11

Lecture

Models for heavy-ion collisions:

(Part 2): transport models –

BUU

SS2021: ‚Dynamical models for relativistic heavy-ion collisions‘

2

Vlasov equation-of-motion

0)t,p,r(f)t,r(U)t,p,r(fm

p)t,p,r(f

tprr =−+

Vlasov equation

- free propagation of particles in the self-generated HF mean-field potential:

Here U is a self-consistent potential associated with f phase-space distribution:

),,(),()2(

1),(

33

3tprftrrpVdrdtrU

−=

➔ Classical equations of motion :

)t,r(Udt

pdp

m

p

dt

rdr

r

−==

==

)t(r:trajectoty

1

2

From the Vlasov equation of motion

to Boltzmann-Uehling-Uhlenbeck equation

(BUU) – collision term

3

Dynamical transport models with collisions

add 2-body collisions:

dt 1

2

3

5

6

4

t

Interaction 1+2→3+4

12V34 coll

d - average distance

In cms:

*

3p

*

4p

*

1p *

2p

*

0pppp *

4

*

3

*

2

*

1 =+=+

❑ If the phase-space around

is essentially empty then the scattering is allowed,

❑ if the states are filled → Pauli suppression

= Pauli principle

)p,r()p,r()p,r()p,r( 44332211

→

)p,r(and)p,r( 4433

➔ In order to describe the collisions between the individual(!) particles, one has to

go beyond the mean-field level ! (See Part 2: Correlation dynamics)

4

BUU (VUU) equation

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

=−+

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:

)m2

p

m2

p

m2

p

m2

p()2()pppp()2(

)4321(wpdpdpd))2((

1

t

fI

4

4

3

3

2

2

1

14321

33

4

3

3

3

2

3

33

coll

coll

−−+−−+

+→+

Collision integral for 1+2→3+4 (let‘s consider fermions) :

Transition probability for 1+2→3+4:qd

d)4321(w

3

3

12

+→+

|pp|m

2112

−=where - relative velocity of the colliding nucleons

Probability including

Pauli blocking of fermions

qd

d3

3- differential cross section, q – momentum transfer 31 ppq

−=

5

BUU: Collision integral

P)4321(d

d)pppp(||dpdpd

)2(

4I 4321

3

123

3

2

3

3coll +→+−−+=

)f1)(f1(ff)f1)(f1(ff

)t,p,r(f1)t,p,r(f1)t,p,r(f)t,p,r(f

)t,p,r(f1)t,p,r(f1)t,p,r(f)t,p,r(fP

43212143

4321

2143

−−−−−

−−−

−−=

Probability including Pauli blocking of fermions:

Pauli blocking factors

for fermions *

*Note: for bosons – enhancement factor 1+f (where f<<1);

often one neglects bose enhancement for HIC, i.e. 1+f →1

Gain term

3+4→1+2Loss term

1+2→3+4

LGIcoll −=For particle 1 and 2:

Collision term = Gain term – Loss term

6

In equilibrium collision term = 0

→ Gain term = Loss term

i.e. number of forward and backward reactions in the system is the same

Collision integral for system in equilibrium

Consider fermion gas in equilibrium - described by Fermi-Dirac distribution:

7

T – temperature,

m – baryon chemical potential

Collision interal of fermion system:

➔ nF(e) is stationary solution of I(p1,p2;t)=0

Collision integral for system in equilibrium

8

To show that we have to demonstrate that

Consider

Since the denominators on both sides are the same one has to proof only

Since due to energy conservation δ(ϵ1 +ϵ2−ϵ3−ϵ4) we have ϵ1 +ϵ2 = ϵ3 +ϵ4, what proofs that nF (ϵ) is a stationary solution of collision integral for system in equilibrium

➔ Transport equations have a correct thermodynamic limit!

Dynamical transport model: collision terms

(20)

❑ BUU eq. for different particles of type i=1,…n

n21collii f,...,f,fIf

dt

dDf =

,...,/J,D,D,...,a,,,K,,,K,K,,:Mesons

;,,,,),...,1535(N),1440(N),1232(,n,p:Baryons:i

1

*

C

*

Drift term=Vlasov eq. collision term

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

...

,...f,f,...,f,f,fIDf

...

,...f,f,...,f,f,fIDf

,...f,f,...,f,f,fIDf

)1440(NNcoll

)1440(NNcoll

)1440(NNcollN

=

=

=

9

10

E.g., Nucleon transport in N,, system : DfN=Icoll

(only 1→2, 2→2

reactions indicated here)

Full collision term consists of >10000

different particle combinations

➔ set of transport equations coupled via Icoll and mean field

Dynamical transport model: collision terms

decayproduction

)N(Loss)N(Gain

))p(f1))(p(f1()p(f)ppp(|M|E

pd

E

pd

)2(

g

))p(f1)(p(f)p(f)ppp(|M|E

pd

E

pd

)2(

gDf

NNN

42

N

N

N

33

N,3

NNN

42

N

N

N

33

N,3

→−→=

+−−+−

−−+=

nucleonbydecayby

absorbtionproduction

)N(Loss)N(Gain

))p(f1)(p(f)p(f)ppp(|M|E

pd

E

pd

)2(

g

))p(f1))(p(f1()p(f)ppp(|M|E

pd

E

pd

)2(

gDf

NNN

42

N

N

N

33

N,3

NNN

42

N

N

N

33

,N3

→−→=

−−+−

−+−+=

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

Eq

. fo

r

Eq

. fo

r

11

→N

N→

→N

N→

Dynamical transport model: possible interactions

Consider possible interactions for the sytem of (N,R,m),

where N-nucleons, R- resonances, m-mesons

❑ elastic collisions:

RRRR

NRNR

NNNN

→

→

→

❑ inelastic collisions:

mRmR

mNmN

→

→ mmmm →

XBB

...

RRNN

RNNR

NRNN

→

XmB

...

BmmB

RmR

RmN

→

Baryon-baryon (BB): meson-Baryon (mB) meson-meson (mm)

Baryon-baryon (BB): meson-Baryon (mB) meson-meson (mm)

Xmm

...

mmmm

m~mm

→

X - multi-particle state

dcba

cba

++

+

Detailed balance:

12

Elementary hadronic interactions

Low energy collisions:

▪ binary 2→2 and

2→3(4) reactions

▪ 1→2 : formation and

decay of baryonic and

mesonic resonances

BB → B´B´

BB → B´B´m

mB → m´B´

mB → B´

mm → m´m´

mm → m´ . . .

Baryons:

B = p, n, (1232),

N(1440), N(1535), ...

Mesons:

M = , , , , , ...

+p

pp

High energy collisions:

(above s1/2~2.5 GeV)

Inclusive particle

production:

BB→X , mB→X, mm→X

X =many particles

described by

string formation and decay

(string = excited color

singlet states q-qq, q-qbar)

using LUND string model

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

25

14

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:

total width

15

Elementary reactions with resonances

a

b

R Partial decay 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| →→→ =

16

Spectral function

Production of resonance with effective mass m ➔

❑ spectral function = Breight-Wigner distribution:

)()M(

)(2)(A

2

tot

222

R

2

tot

2

mmm

mm

m

+−=

1)(Ad0

=

mmNormalization condition:

)()(j

jtot mm =

The total width = the sum over all partial channels:

❑ Life time of resonance with mass m:)(

c)(

tot mm

=

Note: Experimental life time ➔ with pole mass m=MR

)M(

c

Rtot

R=

=m

17

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

jm

m=

pole mass

18

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

1919

Detailed balance on the level of 2→n:

treatment of multi-particle collisions in transport approaches

W. Cassing, NPA 700 (2002) 618

Generalized collision integral for n → m reactions:

is Pauli-blocking or Bose-enhancement factors;

=1 for bosons and =-1 for fermions

is a transition probability A(x,p) - spectral function

2020

Antibaryon production in heavy-ion reactions

0 2 4 6 8 10

100

101

102

Pb+Pb, 160 A GeV

central

__

BB->X

__

3 mesons -> BB

dN

/dt

[arb

. u

nit

s]

t [fm/c]

W. Cassing, NPA 700 (2002) 618

E. Seifert, W. Cassing, 1710.00665, 1801.07557Multi-meson fusion reactions

m1+m2+...+mn → B+Bbar

m=,,,.. B=p,,,, (>2000 channels)

❑ important for anti-proton, anti-lambda,

anti-Xi, anti-Omega dynamics !2→3

→ approximate equilibrium of annihilation and recreation

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Pierre Moreau 32

Stages of a collision in PHSD

Processes in collisions of heavy ions

Introduction PHSD ingredients Chiral symmetry restoration Strangeness in A+A Summary

Useful literature

L. P. Kadanoff, G. Baym, ‚Quantum Statistical Mechanics‘, Benjamin, 1962

M. Bonitz, ‚Quantum kinetic theory‘, B.G. Teubner Stuttgart, 1998

W. Cassing and E.L. Bratkovskaya, ‘Hadronic and electromagnetic probes of

hot and dense nuclear matter’, Phys. Reports 308 (1999) 65-233. http://inspirehep.net/record/495619

29