Isospin Matter in the NJL Model
Pengfei Zhuang (zhuangpf@mail.tsinghua.edu.cn)
Tsinghua University, Beijing
● Phase Diagram at finite μI
● BCS-BEC Crossover in pion superfluid
Heavy Ion Collisions at the LHC Era,
July 2012，Quy Nhon, Vietnam
1
QCD Phase Diagram
Questions:
1) What is the phase of quark matter at finite μI ?
2) Is μI effect similar to μB effect?
D. T. Son and M. A. Stephanov, Phys. Rev. Lett. 86, 592(2001)
J. B. Kogut, D. K. Sinclair, Phys. Rev. D 66, 034505(2002)
K. Splittorff, D. T. Son, and M. A. Stephanov, Phys. Rev. D64, 016003 (2001)
M. Loewe and C. Villavicencio, Phys. Rev. D 67, 074034(2003)
Michael C. Birse, Thomas D. Cohen, and Judith A.McGovern, Phys. Lett. B 516, 27 (2001)
D. Toublan and J. B. Kogut, Phys. Lett. B 564, 212 (2003)
A. Barducci, R. Casalbuoni, G. Pettini, and L. Ravagli, Phys. Rev. D 69, 096004 (2004)
M. Frank, M. Buballa and M. Oertel, Phys. Lett. B 562, 221 (2003)
L. He and P. Zhuang, Phys. Lett. B 615, 93 (2005)
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BCS and BEC Pairing
in BCS, Tc is determined by thermal excitation of fermions,
in BEC, Tc is controlled by thermal excitation of collective modes.
Is there a similar BCS-BEC structure for QCD condensed matter ?
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NJL Model
●there is reliable lattice QCD result at finite T, but not yet precise
lattice simulation at finite μ, we have to consider effective models.
● the physics at finite T is vacuum excitation, but the physics at
at finite μ is vacuum condensation.
●the BCS inspired Nambu-Jona-Lasinio (NJL) model successfully
describes the chiral condensation and color condensation.
chiral thermodynamics
chiral and color condensate
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NJL at Finite μI
He, Jin and PZ, PRD71, (2005)116001
NJL with isospin symmetry breaking

LNJL    i     m0   0   G     i i 5 
2
2

quark chemical potentials

  u
 0
0   B / 3  I / 2
0


d  
0
 B / 3   I / 2 
chiral and pion condensates with finite pair momentum
     u   d ,  u  uu ,  d  dd
   2 ui 5 d 

2
e2iqx (for  I  0),
   2 di 5u 

2
e 2iqx ( for  I  0)
quark propagator in MF




p    q   u  0  m
2iG 5
1
S ( p, q )  



2
iG


p



q




m
5

d 0


T
gap equations:
  G( 2   2 )  Tr Ln S 1
V

 2

 2

 2
 0,
 0,
 0,
 0,
 0,
 0,
 u
 u2
 d
 d2

 2
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m  m0  2G

 2
 0,
0
q
q 2
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Phase Structure of Pion Superfluid
He and PZ, PRD74, (2006)036005
Jiang and PZ, PRCC83 (2011) 038201
μB
μI
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Quantum Fluctuations (Mesons) in RPA
He, Jin, and PZ, PRD71, (2005)116001
meson propagator D at RPA
considering all possible channels in the bubble summation
meson polarization functions
d4 p
 mn (k )  i 
Tr  *m S ( p  k )n S ( p) 
4
(2 )
mixing among normal
 ,  , 
 1, m  
i  , m  


m    5
i  5 , m   
 i 3 5 , m   0
in pion superfluid phase
pole of the propagator determines meson masses M m
2G  (k )
2G 0 (k ) 
1  2G (k ) 2G  (k )



2
G

(
k
)
1

2
G

(
k
)

2
G

(
k
)

2
G

(
k
)
 
  
  
  0


det 
0
2G  (k ) 2G   (k ) 1  2G   (k ) 2G  0 (k ) 


 2G  (k ) 2G  (k )
2G 0  (k ) 1  2G 0 0 (k ) 
0
0 

k0  M m , k  0
the new eigen modes  ,   ,   are linear combinations of
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 ,  , 
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Meson Mixing and Goldstone Mode
Hao and PZ, PLB652, (2007)275
mixing angles α between σ and π+ ,
β between σ and π_ ,
and γ between π+ and π_
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Meson Spectral Functions
Sun, He and PZ, PRD75, (2007)096004
meson spectra function between the temperatures Tc and T*
 (, k )  2Im DR (, k )
BEC
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BCS
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π – π scattering and BCS-BEC Crossover
BEC
BCS
Normal Phase
BEC
BCS
Pion Superfluid
Mao and Zhuang: arXiv:1202.3490
BCS: overlapped molecules, large π – π cross section
BEC: identified molecules, ideal Boson gas limit
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Yukawa Potential in Quark Matter
Mu and PZ, Eur. Phys. J C58, (2008)271
meson exchange in nucleon scattering (1934-1935)
p
p
V (r )
n
U
n
e Mr
r
meson exchange in NJL at quark level
stable quark potential
d 3k ik r
2
V (r )  
e
TrD
(0,
k
)
3
(2 )
screen mass
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Meson Screening Mass
Jiang and PZ, Eur.Phys.J. C71 (2011) 1822
corresponding to the global isospin symmetry breaking, there is a
Goldstone mode in the pion superfluid, which leads to a long
range force between two quarks !
Heavy Ion Collisions at the LHC Era,
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Quark Potential in Pion Superfluid
Jiang and PZ, Eur.Phys.J. C71 (2011) 1822
1) the maximum potential is located at the phase boundary .
2) the potential in pion superfluid is non-zero at extremely high isospin
density, totally different from the temperature and baryon density effects !
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Conclusion and Outlook
1)There exists a pion superfluid at high isospin density, and the
Goldstone mode controls the thermodynamics of the system.
2) There exists a BCS-BEC crossover in the pion superfluid.
3) The maximum coupling is located at the phase transition
boundary, similar to the temperature and baryon density effects.
4) The coupling is non-zero even at extremely high isospin density,
totally different from the temperature and baryon density effects.
Heavy Ion Collisions at the LHC Era,
July 2012，Quy Nhon, Vietnam
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July, 2008
Summer School on Dense Matter and HI
Dubna
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