Transcript Diapositiva 1

Phase diagram of neutron star quark matter
in nonlocal chiral models
A. Gabriela Grunfeld
Tandar Lab. – Buenos Aires
ARGENTINA
In collaboration with
PLAN OF THE TALK
• Motivation
• Non-local chiral quark models
• Numerical results
• Conclusions
D. Gomez Dumm
N. N. Scoccola
D. Blaschke
Motivation
The understanding of the behavior of strongly interacting matter at finite
temperature and/or density is a subject of fundamental interest
Applications in:
 Cosmology (early Universe)
 Physics of RHIC
 Physics of neutron stars
Problem: the study of strong interactions at finite temperature and/or density is
a nontrivial task; rigorous theoretical approaches are not available yet
• 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 : the most simple and widely used model of this type
The extension to NJL-like theories including nonlocal quark interactions
represents a step towards a more realistic modeling of QCD.
Several advantages over the NJL model: consistent description of loops and
anomalies, some description of confinement, etc. Successful description of
meson properties at T =  = 0
Bowler, Birse, NPA(95); Plant, Birse, NPA(98); Scarpettini, DGD, Scoccola, PRD(04)
Formalism
Euclidean action at T,  = 0 – Case of two active flavors

SE   d 4 x  (i   mc ) 

G
H
js ( x) js ( x) 
2
2

A2,5,7

jdA †( x) jdA ( x) 

mc (current quark mass), G and H  parameters of the model
We consider two alternative ways of introducing nonlocality :
Model I
(Instanton Liquid
Model inspired)
Model II
(One Gluon Exchange
interaction inspired)
js ( x)   d 4 y d 4 z r ( y  x) r ( x  z )  ( y ) ( z )
jdA ( x)   d 4 y d 4 z r ( y  x) r ( x  z )  C ( y ) i  5  2 A  ( z )
js ( x)   d 4 z g ( z )  ( x  z / 2)  ( x  z / 2)
jdA ( x)   d 4 z g ( z )  C ( x  z / 2) i  5  2 A  ( x  z / 2)
t
Here, r(x) and g(x) are nonlocal form factors, and  C     i  2  0 
1
The partition function for the model at T,  = 0 is given by
Z 0   D D e SE
We proceed by bosonizing the model, thus we introduce  and  bosonic fields
and integrate out the quark fields
Mean field approximation (MFA) : the bosonic fields are written as
 ( x)     ( x)
( x)    ( x)
and fluctuations are dropped :
 ( x)  ( x)  0
The corresponding potential WMFA(T,fc) for finite temperature T and chemical
potential  is obtained by replacing
d4 p
 (2 )4 f ( p)  T

d3 p
f (n  iˆ , p)

3

n (2 )
Matsubara frequencies ωn=(2n+1)  T
The thermodynamical potential per unit volume reads then

2

2
T
WMFA (T ,  ) 


2G 2 H 2

d3 p
 1 1

ln
det
S
(

,

)
 3
 T

n (2 )
Here S1 (inverse of the propagator) is a 48 x 48 matrix in Dirac, flavor, color and
Nambu-Gorkov spaces
(4 x 2 x 3 x 2)
The determinant can be analytically calculated :

2

2

Ac
d3 p
MFA
W (T ,  fc ) 

T  
ln 8
3
2G 2 H
T
n (2 )
2
where
Ac   p
2
uc

p2
uc
 p
2
dc
 
p 2
dc
   (1  
*
pfc  (n
Here we have defined
bc
)
i fc , p)
p2
 p 2  2( p  . p  )  2  p   p
uc
dc
uc
dc

2
 
*
with f = u,d , c = r,g,b
and
  mc   h
p
fc
  h
p
(1)
fc
(2)
Due to the nonlocality,
h(1)   r  p  2   2
fc

 fc 

 (1)
h fc  g  p fc 2 

; h(2)  r  pur 2  r  pdr 2 
; h(2)
For Model I
  p  p 2 
dr
 g   ur
  For Model II

2
 

 pfc and p are here momentum-dependent quantities
The mean fields  and  can be obtained from the coupled gap equations :
MFA
d Wreg
(T ,  fc )
d
0
MFA
d Wreg
(T ,  fc )
;
d
0
In quark matter, the chemical potential matrix ˆ can be expressed in terms of the
quark chemical potential  ( = B/3), the quark electric charge chemical potential Q
and the color chemical potential 8 required to impose color neutrality
ˆ    Q Q  8 T8
q
Q acts on flavor space, T8 over color space. The chemical potentials fc are given by
2
3 q
2
   Qq 
3
ur  ug    Q 
ub
1
1
1
8 ;  dr   dg    Qq  8
3
3
3
2
1
2
8 ; db    Qq  8
3
3
3
NEUTRON STARS
(quark matter + electrons)
MFA
W full  Wreg
 We
1 4
7 4 4 
2 2 2
with W    e  2 T e 
T 
12 
15

e
Electric and color charge neutrality
Q  Q   e 
tot
1
8 
3
1
2



 dc   e  0

uc

3

c  r , g ,b  3

f u , d
fr
  fg  2  fb   0
+
Beta equilibrium :
dc  uc  e
(no neutrino trapping assumed)
where

W
We
 e      
e
e


MFA
W
W
reg
  

 fc


 fc
fc

Numerical results
For the nonlocal regulators we choose a smooth Gaussian form:
2
g ( p )   r ( p )   exp( p 2 /  2 )
2
2
Model parameters mc , G and  fixed so as to reproduce the empirical values of the
pion mass and the pion decay constant at T =  = 0, and leading to a condensate
  qq 1/ 3  250 MeV
In this way, one obtains:
→
G2 = 15.41
mc = 5.1 MeV
 = 971 MeV
Model II →
G2 = 18.78
mc = 5.1 MeV
 = 827 MeV
Model I
There is no strong phenomenological constraint on the parameter H . Hence we
consider here values of H/G in the range from 0.5 to 1.0
Typical behavior of MF quantities as functions of  for some
representative values of T ( case H/G = 0.75 )
Model I
Model II
Phase diagrams for different values of the ratio H/G
Model I
Model II
Summary
We have studied some chiral quark models with effective nonlocal covariant separable
quark-quark and quark-antiquark interactions at finite T and . We find that:
• Different quark matter phases can occur at low T and intermediate  : normal quark matter
(NQM), superconducting quark matter (2SC) and mixed phase (NQM-2SC) states.
• In the region of interest for applications to compact stars (medium  and low T), for the
standard value H/G = 0.75 models I and II predict the presence of a mixed 2SC-NQM
phase. The latter turns into a pure 2SC phase for larger H/G ratios.
• Model I (ILM inspired) shows stronger SC effects. However, it leads to a relatively low
value of TC ( = 0) in comparison with lattice expectations. Model II leads to a larger quark
mass gap, and a larger value of TC (about 140 MeV), though SC region is smaller.
• The critical T for the second order 2SC-NQM phase transitions rises with  for Model I
while it is approximately -independent for Model II. This can be understood from the
different  dependences associated with the diquark gaps in both models.
To be done: –
–
–
–
Extension to larger  (strangeness, color-flavor locked phase)
Inclusion of vector-vector channels, hadronic matter effects, …
Implications on compact star sizes & radii
…