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


collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


1 Facets of HIC PhysicsPhase diagramGeneral remarksInteraction scales

2 Market of transport modelsBBHKY-hierarchyNon-relativistic BERMFRelativistic BENuclear kinetics - conclusions



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


Phase diagram

Phases of strongly interacting matter

http://www.gsi.de/

Nuclotron

P.Crochet P.Senger A.Sorin



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


General remarks


| < f |An|i > |2 ' const

σ ∼∫ ∏

j






dρN

dt=

∂

∂tρN +

1

ı~[H, ρN ] = 0

to solve it, justified approximations are needed

D.Voskresensky



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


General remarks


| < f |An|i > |2 ' const

σ ∼∫ ∏

j






dρN

dt=

∂

∂tρN +

1

ı~[H, ρN ] = 0

to solve it, justified approximations are neededD.Voskresensky



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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

]



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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


collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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)



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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 ⇑



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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


collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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)



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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



collisions

V. Toneev


Phase diagram

General remarks

Interactionscales


BBHKY-hierarchy

Non-relativisticBE

RMF

Relativistic BE


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


dynamics of relativistic heavy ion collisions i

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