Transcript Diapositiva 1 - University of Wrocław

Color neutrality effects in the phase
diagram of the PNJL model
A. Gabriela Grunfeld
Tandar Lab. – Buenos Aires - Argentina
In collaboration with
D. Blaschke
D. Gomez Dumm
N. N. Scoccola
Motivation
Understanding of the behavior of strongly interacting matter at finite T and/or
density is of fundamental interest and has important applications in
cosmology, in the astrophysics of neutron stars and in the physics of URHIC.
From RHIC
CBM@FAIR
(from Jürgen Schaffner-Bielich)
For a long time,
QCD phase diagram
restricted to 2 phases
HADRONIC PHASE: “our world”
color neutral hadrons, SB
QGP: S is restored
In recent years phase diagram
richer and more complex
structure
T
E
QG Plasma
Hadrons
2SC
CFL

Rajagopal
The treatment of QCD at finite densities and temperatures is a problem of
very high complexity for which rigorous approaches are not yet available
• Development of effective models for interacting quark matter
that obey the symmetry requirements of the QCD Lagrangian
• Inclusion of simplified quark interactions in a systematic way
NJL model is the most simple and widely used model of this type.
local interactions
Lattice results at μ -> 0
Effective theories
Reproduce ?
Chiral symmetry
breaking
extrapolate at high μ
U (, )
Lattice simulations of
P in a pure gauge
theory
confinement
Nambu Jona-Lasinio model + Polyakov loop dynamiccs
Higher Tc than NJL
It reasonable to ask what happens with color neutrality in presence of PL
• important in URHIC
• could be extended to compact stars imposing electric charge neutrality + β decay
The model
In our case: SU(2) flavor + diquarks + color neutrality
NJL SU(2) flavor + quarks with a background color field related to the Polyakov loop Φ:
*S. Rößner, C. Ratti and W. Weise, PRD75, 034007 (2007)

SE  d x  (i   mc )


4
mc (current q mass), G and
H parameters of the model
H/G = ¾ from Fierz tr. OGE
G
H
 js ( x) js ( x) 
2
2

j

A2,5,7
A†
d
A
d
( x) j

( x)   SE ( )

diquarks
 C i  5  2 2 
 C   t i  2  0 
1
quarks with a background color field
Polyakov loop:
then
1

TrC L
NC

1
TrC e i / T
3
  A4  i A0 i   0 g Αa t a
 


L  exp i  d A4 
 0

order parameter for confinement
  0  deconf
  0  conf
Polyakov gauge => diag representation
 3 3  88
gluon dynamics,
δSE (Φ,T) -> (V/T) U(Φ,T)
effective potential, confinement-deconf. transition
We considered the polynomial form for the effective potential *:

1
*
*
*3
U (, T )  a(T )    b(t ) ln 1  6    4 (    3 )  3 (*) 2
2
with

 T0
a(T )  a0  a1 

T

3
 T0 

b(T )  b3  T 
 


 T0 
  a2  T 

 

2
T0 = 270 MeV from lattice
crit temp for deconf.
Then, we obtain the Euclidean effective action
2
SEMFA
2 
T
MFA



U
(

,
T
)



VE(4)
2G 2H 2

n
d3 p
1

Tr
ln

S
(in ; p) 

 (2 )3 j

over Dirac, flavor and color indices
where
• MFA -> drop the meson fluctuations
Matsubara
frequencies
ωn=(2n+1) π T
• (+ Usual 2SC ansatz Δ5 = Δ7= 0 and Δ2 = Δ)
 i 0n   . p     0 (   i )
S (in , p)  
*





5 2 2

1
Matsubara frequencies ωn=(2n+1) π T

 5 22 

i 0n   . p     0 (   i ) 
 n2  E 2
T  ln 
2
n 
 T




  E  2T ln 1  exp( E / T ) 

Then, the thermodynamic potential per volume reads:
2

S

MFA
   U ( , T ) 

2Nf
VE(4)
2G 2 H
2
MFA
E
d3 p
 
1 
 (2 )3 j  ln  1 exp ( E j / T )  2 E j  
E1,2   b
E3,4  (  ) 2   2
i 3
E5,6  (  ) 2   2
i 3
where
  p 
2
     r
2
r   
8
3
b    2
8
3
Thermodynamic equilibrium -> minimum of thermodynamic potential. The
mean fields  and  are obtained from the coupled gap equations
 MFA (T ,  fc )

together with
0
MFA (T ,  fc )
3
8  0
To Ω be real

0
0
We impose color charge neutrality
We consider *
;
 MFA (T ,  fc )
 MFA (T ,  fc )
8
0
1
  
  1  2 cos  3  
3
 T 
=> μ3 = 0
NUMERICAL RESULTS
we use the set of parameters from PRD75, 034007 (2007)
• G = 10.1 GeV-2
• Λ = 0.65 GeV
effective theory, fluctuations, at T = μ = 0
• H = ¾ G, 0.8G
• mc = 5.5
• a0 = 3.51
• a1 = -2.47
• a2 = 15.2
from lattice
• b3 = -1.75
• T0 = 270.
Phase diagram:
Low μ -> XSB and XSB + 2SC
High μ -> 2SC
Phase diagrams
Low temperature expansion

T=0
 Sg(  b ) Sg(  b )    r    r 




2
2

  



2
  r   r 
2 
0 2
dp
p
Sg
(



)

Sg
(



)


b
b

8

  
 3 0


1
 1 1 

2
 0     2  dp p 2   

     
H  0
M mc 2M

p2
0
 2  dp

G
 0
E
• for μ = 0, Δ = μ8= μr = μb = 0, Mo = 324.11 Mev
• for μ ≠ 0 (Δ still 0)


2
0 2
dp p 2 Sg(  b )  Sg(  b )  Sg(  r )  Sg(  r ) 

8
 30
Trivially satisfied for a wide range of μ8
Step beyond: μ8 from fin T and then T -> 0
For μ < M0/3
M  
if   M 0 / 3
M 0   8  0


 2
2
3 
M 0   if   M 0 / 3
8
3
 2   T ln 2
For μ > M0 (before 1st order ph.tr)
G 
f () 
 
2SC ->
•T=0

1 G
2 1

 
dp
p

2 

 0
    H
2G
in region μr = cte f(Δ) ≠ 0
•T≠0
in region μr = cte f(Δ) ≠ 0 until
T = 20 MeV, 2nd order
If H/G > 0.783 f(Δ) ≠ 0
Summary and outlook
• we have studied a chiral quark model at finite T and µ
NJL + diquarks + Polyakov loop + color neutrality
• ansatz PRD75, 034007 (2007) ϕ8 = 0 => μ8 ≠ 0, then μ3 = 0
• to enforce color neutr color neutrality => μ8 ≠ 0
• without PL, symmetric case, with PL non symmetric densities in color space
• different quark matter phases can occur at low T and intermediate µ
• coexisting phase XSB + 2SC region
• Next step: starting with ϕ3 ϕ8 ≠ 0, => μ3 μ8 ≠ 0 more general…
Some References
S. Rößner, C. Ratti and W. Weise, PRD75, 034007 (2007)
F. Karsch and E. Laermann, Phys. Rev. D 50, 6954 (1994) [arXiv:hep-lat/9406008].
C. Ratti, M. A. Thaler and W. Weise, Phys. Rev. D 73, 014019 (2006) [arXiv:hep-ph/0506234].
M. Buballa, Phys. Rept. 407, 205 (2005) [arXiv:hep-ph/0402234].
K. Fukushima Physics Letters B 591 (2004) 277–284
S. Rößner, T. Hell, C. Ratti and W. Weise, arXiv:0712.3152v1 hep-ph
THANKS!
‫فرامرز‬