→−===
Propagation between scattering V(12) with mean field hamiltonian h(1), h(2)
! only allowed if intermediate states 3,3‘ are not accupied !
(19)
22
Elementary reactions with resonances
dcRba +→→+❑ Consider the reaction
intermediate resonance
d
d
3
c
c
3
23
2
cdabdcba
4
a
4
cdabE2
pd
E2
pd
))2((
1|M|)pppp(
sp4
)2(d
−−+= →→
a
b
c
d
R
cdRRRabcdab MPMM →→→ =Matrix element:
Propagator:+−
=2
R
RMs
1P
where self-energy ==j
jtottot )s(),s(si
Cross section:
(26)
(27)
(28)
(29)
total width
23
Elementary reactions with resonances
a
b
R Partial width:
2
abRa
abR |M|s8
p)s( →→ =
pa – momentum of a in the rest frame of R or cms a+b
)s(s)Ms(
)s()s(s
p
4
)1J2)(1J2(
1J2)s(
2
tot
22
R
cdRabR
aba
RcdRab
+−
++
+= →→
→→
The spin averaged/sumed matrix element squared is
2
cdR
2
R
2
Rab
2
cdab |M|P|M||M| →→→ = (30)
(31)
(32)
24
Spectral function
Production of resonance with effective mass ➔
❑ spectral function = Breight-Wigner distribution:
)()M(
)(2)(A
2
tot
222
R
2
tot
2
+−=
1)(Ad0
=
Normalization condition:
)()(j
jtot =
The total width = the sum over all partial channels:
❑ Life time of resonance with mass :)(
c)(
tot
=
Note: Experimental life time ➔ with pole mass =MR
)M(
c
Rtot
R=
=
(33)
(34)
25
Decay rate
Decay rate:
t
e~)t(N
N1
~dt
dN
−
−
tto te1P−
−=
Total probability to survive:
Branching ratio= probability to decay to channel j:
Total probability to decayt
decayto teP
−=
)(
)(PBr
tot
j
j
=
pole mass
26
Detailed balance
Note: DB is important to get the correct equilibrium properties
dcba ++Detailed balance:
)s()s(p
)s(p
)1J2)(1J2(
)1J2)(1J2()s( dcba2
cd
2
ab
dc
babadc +→++→+
++
++=
s2
))mm(s())mm(s()s(p
2/12
ba
2
ba
ab
−−+−=
Momentum of particle a (or b) in cms:
Momentum of particle c (or d) in cms:
s2
))mm(s())mm(s()s(p
2/12
dc
2
dc
cd
−−+−=
J- spin
(23)
(24)
(25)
