Non-equilibrium Dynamics of the Chiral/Deconfinement Phase - - PowerPoint PPT Presentation

Igor N. Mishustin

Frankfurt Institute for Advanced Studies, J.W. Goethe Universität, Frankfurt am Main and National Research Centre, “Kurchatov Institute” Moscow

BLTP seminar, JINR, Dubna, September 27, 2014

Non-equilibrium Dynamics of the Chiral/Deconfinement Phase Transition

Contents ntents

• Introduction: Effects of fast dynamics
• Effective potential and fluctuations of order parameter
• Chiral fluid dynamics with damping and noise
• Extension to finite baryon densities
• Dynamical domain formation in 1st order tansition
• Conclusions

Recent cent publi blicat cation ions

• M. Nahrgang, C. Herold, S. Leupold, I. Mishustin, M. Bleicher, The impact of

dissipationand noise on fluctuations in chiral fluid dynamics, J. Phys. G 40 (2013) 055108;

• C. Herold, M. Nahrgang, I. Mishustin, M. Bleicher, Chiral fluid dynamics with

explicit propagation of the Polyakov loop, Phys. Rev. C 87 (2013) 014907;

• C. Herold, M. Nahrgang, I. Mishustin, M. Bleicher, Formation of droplets

with high baryon density at the QCD phase transition in expanding matter,

• Nucl. Phys. A 925 (2014) 14;
• I. Mishustin, T. Koide, G. Danicol, G. Torrieri, Dynamics and stability of

chiral fluid, Phys. Atom. Nucl. 77, 1130-1144, arXiv:1401.4103.

Phase ase diagr agram am of f str trong

• ngly

ly-inter interact acting ing matter tter

Such a phase diagram is still a beautiful dream! We hope that future FAIR-NICA experiments will help to establish what is the reality.

• NICA

Eff ffects ects of f fa fast t dynami namics cs

expansion Q H 1 2 3 T>T

c

T<T

c

T=T

c

4

T

1 2 3 C spinoidal lines critical line 4

2 4 6

( ; , ) ( , ) 2 4 6 a b c T T

Effective thermodynamic potential for a 1st order transition

In rapidly expanding system 1-st order transition is delayed until the barrier between two competing phases disappears - spinodal decomposition

• I. Mishustin, Phys. Rev. Lett. 82 (1999) 4779; Nucl. Phys. A681 (2001) 56

a,b,c are functions of

and T ( )

eq

P

crossover

cp 1-st order

Equilibrium is determined by

Equ quil ilib ibrium rium fl fluc uctua tuatio tions ns of

• f or
• rde

der pa param ameter eter in in 1st

st or

• rde

der ph phas ase tr e transi ansition tion

P( ) T>T

c

P( ) T<T

c

T=T

c

P( ) Phase I Mixed phase Phase II In an equilibrated system fluctuations of the order parameter, i.e. Polyakov loop, should demonstrate bi-modal distributions (lattice calculations?); In a rapidly evolving system these fluctuations will be out of equilibrium; During supercooling process strong fluctuations may develop in the form

• f droplets of a metastable phase.

Probability distribution for fluctuations

( ) ( ) exp , V P T

Rap apid id ex expa pans nsio ion n th throu

• ugh

gh a 1 a 1st

st or

• rde

der ph phas ase e tr tran ansition sition

The system is trapped in a metastable state until it enters the spinodal instability region, when Q phase becomes unstable and splits into droplets

Csernai&Mishustin, 1995; Mishustin, 1999; Rafelski et al. 2000; Randrup, 2003; Steinheimer&Randrup 2013; …

Evolution of equilibrium fluctuations in 2nd order phase transition

T>T

c

T<T

c

T=T

c

2 2 4

1 1 ( ) ( ) ( ) , ( ) ( ) 2 2 4 ( ) , and 0, , ( ) Distribution of fluctuations ( ) exp

c c c

a T b a T a T T a T T T T T V P T

In rapidly expanding system critical fluctuations have not sufficient time to develop

rel

1 ( ) , 2

c

T T T

Critical slowing down in the 2nd order phase transition

T=0 T>T

c

T=T

c

B In the vicinity of the critical point the relaxation time for the order parameter diverges - no restoring force (Landau&Lifshitz, vol. X, Physical kinetics)

rel

d dt

“Rolling down” from the top of the potential is similar to spinodal decomposition (Csernai&Mishustin 1995)

( )

eff

U f

Fluctuations of the order parameter evolve according to the relaxation equation

Critical itical slow

• wing

ing down wn 2

• B. Berdnikov, K. Rajagopal, Phys. Rec. D61 (2000)

Critical fluctuations have not enough time to build up. One can expect only a factor 2 enhancement in the correlation length even for slow cooling rate, dT/dt=10 MeV/fm.

Simple model for chiral phase transition

Linear sigma model (LσM) with constituent quarks Effective thermodynamic potential Phase diagram contains contributions of mean field σ and quark-antiquark fluid: CO, 2nd and 1st order chiral transitions are obtained in T-μ plane.

5 2 2 2 2 2 2 vac

1 [ ( )] [ ] ( , ), 2 ( , ) ( ) , < > 4 L q i g i q U U v H f H f m

eff 2 2 2

( ; , ) ( , ( ; , ) ( )

q

U T U m T m g

Scavenius, Mocsy, Mishustin&Rischke, Phys. Rev. C64 (2001) 045202

Effective thermodynamic potential

First we consider μ=0 system but tune the order of the chiral phase transition by changing the coupling g.

2 2 3 f

• ( ; , )

ln 1 exp ( ) , =2

q q c

m p d m T T N N T

3

p (2 )

Equilibrium order parameter field

g=4.5 g=3.3 crossover 1-st order 3 solutions at 122 MeV<T<132 MeV unstable states - spinodal instability Only 1 equilibrium solution at each T

q 2 2

( ; , ) ( ) 0, ( )=

s s

m T g

2 2 2 2 2 2

( ) (3

• )

eff s

U m g

Spectrum of plane-wave fluctuations

T=120 MeV T=125 MeV T=131 MeV

( ) ( , )

i t ikx

x k e

Solutions with ω2<0 indicate instability

sound waves

Generally two branches: 1) sound branch 2) sigma branch

2 2 2 s

c k

2 2 2

m k

• I. Mishustin, T. Koide, G. Danicol, , G. Torrieri, Phys. Atom. Nucl.; arXiv:1401.4103.

σ-meson excitations

Fluctuations in Bjorken background

2 2 2 2 2 2 2

( , ) ( ) ,

• 1

( ) ( ) ( ) / ( )

• ( )

/

ik k k s

e t z k f T s m g s T T

Crossover transition (g=3.3) 1st order transition (g=4.5)

Non-equilibrium Chiral Fluid Dynamics

Fluid is formed by constituent quarks and antiquarks which interact with the chiral field via quark effective mass CFD equations are obtained from the energy momentum conservation for the coupled system fluid+field

I.N. Mishustin, O. Scavenius, Phys. Rev. Lett. 83 (1999) 3134;

• K. Paech, H. Stocker and A. Dumitru, Phys. Rev. C 68 (2003) 044907;
• M. Nahrgang, C. Herold, S. Leupold, , C. Herold, M. Bleicher, Phys. Rev. C 84 (2011) 024912;
• M. Nahrgang, C. Herold, S. Leupold, I. Mishustin, M. Bleicher, J. Phys. G40 055108.

m g

fluid field fluid field 2 eff

( ) ( ) ( )

s t

T T T T S U S g We solve generalized e. o. m. with friction (η) and noise (ξ): Langevin equation for the order parameter (f

eff t

U g qq

' ' ' '

1 ( , ) 0, ( , ) ( , ) ( ) ( )coth 2 m t r t r t r m t t r r V T

Calculation of damping term

T.Biro and C. Greiner, PRL, 79. 3138 (1997)

• M. Nahrgang, S. Leupold, C. Herold, M. Bleicher, PRC 84, 024912 (2011)

The damping is associated with the processes: It has been calculated using 2PI effective action Around Tc the damping is due to the pion modes, η=2.2/fm

, qq

3/2 2 2 2 2 1 2

2 4

q F q

m m g n m m

Dynamic simulations: Bjorken-like expansion

Initial state: cylinder of length L in z direction, with ellipsoidal cross section in x-y direction

x

2 At 0: ( ) 0.2 , - ; v 0; 160 2 2

y

z L L t v z c z v T MeV L Mean an value lues s and d standa andard d dev eviati ation

• n of T

T f for r the whole le sys ystem tem and d for r a c centra ntral cell l (1 fm3) ) are re show

• wn as a fu

func nction

• n of ti

time. e. Superc ercooling

• oling and

d rehea eheating ing effec ects ts are e clearly early seen en in the 1-st st order der trans ansitio ition, n, fluct ctuat uation ions s become ecome espec pecial ially ly stron rong g after er 4 fm fm/c. c.

Criti tical cal point t (g=3.6 3.63) 3) First st order r (g=5.5 5.5)

Sigma fluctuations in expanding fireball

eq

2 . 2 2 2 2 2 2 eff 3 3 2

1 1 [ | | | | ], , (2 ) 2

k k k k k

dN U m k m d k

Fluctuat ctuations ions are e rather her weak ak at c cri ritic ical al point nt (lef eft), t), but incr crea ease se stro rongly ngly at the 1st

st order

der transi ansition ion (righ right) t) after er 4 fm fm/c

Critical point (g=3.63) First order (g=5.5)

Extension to finite baryon densities

• Include µ-dependence in Polyakov loop potential,

(cf. Schäfer, Pawlowski, Wambach Fukushima)

• Calculate grand canonical potential for finite chemical potential
• Propagate (net) baryon density in the hydro sector
• C. Herold, M. Nahrgang, I. Mishustin, M. Bleicher, Nucl. Phys. A 925 (2014) 14;

Trajectories on the T-μ plane

Isentropic expansion Hydrodynamic evolution

• Trajectories are close to isentropes for crossover and CP;
• Non-equilibrium “back-bending” is clearly seen in FO case;
• In the case of strong FO transition (solid lines) the system is trapped

in spinodal region for a significant time

CFD calculations are done for spherical fireball of R=4 fm

Dynamical droplet formation

First order Critical point

Splash of a milk drop

HEE-NC-57001

Observable signatures of high- density domains

Azimuthal fluctuations of net-B In single events: strong enhancement at first order PT High harmonics of baryonic flow (averaged over many events): vn=<cos[n(φ-φn)]>

New developments

• In the previous calculations the EOS had a P=0

point at a finite baryon density (like the MIT bag model), that makes possible stable quark droplets

• It is interesting to see what happens in a more

realistic case when quark droplets are unstable at zero pressure (J. Steinheimer et al, PRC 89 (2014) 034901)

• There exist several models which have such a

property, in particular so called Quark-Hadron Model (S. Schramm et al. ) or Quark-Dilaton Model (C. Sasaki et al.).

SU(3) chiral quark-hadron (QH) model

Includes: a) 3 quarks (u,d,s) plus baryon octet, b) scalar mesons (σ, ς), vector meson (ω) c) Polyakov loop (l) Effective masses:

• V. Dexheimer, S. Schramm,Phys, Rev. C 81 (2010) 045201

PQM vs. QH: phase diagram

Nuclear ground state at µN=3µ≈mN is reproduced correctly QH predicts two phase transitions: 1) liquid-gas PT at µ≈300 MeV, and 2) deconfinement/chiral PT at higher µ≈450 MeV

Herold, Limphirat, Kobodaj, Yan, Seam Pacific Conference 2014

PQM vs. QH: domain formation

QH predicts domains with much higher densities!

Herold, Limphirat, Kobodaj, Yan, Seam Pacific Conference 2014

PQM vs. QH: density moments

Strong clustering effect survives even at late times, t>15 fm/c

Experi perimental mental signatures gnatures of f drople

• plets

ts

The bumps correspond to the emission from individual domains. Look for bumpiness in distributions of net baryons in indi- vidual events, i. e. in azimuthal angle, rapidity, transverse momentum

Conclusions

• Phase tran

ansiti sitions

• ns in relativi

tivist stic c heavy-ion ion collisions ns wi will most st likely y proce ceed ed out of equilibri rium um

• 2nd

nd order

er phase tra ransitio nsition n (wi with h CE CEP) is too we weak to produce uce signifi ficant cant observa rvable ble effe fect cts s in f fast st dynami mics cs

• No

Non-equ quilibri librium um effe fect cts s in a1 a1st

st order

r tran ansition sition (spinod inodal al decompo mpositi sition,

• n, dynami

mical cal domain form rmation) ation) may help to id identify tify the chir iral al/deconf econfine inement ment phase transiti ansition

• n
• If

f large e QGP GP domains ins survi vive ve until l the free eeze ze-out

• ut stage

ge they y wi will sh show w up by large non-statistica statistical l fluct ctuat ations ns of

• bservab

rvables les in in sin ingle le events ts

• Exoti

tic c objects cts like stran range gele lets ts have a better ter chance e to

• be form

rmed d in su such a no non-equi equilib ibri rium um scena nario rio