1) Introduction

1
Relativistic BCS-BEC Crossover at High Density
Pengfei Zhuang

Physics Department, Tsinghua University,
Beijing 100084

• 1) Introduction
• 2) Mean Field Theory
• 3) Fluctuations
• 4) Applications to Color Superconductivity and
  Pion Superfluidity
• 5) Conclusions

based on the works with Lianyi He, Xuguang Huang,
Meng Jin, Shijun Mao, Chengfu Mu and Gaofeng Sun
2
introduction pairing
Tc in the BEC region is independent of the
coupling between fermions, since the coupling
only affects the internal structure of the bosons.
in BCS, Tc is determined by thermal excitation of
fermions, in BEC, Tc is controlled by
thermal excitation of collective modes
3
introduction BCS-BEC in QCD
QCD phase diagram
strongly coupled quark matter with both quarks
and bosons
rich QCD phase structure at high density, natural
attractive interaction in QCD, possible BCS-BEC
crossover ?
new phenomena in BCS-BEC crossover of QCD
relativistic
systems, anti-fermion contribution, rich inner
structure (color, flavor), medium dependent mass,

4
introduction theory of BCS-BEC
) Leggett mean field theory (Leggett, 1980)
)NSR
scheme (Nozieres and Schmitt-Rink, 1985)
extension of of
BCS-BEC crossover theory at T0 to T?0 (above Tc
) Nishida and Abuki (2006,2007)

extension of non-relativistic NSR theory to
relativistic systems, BCS-NBEC-RBEC crossover
) G0G scheme (Chen, Levin et al., 1998, 2000,
2005)
asymmetric pair susceptibility

extension of non-relativistic G0G scheme to
relativistic systems (He, Jin, PZ, 2006,
2007)
) Bose-fermion model (Friedderg, Lee, 1989,
1990)
extension to relativistic systems (Deng, Wang,
2007) Kitazawa,
Rischke, Shovkovy, 2007, NJLphase diagram
Brauner, 2008, collective excitations

5
mean field non-relativistic BCS-BEC at T0
A.J.Leggett, in Modern trends in the theory
of condensed matter, Springer-Verlag (1980)
BCS limit
universality behavior
BEC limit
BCS-BEC crossover
6
mean field broken universality in relativistic
systems
Lianyi He, PZ, PRD75, 096003(2007)
NJL-type model at moderate density
order parameter
mean field thermodynamic potential
fermion and anti-fermion contributions
gap equation and number equation
broken universality
extra density dependence
7
mean field relativistic BCS-BEC
plays the role of non-relativistic chemical
potential
?
BCS-NBEC crossover
fermion and anti-fermion degenerate,
NBEC-RBEC crossover
?
?
in non-relativistic case, only one dimensionless
variable , changing the
density can not induce a BCS-BEC crossover.


however, in relativistic case, the extra density
dependence may induce a BCS-BEC.
?
QCD
atom gas
8
fluctuations scheme
Lianyi He, PZ, 2007
bare fermion propagator
mean field fermion propagator
pair propagator
pair feedback to the fermion self-energy
fermions and pairs are coupled to each other
approximation
the pseudogap is related to the uncondensed
pairs, in
G0G scheme the pseudogap does not change the
symmetry structure
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fluctuations BCS-NBEC-RBEC
?
?
BCS no pairs
NBEC
heavy pairs, no anti-pairs
RBEC light
pairs, almost the same number of pairs and
anti-pairs
10
applications BCS-BEC in asymmetric nuclear
matter
Shijun Mao, Xuguang Huang, PZ, PRC79, 034304(2009)
asymmetric nuclear matter with both np and nn
and pp pairings
density-dependent contact interaction (Garrido et
al, 1999)
and density-dependent nucleon mass (Berger,
Girod, Gogny, 1991)
by calculating the three coupled gap equations,
there exists only np pairing BEC state at low
density and no nn and pp pairing BEC states.
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applications color superconductivity in NJL
Lianyi He, PZ, 2007
order parameters of spontaneous chiral and color
symmetry breaking
color breaking from SU(3) to SU(2)
quarks at mean field and mesons and diquarks at
RPA
quark propagator in 12D Nambu-Gorkov space
diquark meson polarizations
diquark meson propagators at RPA
12
applications BCS-BEC and color neutrality
Lianyi He, PZ, PRD76, 056003(2007)
gap equations for chiral and diquark condensates
at T0
to guarantee color neutrality, we introduce color
chemical potential
?
?
?
?
color neutrality speeds up the chiral restoration
and reduces the BEC region
there exists a BCS-BEC crossover
going beyond MF, see the talk by He, May 24

13
applications BCS-BEC in pion superfluid
Gaofeng Sun, Lianyi He, PZ, PRD75, 096004(2007)
meson mass, Goldstone mode
meson spectra function
BCS
BEC
going beyond MF, see the talk by Mu, May 24

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conclusions

• BCS-BEC crossover is a general phenomena
  from cold atom gas to quark matter.
• BCS-BEC crossover is closely related to QCD
  key problems vacuum, color symmetry, chiral
  symmetry, isospin symmetry
• BCS-BEC crossover in color
  superconductivity and pion superfluid is not
  induced by simply increasing the coupling
  constant of the attractive interaction, but by
  changing the corresponding charge number.
• there are potential applications in heavy
  ion collisions (at CSR/Lanzhou, FAIR/GSI and
  RHIC/BNL) and compact stars.

thanks for your patience
15
backups
16
vector meson coupling and magnetic instability
vector-meson coupling
vector condensate
gap equation
vector meson coupling slows down the chiral
symmetry restoration and enlarges the BEC region.
Meissner masses of some gluons are negative for
the BCS Gapless CSC, but the magnetic instability
is cured in BEC region.
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beyond mean field
is determined by the coupling and chemical
potential
? going beyond mean field reduces the
critical temperature of color superconductivity

?
pairing effect is important around the critical
temperature and dominates the symmetry restored
phase
18
pion superfluid

• NJL with isospin symmetry breaking

quark chemical potentials
chiral and pion condensates with finite pair
momentum
quark propagator in MF
thermodynamic potential and gap equations
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mesons in RPA
meson propagator at RPA
considering all possible channels in the bubble
summation
meson polarization functions
pole of the propagator determines meson masses
mixing among normal in
pion superfluid phase, the new eigen modes
are linear combinations of
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phase diagram of pion superfluid
chiral and pion condensates at
in NJL, Linear Sigma Model and
Chiral Perturbation Theory, there is no
remarkable difference around the critical point.
analytic result
critical isospin
chemical potential for pion superfluidity is
exactly the pion mass in the vacuum
pion superfluidity phase diagram in
plane at T0 average Fermi
surface Fermi surface
mismatch homogeneous (Sarma, )
and inhomogeneous pion superfluid (LOFF,
) magnetic instability of Sarma state at
high average Fermi surface leads to the LOFF state