• basic hadronic SU(3) model • generating a critical end point in a hadronic model revisited• including quark degrees of freedom phase diagram – the QH model• excluded volume corrections, phase transition

Modeling of the Parton-Hadron Phase Transition Villasimius 2010

J. Steinheimer, V. Dexheimer, H. Stöcker, SWSGoethe University, Frankfurt

OUTLINE

Hot and dense matter and the phase transition in quark-hadron approaches

A) SU(3) interaction

~ Tr [ B, M ] B , ( Tr B B ) Tr M

B) meson interactions ~ V(M) <> = 0 0 <> = 0 0

C) chiral symmetry m = mK = 0 explicit breaking ~ Tr [ c ] ( mq q q )

light pseudoscalars, breaking of SU(3)

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hadronic model based on non-linear realization of chiral symmetry

degrees of freedom SU(3) multiplets:

~ <u u + d d> < ~ <s s> 0 ~ < u u - d d>

baryons (n,Λ, Σ, Ξ) scalars (, , 0) vectors (ω, ρ, φ) , pseudoscalars, glueball field χ

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fit parameters to hadron masses

’

mesons

Model can reproduce hadron spectra via dynamical mass generation

p,n

K

K*

*

*

Lagrangian (in mean-field approximation)

L = LBS + LBV + LV + LS + LSB

baryon-scalars:

LBS = - Bi (gi + gi

+ gi ) Bi

LBV = - Bi (gi + gi

+ gi ) Bi

baryon-vectors:

meson interactions:

LBS = k1 (2 + 2 + 2 )2

+ k2/2 (4 + 2 4 + 4 + 6 2 2 ) + k3 2 - k4 4 - 4 ln /0 + 4 ln [(2 - 2) / (0

20)]

explicit symmetry breaking: LSB = c1 + c2

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_

LV = g4 (4 + 4 + 4 + β 22)

/ / /

I

I

important reality check

compressibility ~ 223 MeV asymmetry energy ~ 31.9 MeV

equation of state E/A () asymmetry energyE/A (p- n)

nuclear matter properties at saturation density

binding energy E/A ~ -15.2 MeV saturation (B)0 ~ .16/fm3

phenomenology: 200 - 250 MeV 30 - 35 MeV

+ good description of finite nuclei / hypernucleiSWS, Phys. Rev. C66, 064310

Task: self-consistent relativistic mean-field calculationcoupled 7 meson/photon fields + equations for nucleons in 1 to 3 dimensions

parameter fit to known nuclear binding energies and hadron masses

2d calculation of all measured (~ 800) even-even nuclei

error in energy (A 50) ~ 0.21 % (NL3: 0.25 %) (A 100) ~ 0.14 % (NL3: 0.16 %)

good charge radii rch ~ 0.5 % (+ LS splittings)

SWS, Phys. Rev. C66, 064310 (2002)

relativistic nuclear structure models

correct binding energies of hypernuclei

phase transition compared to lattice simulations

heavy states/resonance spectrum is effectively

described by single (degenerate) resonance with

adjustable couplings

mR m0 g R

g R rv g N

reproduction of LQCD phase diagram, especially T

c, μ

c

+successful description of nuclear matter saturation

phase transition becomes first-order for degenerate baryon octet ~ Nf = 3with Tc ~ 185 MeV

Tc ~ 180 MeVµc ~ 110 MeV

D. Zschiesche et al. JPhysG 34, 1665 (2007)

Isentropes, UrQMD and hydro evolution

J. Steinheimer et al. PRC77, 034901 (2008)

lines of constant entropy per baryon, i.e. perfect fluid expansion E/A = 5, 10, 40, 100, 160 GeV E/A = 160 GeV goes through endpoint

P. Rau, J. Steinheimer, SWS, in preparation

Including higher resonances explicitly

Add resonances up to 2.2 GeV. Couple them like the lowest-lying baryons

order parameter of the phase transition

confined phase

deconfined phase

effective potential for Polyakov loop, fit to lattice data

quarks couple to mean fields via gσ, gω

connect hadronic and quark degrees of freedom

minimize grand canonical potential

baryonic and quark mass shift δ mB ~ f(Φ) δ mq ~ f(1-Φ)

V. Dexheimer, SWS, PRC 81 045201 (2010)Ratti et al. PRD 73 014019 (2006)Fukushima, PLB 591, 277 (2004)

U = ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2]

a(T) = a0T4 + a1 µ4 + a2 µ2T2

qq

hybrid hadron-quark model critical endpoint tuned to lattice results

Phase Diagram for HQM model

µc = 360 MeVTc = 166 MeV

µc. = 1370 MeVρc ~ 4 ρo

V. Dexheimer, SWS, PRC 81 045201 (2010)

Mass-radius relation using Maxwell/Gibbs construction

Gibbs construction allows for quarks in the neutron starmixed phase in the inner 2 km core of the star

V. Dexheimer, SWS, PRC 81 045201 (2010)R. Negreiros, V. Dexheimer, SWS, PRC, astro-ph:1006.0380 

Csph ~ ¼ Cs,ideal

isentropic expansion

overlap initial conditionsElab = 5, 10, 40, 100, 160 AGeV

averaged Cs significantly higher than 0.2

Temperature distribution from UrQMD simulation as initial state for (3d+1) hydro calculation

dip in cs is smeared out

Speed of sound - (weighted) averageover space-time evolution

initial temperature distribution

Include modified distribution functions for quarks/antiquarks

Following the parametrization used in PNJL calculations

The switch between the degrees of freedom is triggered by excluded volume corrections

thermodynamically consistent -

D. H. Rischke et al., Z. Phys. C 51, 485 (1991)J. Cleymans et al., Phys. Scripta 84, 277 (1993)

U = - ½ a(T) ΦΦ* + b(T) ln[1 – 6 ΦΦ* + 4 (ΦΦ*)3 – 3 (ΦΦ*)2]

a(T) = a0T4 + a1 T0T3 + a2 T02T2 , b(T) = b3 T0

3 T

χ = χo (1 - ΦΦ* /2)

Vq = 0Vh = vVm = v / 8

µi = µ i – vi P ~

different approach – hadrons, quarks, Polyakov loop and excluded volume

e = e / (1+ Σ vi ρi )~ ~

Steinheimer,SWS,Stöcker hep-ph/0909.4421

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quark, meson, baryon densities at µ = 0

natural mixed phase, quarks dominate beyond 1.5 Tc

densities of baryon, mesons and quarks

Energy density and pressure compared to lattice simulations

Interaction measure e – 3p

Temperature dependenceof chiral condensate and Polaykov loop at µ = 0

lattice data taken from Bazavov et al. PRD 80, 014504 (2009)

speed of sound shows a pronounced dip around Tc !

subtracted condensate and polyakov loop different lattice groups and actions

From Borsanyi et al., arxiv:1005:3508 [hep-lat]

Lattice comparison of expansion coefficients as function of T

expansion coefficients

lattice data from Cheng et al., PRD 79, 074505 (2009)

lattice results

Steinheimer,SWS,Stöcker hep-ph/0909.4421 suppression factor peaks

Φ

Dependence of chiral condensate on µ, T

Lines mark maximum in T derivative

σ

Separate transitions in scalar field and Polyakov loop variable

Φ

Dependence of Polyakov loop on µ, T

Lines mark maximum in T derivative

Separate transitions in scalar field and Polyakov loop variable

σ

Susceptibilitiy c2 in PNJL and QHM for different quark vector interactions

Steinheimer,SWS, hepph/1005.1176

gqω = gnω /3

gqω = 0

PNJL

QH

At least for µ = 0 –small quark vector repulsion

σ Φ

UrQMD/Hydro hybrid simulation of a Pb-Pb collision at 40 GeV/A

red regions show the areas dominated by quarks

fs

If you want it exotic …

follow star calcs by J. Schaffner et al., PRL89, 171101 (2002)

E/A-mN

additional coupling g2 of hyperons to strange scalar field

g2 = 0

g2 = 2

g2 = 4

g2 = 6

barrier at fs ~ 0.4 – 0.60 0.5 1 1.5

0

100

200

300

simple time evolution including π, K evaporation (E/A = 40 GeV)

C. Greiner et al., PRD38, 2797 (1988)

with evaporation

SUMMARY

• general hadronic model as starting point• works well with basic vacuum properties, nuclear matter, nuclei, …• phase diagram with critical end point via resonances• implement EOS in combined molecular dynamics/ hydro simulations• quarks included using effective deconfinement field• „realistic“ phase transition line• implementing excluded volume term, natural switch of d.o.f.

If you want to see hadronization,grab your Iphone -> Physics to Go! Part 3

Hypernuclei - single-particle energies

Model and experiment agree well

Nuclear matter

Evolution of the collision system

Elab

≈ 5-10 AGeV sufficient to overshoot phase border, 100-160 AGeV around endpoint

amount of volume scanning the critical endpoint (lattice)

Overlap of projectile and target

Comparison of gross properties of initial conditions

time integrated volume around the critical end point

effective volume sampling the critical end point

window T = Tc ± 10 MeV µ = µc ± 10 MeV

maximum shifts in time