Relativistic BCS-BEC Crossover at Quark Level

July, 2008 Summer School on Dense Matter and HI DubnaJuly, 2008 Summer School on Dense Matter and HI Dubna 11

Relativistic BCS-BEC Crossover at Quark LevelRelativistic BCS-BEC Crossover at Quark Level Pengfei Zhuang Physics Pengfei Zhuang Physics

Department, Tsinghua University, Beijing 100084Department, Tsinghua University, Beijing 100084

1) Motivation1) Motivation

2) Mean Field Theory at T = 0 2) Mean Field Theory at T = 0

3) Fluctuations at T≠ 03) Fluctuations at T≠ 0

4) Application to QCD: 4) Application to QCD: Color Superconductivity and Pion Superfluid Color Superconductivity and Pion Superfluid

5) Conclusions5) Conclusions


1) Motivation1) Motivation


BCS-BECBCS-BEC

BCS BCS (Barden, Cooper and Schrieffer, 1957):(Barden, Cooper and Schrieffer, 1957): normal superconductivitynormal superconductivity weak coupling, large pair size, k-space pairing, overlaping Cooper weak coupling, large pair size, k-space pairing, overlaping Cooper

pairspairs BEC BEC (Bose-Einstein-Condensation, 1924/1925):(Bose-Einstein-Condensation, 1924/1925): strong coupling, small pair size, r-space pairing, ideal strong coupling, small pair size, r-space pairing, ideal gas of bosons, first realization in dilute atomic gas with bosons in gas of bosons, first realization in dilute atomic gas with bosons in

1995.1995. BCS-BEC crossoverBCS-BEC crossover (Eagles, Leggett, (Eagles, Leggett,

1969, 1980):1969, 1980): BCS wave function at T=0 can be generalized to arbitrary BCS wave function at T=0 can be generalized to arbitrary

attraction: aattraction: a smoothsmooth crossover from BCS to BEC!crossover from BCS to BEC!

BEC of moleculesBEC of molecules BCS fermionic superfluidBCS fermionic superfluid

2/12

2/1

1 kB

kBk

N

Nv

BCSBCS

0exp 2/10

kkk k ccNB

BECBEC


pairingspairings

in BCS, Tin BCS, Tcc is determined by thermal excitation of fermions, is determined by thermal excitation of fermions,

in BEC, Tin BEC, Tcc is controlled by thermal excitation of collective modes is controlled by thermal excitation of collective modes


in order to study the question of ‘vacuum’, we must turn to a different in order to study the question of ‘vacuum’, we must turn to a different direction: we should investigate some ‘bulk’ phenomena by distributing direction: we should investigate some ‘bulk’ phenomena by distributing

high energy over a relatively large volume.high energy over a relatively large volume.

BCS-BEC in QCDBCS-BEC in QCD

T. D. Lee, Rev. Mod. Phys. 47, 267(1975)T. D. Lee, Rev. Mod. Phys. 47, 267(1975)

pair dissociation line

BCSBEC

sQGP

QCD phase diagramQCD phase diagram

rich QCD phase structure at high density, natural attractive interaction rich QCD phase structure at high density, natural attractive interaction in QCD, possible BCS-BEC crossover ?in QCD, possible BCS-BEC crossover ?

new phenomena in BCS-BEC crossover of QCD:new phenomena in BCS-BEC crossover of QCD: relativistic systems, anti-fermion contribution, rich inner structure relativistic systems, anti-fermion contribution, rich inner structure (color, flavor), medium dependent mass, ……(color, flavor), medium dependent mass, ……


theory of BCS-BEC crossovertheory of BCS-BEC crossover

*) Leggett mean field theory (Leggett, 1980) *) Leggett mean field theory (Leggett, 1980) *)NSR scheme (Nozieres and Schmitt-Rink, 1985) *)NSR scheme (Nozieres and Schmitt-Rink, 1985) extension of of BCS-BEC crossover theory at T=0 to T extension of of BCS-BEC crossover theory at T=0 to T≠0 (above T≠0 (above Tcc

) ) Nishida and Abuki (2006,2007) Nishida and Abuki (2006,2007) extension of non-relativistic NSR theory to relativistic systems, extension of non-relativistic NSR theory to relativistic systems, BCS-NBEC-RBEC crossover BCS-NBEC-RBEC crossover

4

0 04

1ln ( ) ,

(2 )fl

d qq G G

G

*) *) GG00G scheme (Chen, Levin et al., 1998, 2000, 2005)G scheme (Chen, Levin et al., 1998, 2000, 2005)

asymmetric pair susceptibility asymmetric pair susceptibility 0 G G extension of non-relativistic extension of non-relativistic GG00G scheme to relativistic systems (He, G scheme to relativistic systems (He,

Zhuang, 2006, 2007)Zhuang, 2006, 2007)

*) Bose-fermion model (Friedderg, Lee, 1989, 1990)*) Bose-fermion model (Friedderg, Lee, 1989, 1990) extension to relativistic systems (Deng, Wang, 2007) extension to relativistic systems (Deng, Wang, 2007) Kitazawa, Rischke, Shovkovy, 2007, NJL+phase diagram Kitazawa, Rischke, Shovkovy, 2007, NJL+phase diagram Brauner, 2008, collective excitations ……Brauner, 2008, collective excitations ……


2) 2) Leggett Leggett Mean Field Theory at T = 0Mean Field Theory at T = 0 A.J.Leggett, in Modern trends in the theory of condensed matter, A.J.Leggett, in Modern trends in the theory of condensed matter, Springer-Verlag (1980) Springer-Verlag (1980)


non-relativistic mean field theorynon-relativistic mean field theory

fermion number: fermion number: Fermi momentum: Fermi momentum:

Fermi energy:Fermi energy:

3 2, / 3F Fk n k p=

2

32tL i h gm

2 2g g

2 3 2

3(2 ) 4k kk

d kE

g

order parameterorder parameter

2 2 2, , / 2k k k k kE k mx x e m e= +D = - =

n

2 / 2F Fk me =

thermodynamics in mean field approximationthermodynamics in mean field approximation

quasi-particle energyquasi-particle energy

renormalization to avoid the integration divergence renormalization to avoid the integration divergence 3

3

1 1,

4 (2 ) 2

0 for attractive coupling, 0 for repulsive coupling

ss k

s s

m d kg a

a g

a a


universalityuniversality

0

gap equationgap equation

n

number conservationnumber conservation

( ), ( )n n

3

3

3

3

1 1

4 (2 ) 2 2

1(2 )

s k k

k

k

m d k

a E

d kn

E

2222 20

22

22 20

1ˆ / , / ,

1 1

2

ˆ21

3

F FF sk a

dxxxx

xdxx

x

universalityuniversality

ˆ ( ), ( )

effective couplingeffective coupling


BCS limit

BEC limit

non-relativistic BCS-BEC crossovernon-relativistic BCS-BEC crossover

2 /2

8ˆ, , 1e

e

2

2

( ) /

16ˆ, ,

31

, 2

1( ) 0

1

bb

s

T

ma

n pe

BCS-BEC crossover0 0,

small large ,

0 0


relativistic BCS-BEC crossoverrelativistic BCS-BEC crossover

0, m mm m- = =

( )2 2 2 2, k k kE k m

m

x x m

m

± ± ±= +D = + ±

-

2

, 02

0

b

b

m

m m

e

em m

m

<

- >- ³ - ³

®

BCS-BEC crossover around BCS-BEC crossover around

plays the role of non-relativistic chemical potentialplays the role of non-relativistic chemical potential

pair binding energypair binding energy

atat fermion and anti-fermion degenerate, fermion and anti-fermion degenerate, relativistic effectrelativistic effect

0 m m

BCS BCS limitlimit

NBEC-BCS NBEC-BCS crossovercrossover

NBECNBECRBEC-NBEC RBEC-NBEC crossovercrossover

RBEC RBEC limitlimit


relativistic mean field theoryrelativistic mean field theory

5 5

0 2

4T Tg

L i m i C iC

C i

52TgiC

2 3

3(2 ) k k k k

d kE E

g

order parameterorder parameter

antifermionantifermion

extremely high T and high density: pQCD extremely high T and high density: pQCD finite T (zero density): lattice QCD finite T (zero density): lattice QCD moderate T and density: models like 4-fermion interaction (NJL)moderate T and density: models like 4-fermion interaction (NJL)

charge conjunction matrixcharge conjunction matrix

mean field thermodynamic potentialmean field thermodynamic potential

the most important thermodynamic contribution from the uncondensed the most important thermodynamic contribution from the uncondensed pairs is from the Goldstone modes, , at T=0, the fluctuation pairs is from the Goldstone modes, , at T=0, the fluctuation contribution disappears, and MF is a good approximation at T=0.contribution disappears, and MF is a good approximation at T=0.

4T

fermionfermion

He, Zhuang, 2007He, Zhuang, 2007


broken universalitybroken universality

gap equation and number equation: gap equation and number equation: 3

3

3

3

1 1 1

(2 ) 2 2

1 1(2 )

k k

k k

k k

d k

g E E

d kn

E E

22 2

0

2

0

1 1 1 1

2 2 2

21 1

3

1,

z

x x x x

zx x

x x

F

F s

dxxE E

dxxE E

k

k a m

renormalization to avoid the integration divergence renormalization to avoid the integration divergence 3

3

41 1 1 1 1,

2 (2 )s

k k

ad kU

U g m m m

the ultraviolet divergence can not be completely removed, and a the ultraviolet divergence can not be completely removed, and a momentum cutoff still exists in the theory.momentum cutoff still exists in the theory.

broken universality:broken universality:

explicit density dependence !explicit density dependence !


m m / 1Fk m

non-relativistic limit: 1) non-relativistic kinetics 2) negligible anti-fermions ●

●

1

2

( , ) ( )

( , ) 0 ( )

c

c

mmh z h z

mh z h z

= Þ

® Þ

● NBEC

● RBEC

non-relativistic limitnon-relativistic limit


density induced crossoverdensity induced crossover

atom gas

QCD

1/ F sk a In non-relativistic case, only one dimensionless variable, , changing the density of the system can not induce a BCS-BEC crossover. However, in relativistic case, the extra density dependence may induce a BCS-BEC crossover.

/Fk mz=


3) Fluctuations at T≠ 0 3) Fluctuations at T≠ 0


● ● the the Landau mean field theory Landau mean field theory is a good approximation only at T=0 is a good approximation only at T=0 where there is no thermal excitation,where there is no thermal excitation, one has to go beyond the mean one has to go beyond the mean field at finite temperature.field at finite temperature.

pair dissociation line

BCSBEC

sQGP

● ● an urgent question in relativistic heavy ion collisionsan urgent question in relativistic heavy ion collisions

introductionintroduction

tto understand the sQGP phase o understand the sQGP phase with possible bound states of with possible bound states of quarks and gluons, one has to quarks and gluons, one has to go beyond the mean field !go beyond the mean field !

● ● going beyond mean field self-consistently is very difficultgoing beyond mean field self-consistently is very difficult

NSR Theory (GNSR Theory (G00GG00 Scheme) above T Scheme) above Tcc: Nishida and Abuki (2005),: Nishida and Abuki (2005),Bose-Fermi Model: Deng and Wang (2006),Bose-Fermi Model: Deng and Wang (2006),GG00G Scheme below TG Scheme below Tcc: He and Zhuang (2007), ……: He and Zhuang (2007), ……


10 0 0( , ) ( )G k k k m

0 0

3

03

1 ( , ) ( , ) 0, ( , ) 2

2 , , ,

(2 1) , (2 )

k k

n n

gi Tr G k G k n i Tr G k

n T bosonsd kiT k i

n T fermi

k n ons

20( ) ( , )mf k G k

5 54T Tg

L i m i C iC

fermion propagator with diagonal and off-diagonal elements in Nambu-Gorkov spacefermion propagator with diagonal and off-diagonal elements in Nambu-Gorkov space

bare propagator and bare propagator and condensate induced self-energycondensate induced self-energy

( , ) ( , )( , )

( , ) ( , )

G k F kS k

F k G k

1 10

5 0

( , ) ( , ) ( )

( , ) ( , ) ( , )

mfG k G k k

F k G k i G k

gap and number equations at finite Tgap and number equations at finite T

3

3

3

3

1 2 ( ) 1 2 ( )1

(2 ) 2 2

1 1 2 ( ) 1 1 2 ( )(2 )

k k

k k

k kk k

k k

f E f Ed k

g E E

d kn f E f E

E E

gap and number equations in terms of the fermion propagatorgap and number equations in terms of the fermion propagator

mean field fermion propagatormean field fermion propagator

Matsubara frequency summation Matsubara frequency summation


2

( ) ( )mft q i qT

0 G Gmean field theory in schememean field theory in scheme

introducing a condensed-pair propagatorintroducing a condensed-pair propagator

0( ) ( ) ( , )mf mfq

k t q G q k

0( ) ( , ) ( , )2 k

iq Tr G k G q k

1(0) 0t

gap equation in condensed phase is determined by uncondensed pairsgap equation in condensed phase is determined by uncondensed pairs

0G

mft

( )1 ( )

igt q

g q

problem: problem: there is no feedback of the uncondensed pairs on the fermion self- there is no feedback of the uncondensed pairs on the fermion self-energy energy

defining an uncondensed pair propagatordefining an uncondensed pair propagator

the name scheme, the name scheme, G is the mean field G is the mean field fermion propagatorfermion propagator

0 G G

( , )T


2

( ) ( ) ( )

( ) ( )

( )1 ( )

mf pg

mf

pg

t q t q t q

t q i qTig

t qg q

0( ) ( , ) ( , )2 k

iq Tr G k G q k

full pair propagatorfull pair propagator

0( ) ( ) ( , )

( ) ( )

q

mf pg

k t q G q k

k k

1 10( , ) ( , ) ( )G k G k k

going beyond mean field in schemegoing beyond mean field in scheme0G G

with full susceptibility and full propagatorwith full susceptibility and full propagator

full fermion self-energyfull fermion self-energy

pgt

1 (0) 0pgt

fermions and pairs are coupled to each otherfermions and pairs are coupled to each other

new gap equationnew gap equation ( , )T a new order parameter which is a new order parameter which is different from the mean field onedifferent from the mean field one

all the formulas look the same as the mean field ones, all the formulas look the same as the mean field ones, but we do not know the expression of the full fermion propagator G.but we do not know the expression of the full fermion propagator G.



approximation in condensed phaseapproximation in condensed phase

1) peaks at 1 (0) 0pgt ( )pgt q

20 0 0

0 0

2

0

( ) ( ) ( , ) ( ) ( , ) ( , )

( )

pg pg pg pgq q

pg pgq

k t q G q k t q G k G k

t q

0q

2 20( ) ( ) ( ) ( , )mf pg pgk k k G k

1 (0) 0pgt

0 0

2 2 21 0 2 0

2 2

1 2 22 20 0 00 0

( )1 ( ) ( ) (0)

1 1, ,

2 2

pg

qq q

ig i it q

g q q Z q Z q q

Z Zq q q

0q

2 2

32

32

mean field gap equation with

( ) ( )1

(2 ) 2

pg

b q b qpg

q

f v f vd q

Z

v

2 2 21 2 q 2/ 2 , /Z Z q Z v

, , ,pgT T

the pseudogap is related to the uncondensed the pseudogap is related to the uncondensed pairs and does not change the symmetry !pairs and does not change the symmetry !

full self-energy

2) expansion around

1 (0) 0pgt gap equation mean field gap equation withgap equation mean field gap equation with 2 2

pg


thermodynamicsthermodynamics

2

2 3

3

3

3

3/ 2

1 23

3

3

1ln 1 1

(2 )

1ln 1 ( ) ln 1 1

(2 )

1ln 1 Z ,

(2 )

2ln 1

(2 )

k k

q q

B

mf fl

E Emf k k k k

flq

q m

d kE E e e

g

d qg q e e

d qe Z non relativistic boson gas

d qe

2 1 Z , cq Z relativistic boson gas

3

2 2132 (2 )

mf flB k k pg B B

n d kn f f Z n n

number of bosonsnumber of bosons

21

/ 2 / 2

mfB

c

n ZP

n n

fraction of condensed pairsfraction of condensed pairs


BCS-NBEC-RBEC crossoverBCS-NBEC-RBEC crossover

*

*

: critical temperature

: 0, condensed phase

: 0, normal or pseudogap phase

: pair dissociation temperature

: 0, pseudogap phase

c

c

c

c pg

T

T T

T T

T

T T T

* : 0 normal phasepgT T

BCS: no pairs BCS: no pairs

NBEC: heavy pairs, no anti-pairsNBEC: heavy pairs, no anti-pairs

RBEC: light pairs, almost the same RBEC: light pairs, almost the same number of pairs and anti-pairsnumber of pairs and anti-pairs

0, m 0 / , 0<Fm k m

/ , 0Fm k

●●

●●

/( / 2)B Br n n

1F Br r

number fractions at Tnumber fractions at Tc c , ,

in RBEC, Tin RBEC, Tcc is large enough and is large enough and

there is a strong competition there is a strong competition between condensation and between condensation and dissociation.dissociation.


discussion on discussion on 0G G

10 51

15 0

( , ) ( , )( )

( , ) ( , )pg

pg

G k k iS k

i G k k

can the symmetry be restored in the pseudogap phase? can the symmetry be restored in the pseudogap phase?

fermion propagator including fluctuations (to the order of ) :fermion propagator including fluctuations (to the order of ) :2 2/

the pseudogap appears in the diagonal elements of the propagator the pseudogap appears in the diagonal elements of the propagator and does not break the symmetry of the system. and does not break the symmetry of the system.

Kandanoff and Martin: Kandanoff and Martin: the scheme can not give a correct symmetry restoration the scheme can not give a correct symmetry restoration picture and the specific heat is wrong. picture and the specific heat is wrong.

GG 2

VC T


BCS-BEC in Bose-fermion modelBCS-BEC in Bose-fermion modelDeng, Wang, 2007Deng, Wang, 2007

mean field thermodynamicsmean field thermodynamics

fluctuation changes the phase fluctuation changes the phase transition to be first-order.transition to be first-order.


4) Application to QCD: 4) Application to QCD: Color Superconductivity and Pion SuperfluidColor Superconductivity and Pion Superfluid


motivationmotivation

* QCD phase transitions like chiral symmetry restoration, color * QCD phase transitions like chiral symmetry restoration, color superconductivity, and pion superfluid happen in non-perturbative superconductivity, and pion superfluid happen in non-perturbative temperature and density region, the coupling is strong. temperature and density region, the coupling is strong.

* relativistic BCS-BEC crossover is controlled by , the BEC-BCS * relativistic BCS-BEC crossover is controlled by , the BEC-BCS crossover would happen when the light quark mass changes in the crossover would happen when the light quark mass changes in the QCD medium. QCD medium.

* effective models at hadron level can only describe BEC state, they * effective models at hadron level can only describe BEC state, they can not describe BEC-BCS crossover. One of the models that enables can not describe BEC-BCS crossover. One of the models that enables us to describe both quarks and mesons and diquarks is the NJL model us to describe both quarks and mesons and diquarks is the NJL model at quark level.at quark level.

disadvantage: no confinementdisadvantage: no confinement

* there is no problem to do lattice simulation for real QCD at finite * there is no problem to do lattice simulation for real QCD at finite

isospin densityisospin density

m

2 2

0 0 5 5 5C ij ij C

NJL S i D i j i jL i m G i G i i


1 2 2 1 1 2 2 1 3 3 3 3 C C C C C Cu d d u d u u d u u d d 1

2

2

1

1

2

2

1

3

3

3

3

u

Cd

d

Cu

d

Cu

u

Cd

u

Cu

d

Cd

3 3

5C iji ji

order parameters of spontaneous chiral and color symmetry breakingorder parameters of spontaneous chiral and color symmetry breaking

quark propagator in 12D Nambu-Gorkov spacequark propagator in 12D Nambu-Gorkov space

A

B

C

D

E

F

S

S

SS

S

S

S

I II

I I

GS

G

0 2q SM m G

, , , , ,I A B C D E F

2 2

2 22

k q

k D

E k M

E E G

color superconductivitycolor superconductivity

color breaking from SU(3) to SU(2)

diquark & meson polarizationsdiquark & meson polarizations

M D

diquark & meson propagators at RPAdiquark & meson propagators at RPA

leading order of 1/Nleading order of 1/Ncc for quarks, and next to leading order for mesons & diquarks for quarks, and next to leading order for mesons & diquarks


BCS-BEC and color neutralityBCS-BEC and color neutrality

gap equations for chiral and diquark condensates at T=0gap equations for chiral and diquark condensates at T=0

3

0 3

3

3

/ 3 / 318 / 3

(2 )

1 18

(2 )

k B k Bs k B

p

d

E Ed km m G m E

E E E

d kG

E E

/ 3 /B d sm m G G

●● ●●

there exists a BCS-BEC crossoverthere exists a BCS-BEC crossover

to guarantee color neutrality, we introduce to guarantee color neutrality, we introduce color chemical potential: color chemical potential:

8 8/ 3 / 3, / 3 2 / 3r g B b B

color neutrality speeds up the chiral color neutrality speeds up the chiral restoration and reduces the BEC regionrestoration and reduces the BEC region

●● ●●

r m



vector meson coupling and magnetic instabilityvector meson coupling and magnetic instability

vector-meson couplingvector-meson coupling 2 2

5V VL G

gap equationgap equation

02V VG

vector meson coupling slows down vector meson coupling slows down the chiral symmetry restoration and the chiral symmetry restoration and enlarges the BEC region. enlarges the BEC region.

vector condensatevector condensate

3

3

/ 3 / 38 / 3

(2 )k B k B

V V k B

E Ed kG E

E E

1

Meissner masses of some gluons Meissner masses of some gluons are negative for the BCS Gapless are negative for the BCS Gapless CSC, CSC, but the magnetic instability but the magnetic instability is cured in BEC region.is cured in BEC region.

r m


beyond men fieldbeyond men field

0 100 200 MeV 300 500 MeVq 0 ( 0)T is determined by the coupling and chemical potentialis determined by the coupling and chemical potential

● going beyond mean field reduces the critical temperature of color going beyond mean field reduces the critical temperature of color superconductivitysuperconductivity ● ● pairing effect is important around the critical temperature and dominates the pairing effect is important around the critical temperature and dominates the symmetry restored phasesymmetry restored phase


NJL with isospin symmetry breakingNJL with isospin symmetry breaking

2 2 1

2 2 2 2

2 2 2 2

( ) Tr Ln

0, 0, 0, 0, 0, 0, 0, 0u u d d

TG S

V

q q

2 2

0 0 5NJL iL i m G i

, , u d u duu dd chiral and pion condensates with finite pair momentumchiral and pion condensates with finite pair momentum

quark propagator in MFquark propagator in MF 0 51

5 0

2( , )

2u

d

p q m iGS p q

iG k q m

0 2m m G

2 25 52 , 2

2 2iq x iq xui d e di u e

quark chemical potentialsquark chemical potentials

0 / 3 / 2 0

0 0 / 3 / 2u B I

d B I

thermodynamic potential and gap equations:thermodynamic potential and gap equations:

pion superfluidpion superfluid


5

5

3 5 0

1,

,

,

,

m

m

i m

i m

i m

4

*4

( ) Tr ( ) ( )(2 )mn m n

d pk i S p k S p

meson polarization functionsmeson polarization functions

meson propagator at RPAmeson propagator at RPA

pole of the propagator determines meson masses pole of the propagator determines meson masses

mixing among normal in pion superfluid phase, mixing among normal in pion superfluid phase,

the new eigen modes are linear combinations of the new eigen modes are linear combinations of

considering all possible channels in the bubble summationconsidering all possible channels in the bubble summation

0

0

0

0 0 0 0 00 , 0

1 2 ( ) 2 ( ) 2 ( ) 2 ( )

2 ( ) 1 2 ( ) 2 ( ) 2 ( )det

2 ( ) 2 ( ) 1 2 ( ) 2 ( )

2 ( ) 2 ( ) 2 ( ) 1 2 ( )mk M k

G k G k G k G k

G k G k G k G k

G k G k G k G k

G k G k G k G k

0

, ,

mM

, ,

D

mesons in RPAmesons in RPA

, ,


I B

chiral and pion condensates at chiral and pion condensates at in NJL, Linear Sigma Model and Chiral in NJL, Linear Sigma Model and Chiral Perturbation Theory, there is no remarkable Perturbation Theory, there is no remarkable difference around the critical point.difference around the critical point.

analytic result: analytic result: critical isospin chemical potential for pion critical isospin chemical potential for pion superfluidity is exactly the pion mass in the superfluidity is exactly the pion mass in the vacuum: vacuum:

0BT q

cI m

( )B BnI

pion superfluidity phase diagram in pion superfluidity phase diagram in plane at T=0plane at T=0

: average Fermi surface: average Fermi surface

: Fermi surface mismatch : Fermi surface mismatch

homogeneous (Sarma, ) and homogeneous (Sarma, ) and inhomogeneous pion superfluid (LOFF, )inhomogeneous pion superfluid (LOFF, )

magnetic instability of Sarma state at high magnetic instability of Sarma state at high average Fermi surface leads to the LOFF stateaverage Fermi surface leads to the LOFF state

0q 0q

phase diagram of pion superfluidphase diagram of pion superfluid


BCS-BEC crossover of pion superfluidBCS-BEC crossover of pion superfluid

( , ) 2 Im ( , )k D k meson spectra functionmeson spectra function

meson mass, Goldstone modemeson mass, Goldstone mode

phase diagramphase diagram

BEC BCS


BCS-BEC crossover in asymmetric nuclear matterBCS-BEC crossover in asymmetric nuclear matter

Mao, Huang, Zhuang, 2008Mao, Huang, Zhuang, 2008

★ ★ transition from BCS pairing to BEC in low-density asymmetric nuclear matter, transition from BCS pairing to BEC in low-density asymmetric nuclear matter, U. Lombardo, P. Nozières:PRC64, 064314 (2001)U. Lombardo, P. Nozières:PRC64, 064314 (2001)

★ ★ spatial structure of neutron Cooper pair in low density uniform matter, spatial structure of neutron Cooper pair in low density uniform matter, Masayuki Matsuo:PRC73, 044309 (2006)Masayuki Matsuo:PRC73, 044309 (2006)

★ ★ BCS-BEC crossover of neutron pairs in symmetric and asymmetric nuclear BCS-BEC crossover of neutron pairs in symmetric and asymmetric nuclear matters, matters, J. Margueron: arXiv:0710.4241J. Margueron: arXiv:0710.4241

asymmetric nuclear matter with both asymmetric nuclear matter with both npnp and and nnnn and and pp pp pairingspairings

considering density dependent Paris potential and considering density dependent Paris potential and nucleon massnucleon mass

there exists a strong there exists a strong Friedel oscillation in BCS Friedel oscillation in BCS region, and it is washed region, and it is washed away in BEC region. away in BEC region.

3

32

ip rd pr e p


near side correlation as a signature of sQGPnear side correlation as a signature of sQGPDD

we take drag coefficient to we take drag coefficient to be a parameter charactering be a parameter charactering the coupling strength the coupling strength

* * c quark motion in QGP:c quark motion in QGP:

* * QGP evolution: QGP evolution: ideal hydrodynamicsideal hydrodynamics

for strongly interacting quark-gluon plasma: for strongly interacting quark-gluon plasma: ● ● at RHIC, the back-to-back correlation is washed out.at RHIC, the back-to-back correlation is washed out.● ● at LHC, c quarks are fast thermalized, the strong at LHC, c quarks are fast thermalized, the strong flow push the D and Dbar to the near side!flow push the D and Dbar to the near side!

Zhu, Xu, Zhuang, PRL100, 152301(2008)Zhu, Xu, Zhuang, PRL100, 152301(2008)

large drag parameter is confirmed by R_AA and v_2 of large drag parameter is confirmed by R_AA and v_2 of non-photonic electrons non-photonic electrons (PHENEX, (PHENEX, 2007; Moore and Teaney, 2005; Horowitz, Gyulassy, 2007).2007; Moore and Teaney, 2005; Horowitz, Gyulassy, 2007).


conclusionsconclusions

* * BCS-BEC crossover is a general phenomena from cold atom gas BCS-BEC crossover is a general phenomena from cold atom gas to quark matter.to quark matter.

* * BCS-BEC crossover is closely related to QCD key problems: BCS-BEC crossover is closely related to QCD key problems: vacuum, color symmetry, chiral symmetry, isospin symmetry ……vacuum, color symmetry, chiral symmetry, isospin symmetry ……

* BCS-BEC crossover of color superconductivity and pion * BCS-BEC crossover of color superconductivity and pion superfluid is not induced by simply increasing the coupling superfluid is not induced by simply increasing the coupling constant of the attractive interaction but by changing the constant of the attractive interaction but by changing the corresponding charge number.corresponding charge number.

* * There are potential applications in heavy ion collisions (at There are potential applications in heavy ion collisions (at CSR/Lanzhou, FAIR/GSI and RHIC/BNL) and compact stars.CSR/Lanzhou, FAIR/GSI and RHIC/BNL) and compact stars.

thanks for your patience thanks for your patience


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Relativistic BCS-BEC Crossover at Quark Level