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basic hadronic SU(3) model generating a critical end point in a hadronic model
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Transcript of basic hadronic SU(3) model generating a critical end point in a hadronic model
• 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
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
**
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
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
Evolution of the collision system
Elab
≈ 5-10 AGeV sufficient to overshoot phase border, 100-160 AGeV around endpoint