lecture models for heavy-ion collisions: (part 2): transport models...
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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
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