Transcript Kapusta

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
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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.
T / T0    / 0 
2
2
1
lattice
QCD
nuclear matter
P  A4T 4  A2T 2  2  A0  4  B2T02T 2  B0T04
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
2

Kn
Model I :
E0 (n)  E0 (n0 )    1
18  n0 
 n

Kn



 (n)  mN  E0 (n0 )    1 3  1
18  n0  n0 
Stiff
2

2 K  n

Model II : E0 (n)  E0 (n0 ) 

1

9  n0



2 K  n
n
 (n)  mN  E0 (n0 ) 
 1 2
 1

9  n0
 n0

Soft
n0  0.153 fm 3 E0 (n0 )  16.3 MeV K  250  30 MeV
 (4n0 )  1230  150 MeV
Parameterize the Helmhotz free energy density to incorporate
critical exponents and amplitudes and to match on to lattice
QCD at µ = 0 and to nuclear matter at T = 0.
f ( , t )  f 0 (t )  f1 (t )  f 2 (t )  f (t )
2
  (n  nc ) / nc and t  (T  Tc ) / Tc
 f 0 (t )  a (t ) 2 if t  0
f 0 (t )  
 f 0 (t )  a t 2
if t  0
2
c
2
0
T
f1 (t )  0 nc 1 
(1  t ) 2
T

Parameterize the Helmhotz free energy density to incorporate
critical exponents and amplitudes and to match on to lattice
QCD at µ = 0 and to nuclear matter at T = 0.
f ( , t )  f 0 (t )  f1 (t )  f 2 (t )  f (t )
2
  (n  nc ) / nc and t  (T  Tc ) / Tc
 f 2 (t )  b (t ) if t  0
f 2 (t )  
 f 2 (t )  b t 
if t  0
f  5Pc  512 MeV/fm 3 along critical curve

Critical exponents and amplitudes



c
(

t
)
when
t

0
s (n, T )  
cV  T
  
T
c t
whe n t  0 
1




(

t
)
when
t

0
B

P
(
n
,
T
)
 





n

 
T
2


n
n 
  t
whe n t  0 

l   g ~ (t )  along coexistenc e curve
P  Pc ~|  | sign ( ) along critical isotherm
exponents are related :   2    2 and    (  1)
  /    5 and c / c  0.5 are universal
term |  | in free energy has     1
  0.11   0.325   1.24   4.815
phase
coexistence
spinodal
  1.24
  0.11
Expansion away from equilibrium states using Landau theory
(  , T ; )  0 (  , T )  [( f1  nc  )  f 2 2  f |  | ]V
 f 0  nc   V
0 along coexistence curve
The relative probability to be at a density
other than the equilibrium one is
P( )
 exp   / T 
P(l )
 



  f 2  2  l2  f |  |  | l | V
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.