Phase Fluctuations near the Chiral Critical Point

Joe Kapusta

University of Minnesota

Winter Workshop on Nuclear Dynamics Ocho Rios, Jamaica, January 2010

Phase Structure of QCD:Chiral Symmetry and Deconfinement

• If the up and down quark masses are zero and the strange quark mass is not, the transition may be first or second order at zero baryon chemical potential.

• If the up and down quark masses are small enough there may exist a phase transition for large enough chemical potential. This chiral phase transition would be in the same universality class as liquid-gas phase transitions and the 3D Ising model.

Phase Structure of QCD: Diverse Studies Suggest a Critical Point

• Nambu Jona-Lasinio model

• composite operator model

• random matrix model

• linear sigma model

• effective potential model

• hadronic bootstrap model

• lattice QCD

Goal: To understand the equation of state of QCD near the chiral critical point and its implications for high energy heavy ion collisions.

Requirements: Incorporate critical exponents and amplitudes and to match on to lattice QCD at µ = 0 and to nuclear matter at T = 0.

Model: Parameterize the Helmholtz free energy density as a function of temperature and baryon density to incorporate the above requirements.

1// 20

20 TT

latticeQCD

nuclear matter

400

2202

40

222

44 TBTTBATATAP

Coefficients are adjusted to:(i) free gas of 2.5 flavors of massless quarks(ii) lattice results near the crossover when µ=0

(iii) pressure = constant along critical curve.

Cold Dense Nuclear Matter

1219

2)()(

19

2)()( :II Model

13118

)()(

118

)()( :I Model

0000

2

0000

0000

2

0000

n

n

n

nKnEmn

n

nKnEnE

n

n

n

nKnEmn

n

nKnEnE

N

N

MeV 30250 MeV 3.16)( fm 153.0 003

0 KnEn

Stiff

Soft

MeV 1501230)4( 0 n

Parameterize the Helmhotz free energy density to incorporatecritical exponents and amplitudes and to match on to latticeQCD at µ = 0 and to nuclear matter at T = 0.

)()()()(),( 2

210 tftftftftf

cccc TTTtnnn /)(and/)(

0tif)(

0tif)()()(

20

20

0

tatf

tatftf

22

0

2

01 )1(1)( tT

Tntf cc

Parameterize the Helmhotz free energy density to incorporatecritical exponents and amplitudes and to match on to latticeQCD at µ = 0 and to nuclear matter at T = 0.

)()()()(),( 2

210 tftftftftf

cccc TTTtnnn /)(and/)(

0tif)(

0tif)()()(

2

22

tbtf

tbtftf

curve critical along MeV/fm 5125 3 cPf

isotherm critical along )(sign|~|

curve ecoexistenc along )(~

0n t whe

0 when )(),(

0n t whe

0 when )(),(

1

2

c

gl

TB

V

PP

t

t

tt

n

TnPn

n

tc

ttc

T

TnsTc

Critical exponents and amplitudes

815.424.1325.011.0

1 hasenergy freein || term

universal are 5.0/ and 5/

)1( and 22 :related are exponents

cc

phasecoexistence

spinodal

24.1

11.0

VffnfTT c ][ ||)(),();,( 2210

Expansion away from equilibrium states using Landau theory

Vnf c0 0 along coexistence curve

The relative probability to be at a density other than the equilibrium one is

Vff

TP

P

ll

l

||||

/exp)(

)(

222

Volume = 400 fm3

Volume = 400 fm3

Future Work• A more accurate parameterization of the equation of

state for a wider range of T and µ. • Incorporate these results into a dynamical simulation

of high energy heavy ion collisions.• What is the appropriate way to describe the

transition in a small dynamically evolving system? Spinodal decomposition? Nucleation?

• What are the best experimental observables and can they be measured at RHIC, FAIR or somewhere else?

Supported by the U.S. Department of Energy under Grant No. DE-FG02-87ER40328.