Dynamics of relativistic heavy ion collisions I
Transcript of Dynamics of relativistic heavy ion collisions I
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Dynamics of relativistic heavy ion collisions I
V. Toneev
The Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna
August 21, 2006
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
1 Facets of HIC PhysicsPhase diagramGeneral remarksInteraction scales
2 Market of transport modelsBBHKY-hierarchyNon-relativistic BERMFRelativistic BENuclear kinetics - conclusions
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Energy stair
10 MeV
1 GeV
10 GeV
100 GeV
10 TeV
ACCELERATORS NEW PHENOMENA
formation time effects
pion (Delta) production
Fermi energy
Coulomb barrier
strangeness productionantibaryon production
GSI Darmstadt
nuclear sound velocity100 MeV
Dubna, MSUGANIL, GSI...
AGS Brookhaven
SPS CERN
Q−
H m
ixed
bary
on
less
pla
sma
RHIC Brookhaven
ph
ase
reso
nan
cepro
du
ctio
n
semihard collisions1 TeV
FAIR Darmstadt (2015)
Lanzhou
charm
beauty
color glass condensate
jets (suppression)
QG
P
LHC(2009)
Nuclotron Dubna
Synchrophasotron Dubna
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Phase diagram
Phases of strongly interacting matter
http://www.gsi.de/
Nuclotron
P.Crochet P.Senger A.Sorin
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Phase diagram
Phases of strongly interacting matter
http://www.gsi.de/
Nuclotron
P.Crochet P.Senger A.Sorin
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Phase diagram
Universe expansion T (τ)τ0
= T0(τ0)τ
τ0 = 3K×1.5·109 years200 MeV ∼ 10−3 s
1
T
MeV
200
100
B /
early
U
niv
erse
Quark Gluon Plasma
Hadron Gas
(resonance matter)
L−G phase
Fluidity CondensateKaon
SuperconductivityColorSuper
Neutron Stars
ρρ0
Color Condensates
G.Ropke M.Buballa
E.Laermann
F.Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
Hydrogen/Heatmosphere
R ~ 10 km
n,p,e, µ
neutron star withpion condensate
quark−hybridstar
hyperon star
g/cm3
1011
g/cm3
106
g/cm3
1014
Fe
−π
K−
s ue r c n d c t
gp
oni
u
p
r ot o
ns
color−superconductingstrange quark matter(u,d,s quarks)
CFL−K +
CFL−K 0
CFL−0π
n,p,e, µquarks
u,d,s
2SCCSLgCFLLOFF
crust
N+e
H
traditional neutron star
strange star
N+e+n
Σ,Λ,
Ξ,∆
n superfluid
nucleon star
CFL
CFL
2SC
G.Bisnovatyi-Kogan,D.Blaschke, H.Grigorian,S.Popov
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Phase diagram
Universe expansion T (τ)τ0
= T0(τ0)τ
τ0 = 3K×1.5·109 years200 MeV ∼ 10−3 s
1
T
MeV
200
100
B /
early
U
niv
erse
Quark Gluon Plasma
Hadron Gas
(resonance matter)
L−G phase
Fluidity CondensateKaon
SuperconductivityColorSuper
Neutron Stars
ρρ0
Color Condensates
G.Ropke M.Buballa
E.Laermann
F.Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
Hydrogen/Heatmosphere
R ~ 10 km
n,p,e, µ
neutron star withpion condensate
quark−hybridstar
hyperon star
g/cm3
1011
g/cm3
106
g/cm3
1014
Fe
−π
K−
s ue r c n d c t
gp
oni
u
p
r ot o
ns
color−superconductingstrange quark matter(u,d,s quarks)
CFL−K +
CFL−K 0
CFL−0π
n,p,e, µquarks
u,d,s
2SCCSLgCFLLOFF
crust
N+e
H
traditional neutron star
strange star
N+e+n
Σ,Λ,
Ξ,∆
n superfluid
nucleon star
CFL
CFL
2SC
G.Bisnovatyi-Kogan,D.Blaschke, H.Grigorian,S.Popov
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Phase diagram
Universe expansion T (τ)τ0
= T0(τ0)τ
τ0 = 3K×1.5·109 years200 MeV ∼ 10−3 s
1
T
MeV
200
100
B /
early
U
niv
erse
Quark Gluon Plasma
Hadron Gas
(resonance matter)
L−G phase
Fluidity CondensateKaon
SuperconductivityColorSuper
Neutron Stars
ρρ0
Color Condensates
G.Ropke M.Buballa
E.Laermann
F.Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
Hydrogen/Heatmosphere
R ~ 10 km
n,p,e, µ
neutron star withpion condensate
quark−hybridstar
hyperon star
g/cm3
1011
g/cm3
106
g/cm3
1014
Fe
−π
K−
s ue r c n d c t
gp
oni
u
p
r ot o
ns
color−superconductingstrange quark matter(u,d,s quarks)
CFL−K +
CFL−K 0
CFL−0π
n,p,e, µquarks
u,d,s
2SCCSLgCFLLOFF
crust
N+e
H
traditional neutron star
strange star
N+e+n
Σ,Λ,
Ξ,∆
n superfluid
nucleon star
CFL
CFL
2SC
G.Bisnovatyi-Kogan,D.Blaschke, H.Grigorian,S.Popov
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Phase diagram
Universe expansion T (τ)τ0
= T0(τ0)τ
τ0 = 3K×1.5·109 years200 MeV ∼ 10−3 s
1
T
MeV
200
100
B /
early
U
niv
erse
Quark Gluon Plasma
Hadron Gas
(resonance matter)
L−G phase
Fluidity CondensateKaon
SuperconductivityColorSuper
Neutron Stars
ρρ0
Color Condensates
G.Ropke M.Buballa
E.Laermann
F.Weber, Prog.Part.Nucl.Phys. 54 (2005) 193
Hydrogen/Heatmosphere
R ~ 10 km
n,p,e, µ
neutron star withpion condensate
quark−hybridstar
hyperon star
g/cm3
1011
g/cm3
106
g/cm3
1014
Fe
−π
K−
s ue r c n d c t
gp
oni
u
p
r ot o
ns
color−superconductingstrange quark matter(u,d,s quarks)
CFL−K +
CFL−K 0
CFL−0π
n,p,e, µquarks
u,d,s
2SCCSLgCFLLOFF
crust
N+e
H
traditional neutron star
strange star
N+e+n
Σ,Λ,
Ξ,∆
n superfluid
nucleon star
CFL
CFL
2SC
G.Bisnovatyi-Kogan,D.Blaschke, H.Grigorian,S.Popov
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
General remarks
Cross section
σ ∼∫ ∏
j
d3pj | < f |An|i > |2 δ(Ef − Ei )
f , i → A,R, ρi , . . . An → gi , . . .limiting cases:
• elastic, inelastic scattering p + A → p′ + A′
λ =~p 1 ψ(x) ∼ exp(ıkz) +A(~q) exp(ı~k~r)/r
with the Glauber-Sitenko amplitude
A(~q) =ı
2πλ
∫d2b exp(ı~q~b) Γ(~b)
Γ(~b) =
∫K(r)d~r =
∑i
ηi (~b −~ri )
r b
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
General remarks
• participant-spectator model (Fermi) phase space
| < f |An|i > |2 ' const
σ ∼∫ ∏
j
d3pj δ(Ef − Ei )b
L.Turko,M.Gorenstein
? Pure state → particle ensemble → statistical consideration
? Adiabatic switching on the interaction ? → time evolution
N-body Liouville equation (time reversible !)
dρN
dt=
∂
∂tρN +
1
ı~[H, ρN ] = 0
to solve it, justified approximations are neededD.Voskresensky
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
General remarks
• participant-spectator model (Fermi) phase space
| < f |An|i > |2 ' const
σ ∼∫ ∏
j
d3pj δ(Ef − Ei )b
L.Turko,M.Gorenstein
? Pure state → particle ensemble → statistical consideration
? Adiabatic switching on the interaction ? → time evolution
N-body Liouville equation (time reversible !)
dρN
dt=
∂
∂tρN +
1
ı~[H, ρN ] = 0
to solve it, justified approximations are neededD.Voskresensky
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
General remarks
• participant-spectator model (Fermi) phase space
| < f |An|i > |2 ' const
σ ∼∫ ∏
j
d3pj δ(Ef − Ei )b
L.Turko,M.Gorenstein
? Pure state → particle ensemble → statistical consideration
? Adiabatic switching on the interaction ? → time evolution
N-body Liouville equation (time reversible !)
dρN
dt=
∂
∂tρN +
1
ı~[H, ρN ] = 0
to solve it, justified approximations are needed
D.Voskresensky
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
General remarks
• participant-spectator model (Fermi) phase space
| < f |An|i > |2 ' const
σ ∼∫ ∏
j
d3pj δ(Ef − Ei )b
L.Turko,M.Gorenstein
? Pure state → particle ensemble → statistical consideration
? Adiabatic switching on the interaction ? → time evolution
N-body Liouville equation (time reversible !)
dρN
dt=
∂
∂tρN +
1
ı~[H, ρN ] = 0
to solve it, justified approximations are neededD.Voskresensky
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Nuclear scales
d - repulsion NN force rangeΛ = 1
σρ - (nucleon) mean free path
ρ0 ' 0.16 fm−3 σ ' 40 mb →Λ ∼1.5 fm
Pauli principlecompression . . .
L - ”macroscopic” length, 2-8 fm
d~0.4 fm
~1.2 fm
r
V(r)
units d Λ L d/Λ Kn = Λ/L
air (10−8 cm ) 1 105 108 10−3 10−3
liquid (10−8 cm ) 1 2-10 108 0.1-0.5 10−7
nuclei ( 0.4/1.2 fm) 1 1.5-2 2-8 0.2-0.6 1-0.2
kinetics ⇐ d Λ L ⇒ hydrodynamics
For nuclear case (intermediate energies) : d < Λ < L
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Molecular dynamics
• A-body problem in a classical picture[Quantum] Molecular Dynamics
~xi =∂
∂~piH(i = 1, . . .A)
~pi = − ∂
∂~xiH(i = 1, . . .A)
with H =−
∑52
pi+
∑i>k Vik
nuclear stabilityV → V Pauli (p)NN-scattering ?
• Fermionic Molecular Dynamics
q = ~p,~x , s . . . wave packet
∑Aµν qν = − ∂
∂qµH
with Aµν =∂2L0
∂qµ ∂qν− ∂2L0
∂qν ∂qµ
CMD limit: Aµν ⇒[0 11 0
]
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Molecular dynamics
• A-body problem in a classical picture[Quantum] Molecular Dynamics
~xi =∂
∂~piH(i = 1, . . .A)
~pi = − ∂
∂~xiH(i = 1, . . .A)
with H =−
∑52
pi+
∑i>k Vik
nuclear stabilityV → V Pauli (p)NN-scattering ?
• Fermionic Molecular Dynamics
q = ~p,~x , s . . . wave packet
∑Aµν qν = − ∂
∂qµH
with Aµν =∂2L0
∂qµ ∂qν− ∂2L0
∂qν ∂qµ
CMD limit: Aµν ⇒[0 11 0
]V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
BBGKY-Hierarchy
• Non-relativistic kinetic models
H = T + V =∑
εia†i ai +
∑V (ij , i ′j ′) a†i a
†j ai ′aj′
n-particle density :ρn(x1, x2, . . . xn) = Vn−N
∫dxn+1 . . . dxN ρ(x1 . . . xN)
i~∂ρ1(1)
∂t= [T1, ρ1(1)] + Tr(2)[V12, ρ2(1, 2)]
i~∂ρ2(1, 2)
∂t= [(T1 + T2 + V12), ρ2(1, 2)]
+ Tr(3)[(V13 + V23), ρ3(1, 2, 3)]
. . . . . . . . . . . . . . . . . . . . . . . . . . .
ρ1 ⇒ f W (~p,~x , t) =< n(~p,~x) >t
with n(~p,~x) =∫
d3k(2π~)3 eı~k~x a†
~p−~k/2a~p+~k/2
and 1∆µ
∫f W (~p,~x , t) dµ = f (~p,~x , t) + O( ~
∆µ )
approximations are needed ρ2(1, 2) = ρ1(1) ρ1(2)
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Vlasov-Boltzmann terms
Generalized kinetic equation :
∂f (~p,~x,t)∂t = −D(f ) + C (ff )
• Driving Vlasov term (classical limit)(Hartree approximation, no exchange terms)
D(~p,~x , t) =~p
m
∂
∂~xf (~p,~x , t)− ∂
∂~xU(x)
∂
∂~pf (~p,~x , t)
with an effective potential
U(x) =
∫d3x1 d3p1
(2π~)3V (~x − ~x1) f (~p1,~x1, t)
phenomenologically (Skyrme) U(x) = −aρ+ bρ2
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Boltzmann terms
• Collision term
~p + ~p2 ⇒ ~p′1 + ~p′2, no correlation and retardation effects
C (~p,~x , t) =
∫d3p2d
3p′1d3p′2
(2π~)6|T2(~p~p2;~p
′1~p′2)− T2(~p~p2;~p
′2~p′1)|2
× δ(Ep + Ep2 − E ′p1− E ′
p2) δ(~p + ~p2 − ~p′1 − ~p′2)
×[fp′
1fp′
2(1− fp)(1− fp2)− fpfp2(1− fp′
1)(1− fp′
2)]
⇑ gain ⇑ lossno exchange, no im-medium effects, ladder approximation for T2
C (~p,~x , t) =
∫d3p2d
3p′2(2π~)3
δ(~p + ~p2 − ~p′1 − ~p′2) v12dσel
dΩ
×[fp′
1fp′
2(1− fp)(1− fp2)− fpfp2(1− fp′
1)(1− fp′
2)]
∂∂t
+ ~pm
∂∂~x
+ ~pm
∂∂~pf (~p, ~x , t) = C (~p, ~x , t) + δC
BUU ⇒ events generators; f 1 ⇒ Boltzmann equationaccount for fluctuations ⇒ B-Langevin equation random force ⇑
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Relativistic kinetic equations
• the Walecka σ − ω model : L = L0 + Lint
L0 = ψ(iγµ∂µ−mN)ψ+ 1
2 (∂µσ ∂µσ−mS σ
2)− 14Fµν Fµν+ 1
2m2V ωµω
µ;
Lint = gS ψψσ − gV ψγµψωµ with Fµν = ∂µων − ∂νωµ.
Equations of motion
(∂µ∂µ + m2
S) σ = gS ψψ
∂Fµν + m2V = gV ψγ
νψ
γµ(i∂µ + gVωµ)− (mN − gSσ)ψ = 0
Klein-Gordon
Proka
DiracIn the mean-field approximation
σ0 = gS
m2S< ψψ >≡ gS
m2Sρs
ω0 = gV
m2V< ψγ0ψ >≡ gV
m2VρB[
pµ∂µ −m∗
N pν ∂∂pν
]f (p, x) = C rel(p, x)
with m∗N pν = gV pµFµν + m∗
N(∂νm∗N) and quasiparticle parameters:
m∗N = mN − gS σ0 - eff. mass, pµ → pµ− gV ωµ - kinetic momentum
RBUU ⇒ events generators
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Relativistic kinetic equations
• the Walecka σ − ω model : L = L0 + Lint
L0 = ψ(iγµ∂µ−mN)ψ+ 1
2 (∂µσ ∂µσ−mS σ
2)− 14Fµν Fµν+ 1
2m2V ωµω
µ;
Lint = gS ψψσ − gV ψγµψωµ with Fµν = ∂µων − ∂νωµ.
Equations of motion
(∂µ∂µ + m2
S) σ = gS ψψ
∂Fµν + m2V = gV ψγ
νψ
γµ(i∂µ + gVωµ)− (mN − gSσ)ψ = 0
Klein-Gordon
Proka
DiracIn the mean-field approximation
σ0 = gS
m2S< ψψ >≡ gS
m2Sρs
ω0 = gV
m2V< ψγ0ψ >≡ gV
m2VρB[
pµ∂µ −m∗
N pν ∂∂pν
]f (p, x) = C rel(p, x)
with m∗N pν = gV pµFµν + m∗
N(∂νm∗N) and quasiparticle parameters:
m∗N = mN − gS σ0 - eff. mass, pµ → pµ− gV ωµ - kinetic momentum
RBUU ⇒ events generatorsV. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Steps towards higher energies
• Relativistic Boltzmann equation (ψ, σ, ω ⇒ 0; f 1)
(pµ∂µ) fi (x , pi ) =
∑j C rel(x , pi ) +
∑r Rr→i
= −fi (x , pi )∑
j
∫dωj fj(x , pj)Qijσ
ij +∑
kj
∫dωkdωjΦ(pjpk | x , pi , τf )
+∑
r
∫dωk′dωr fr (x , pr ) Γr→i+k′
δ(pr − pi − k ′)
hadron production rate
Φ(pjpk | x , pi , τf ) =∫
dx ′ fk(x′, pi )fj(x
′, pj)Qijσij︸ ︷︷ ︸
collision rate
φ(x ′ | x , pi , τf )︸ ︷︷ ︸transition prob.
with dω = d3p/E , Qij = (pipj)2 − p2
i p2j =| vi − vj | EiEj
transition probability for a finite formation time
φ(x ′ | x , p, τf ) = 1σ
dσdω θ(t − t ′ − τf ) δ(3)(~x − ~x ′ − ~p
E (t − t ′)) F (τf )
? multiple particle production ⇒ coupled set of equations forstable hadrons and resonances hi; new flavors
? finite formation time θ(t − τf ), τf = (E/m)τ 0f with τ 0
f ∼ 1 fm;⇒ memory (retarded) effect (non-Markovian process)
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Steps towards higher energies
• Relativistic Boltzmann equation (ψ, σ, ω ⇒ 0; f 1)
(pµ∂µ) fi (x , pi ) =
∑j C rel(x , pi ) +
∑r Rr→i
= −fi (x , pi )∑
j
∫dωj fj(x , pj)Qijσ
ij +∑
kj
∫dωkdωjΦ(pjpk | x , pi , τf )
+∑
r
∫dωk′dωr fr (x , pr ) Γr→i+k′
δ(pr − pi − k ′)
hadron production rate
Φ(pjpk | x , pi , τf ) =∫
dx ′ fk(x′, pi )fj(x
′, pj)Qijσij︸ ︷︷ ︸
collision rate
φ(x ′ | x , pi , τf )︸ ︷︷ ︸transition prob.
with dω = d3p/E , Qij = (pipj)2 − p2
i p2j =| vi − vj | EiEj
transition probability for a finite formation time
φ(x ′ | x , p, τf ) = 1σ
dσdω θ(t − t ′ − τf ) δ(3)(~x − ~x ′ − ~p
E (t − t ′)) F (τf )
? multiple particle production ⇒ coupled set of equations forstable hadrons and resonances hi; new flavors
? finite formation time θ(t − τf ), τf = (E/m)τ 0f with τ 0
f ∼ 1 fm;⇒ memory (retarded) effect (non-Markovian process)
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Strings
• new degrees of freedom (QCD) : quark/gluons , strings,formation of color rope
? Hadron as a stringG.S.Bali, K.Schilling, Phys.Rev.D 46 (1992) 2636
Vqq = −αeffr + κr
yo-yo mode
String breakup
Hyo−yo = | p1 | + | p2 | +κ | x1 − x2 |dp1,2
dt= ±κ , dx1,2
dt= ±1
x+ = p+
κ = E+pκ , x− = p−
κ = E−pκ , S = p+p−
κ2 = E2−p2
κ2 = m2
κ2
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Strings
• new degrees of freedom (QCD) : quark/gluons , strings,formation of color rope
? Hadron as a stringG.S.Bali, K.Schilling, Phys.Rev.D 46 (1992) 2636
Vqq = −αeffr + κr
yo-yo modeString breakup
Hyo−yo = | p1 | + | p2 | +κ | x1 − x2 |dp1,2
dt= ±κ , dx1,2
dt= ±1
x+ = p+
κ = E+pκ , x− = p−
κ = E−pκ , S = p+p−
κ2 = E2−p2
κ2 = m2
κ2
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
String interaction
• Particularities of space-time evolution
CLASSICAL STRING THEORY
? string fusion
time ⇒? string rearrangement
DUAL TOPOL. MODEL? planar diagram
? cylindrical diagram
? leading particle effect
? color rope formation
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Jets
• jet production
Hard parton-parton collision(true two-body scattering)p⊥ ΛQCD , r⊥ ∼ ~c/p⊥
dσ
dp⊥=
C
pn⊥
RHIC physics
HIJING code
• solution ⇒ Monte Carlo Methods:event generators ⇒ UrQMD, QGSM, HSD . . .
• quark-gluon transport (color → dynamical degree of freedom,
in the quasi-classical limit → (p∂x − gpF∂p)W )
? Nambu-Jona-Losinio model V.Yudichev
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Jets
• jet production
Hard parton-parton collision(true two-body scattering)p⊥ ΛQCD , r⊥ ∼ ~c/p⊥
dσ
dp⊥=
C
pn⊥
RHIC physics
HIJING code
• solution ⇒ Monte Carlo Methods:event generators ⇒ UrQMD, QGSM, HSD . . .
• quark-gluon transport (color → dynamical degree of freedom,
in the quasi-classical limit → (p∂x − gpF∂p)W )
? Nambu-Jona-Losinio model V.Yudichev
V. Toneev Dynamics of relativistic HI collisions
Dynamics ofrelativistic HI
collisions
V. Toneev
Facets of HICPhysics
Phase diagram
General remarks
Interactionscales
Market oftransportmodels
BBHKY-hierarchy
Non-relativisticBE
RMF
Relativistic BE
Nuclear kinetics- conclusions
Basic kinetic idea
HIC ⇒ subsequent collisionsbetween quasiparticles(Boltzmann-like equations)
Physics : What is a quasiparticle ?
non-relativistic
( ∂∂t + ~v ~5x + d~p
dt~5p)f (~p,~x , t) = C (~p,~x , t)
⇑ d~pdt = −~5x
d~pdt U(~r , t)
relativistic – QHD
(pµ∂µ + m∗pµ∂
µ)f (p, x) = C rel(p, x)m∗pµ = gV pνFµν + m∗(∂µ
x m∗) + field eqs.
Boltzmann : (pµ∂µ)f (p, x) = C rel(p, x)
non-abelian fields (color) – QCDp, x ⇒ p, x ,Qflow term +source term
extreme case: free rescatt. of quarks and gluonspartons
(p-h)N + V (r)free N
hadrons +ψ(Walecka-like)
resonancesstringscolor ropesjets
quarks/gluons
V. Toneev Dynamics of relativistic HI collisions