1

Relativistic BCS-BEC Crossover in Quark MatterPengfei Zhuang

Physics Department, Tsinghua University, Beijing 100084

1) Introduction2) Mean Field 3) Fluctuations 4) Nuclear Matter and Quark Matter5) Conclusions

2

Introduction: Pairing

in BCS, Tc is determined by thermal excitation of fermions, in BEC, Tc is controlled by thermal excitation of collective modes

Tc in the BEC region is independent of the coupling between fermions, since the coupling only affects the internal structure of the bosons.

3

Introduction: BCS-BEC in Quark Matter

pair dissociation line

BCSBEC

sQGP

QCD phase diagram

there may exist BCS-BEC crossovers in quark matter !new phenomena in BCS-BEC crossover of QCD:relativistic systems, anti-fermion contribution, rich inner structure (color, flavor), medium dependent mass, ……

strongly coupled quark matter with both quarks and bosons

4

Introduction: History of BCS-BEC Theory

*) non-relativistic mean field theory at T=0 (Leggett, 1980)

*) non-relativistic theory at T ≠ 0 (Nozieres and Schmitt-Rink, 1985)

extension to relativistic system (Nishida and Abuki (2006,2007)4

0 04

1ln ( ) , (2 )fld q q G G

Gχ χ

π Ω = − = ∫

*) non-relativistic G0G scheme (Chen, Levin et al., 1998, 2000, 2005)

asymmetric pair susceptibility 0 G Gχ =

extension to relativistic G0G scheme (He, Jin, PZ, 2006, 2007)

*) bose-fermion model (Friedderg, Lee, 1989, 1990)extension to relativistic systems (Deng, Wang, 2007)

*) Kitazawa, Rischke, Shovkovy, 2007, NJL+phase diagram Brauner, 2008, collective excitations ……

5

BCS limit

BEC limit

Non-relativistic Mean Field at T=0

2 /2

1 8 ˆ, , 1F s

ek a e

η πη µ= → −∞ ∆ = =

2

2

( ) /

16 ˆ, , 31,

21( ) 0

1

bb

s

T

ma

n pe ε µ

ηη µ ηπ

εµ ε

µ−

→∞ ∆ = = −

= − =

= ⇒ ≤−

BCS-BEC crossover0 0,

small large , 0 0

η η

µ µ

< → >∆ → ∆

> → <

A.J.Leggett, 1980 universality behavior

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Relativistic Mean Field with Broken Universality

( ) ( )( )5 54T TgL i m i C iCµ

µψ γ ψ ψ γ ψ ψ γ ψ= ∂ − +

52Tg iCψ γ ψ∆ =

2 3

3(2 ) k k k kd k E E

gξ ξ

π+ − + −∆ Ω = − + − − ∫

order parameter

NJL-type model at moderate density

mean field thermodynamic potential

fermion and anti-fermion contributions

Lianyi He, PZ, 2007

22 2

0

2

0

1 1 1 12 2 2

2 1 13

1 ,

z

x x x x

zx x

x x

F

F s

dxxE E

dxxE E

kk a m

π ηε ζ ε ζ

ξ ξ

η ζ

− − + −

− +

− +

− = − + − − +

= − − −

= =

∫

∫

gap equation and number equation:

broken universality extra density dependence

2 2

2 2

k k

k

E

k m

x

x m

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Relativistic Mean Field BCS-BEC

mm plays the role of non-relativistic chemical potential

0 :mm fermion and anti-fermion degenerate, NBEC-RBEC crossover

0 :m BCS-NBEC crossover

in non-relativistic case, there is only one 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

1/ F skh a

/Fk mx

Lianyi He, PZ, 2007

8

0 G GFluctuations: Scheme

bare fermion propagator

pair propagator

10 0 0( , ) ( )G k k k mµ µ γ γ− = + − ⋅ −

1 10( , ) ( , ) ( ) mfG k G k kµ µ− −= −Σ

Lianyi He, PZ, 2007

mean field fermion propagator

fermions and pairs are coupled to each other

( ) ( ) ( )mf flk k kΣ = Σ +Σ =

pair feedback to the fermion self-energy

approximation2

0( ) ( , )fl pgk G k µΣ −∆ −

the pseudogap is related to the uncondensed pairs, in G0G scheme the pseudogap does not change the symmetry structure

9

Fluctuations: BCS-NBEC-RBEC

*

*

*

: critical temperature

: pair dissociation temperature

: 0, 0,

condensed phase: 0, 0,

normal phase with both fermions and pairs:

c

c pg

c pg

TT

T T

T T T

T T

< ∆ ≠ ∆ ≠

< < ∆ = ∆ ≠

> 0, 0

normal phase with only fermionspg∆ = ∆ =

BCS: no pairs

NBEC: heavy pairs, no anti-pairs

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

0, mη µ< >

0 / , 0<Fm k mη µ< < <

/ , 0Fm kη µ>

Lianyi He, PZ, 2007

10

BCS-BEC in Asymmetric Nuclear MatterShijun Mao, Xuguang Huang, PZ, 2009

asymmetric nuclear matter with both np and nn and pp pairingsdensity-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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( )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 dψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψ ψΨ =1

2

2

1

1

2

2

1

3

3

3

3

uCd

dCu

dCu

uCd

uCu

dCd

ψ

ψψ

ψψ

ψψ

ψψ

ψψ

ψ

Ψ =

σ ψψ=3 3

5C iji jiαβα βψ ε ε γ ψ∆ = ∆ =

order parameters of spontaneous chiral and color symmetry breaking

quark propagator in 12D Nambu-Gorkov space

A

B

C

D

E

F

SS

SS

SS

S

=

I II

I I

GS

G

+ −

+ −

Ξ=

Ξ

0 2q SM m G σ= −

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

( ) ( )

2 2

2 22

k q

k D

E k M

E E Gµ±∆

= +

= ± + ∆

BCS-BEC in Color Superconductivity

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

diquark & meson polarizations

M DΠ Π

diquark & meson propagators at RPA

quarks at mean field and mesons and diquarks at RPA

Lianyi He, PZ, 2007

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BCS-BEC with Color Neutrality

gap 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 EE E E

d kGE E

µ µ µπ

π

− +∆ ∆

− +∆ ∆

− +− = + +Θ −

∆ = ∆ +

∫

∫

/ 3 /B d sm m G Gµ η∆ − =

there exists a BCS-BEC crossover

to guarantee color neutrality, we introduce color chemical potential:

8 8/ 3 / 3, / 3 2 / 3r g B b Bµ µ µ µ µ µ µ= = + = −

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

r mµ −

Lianyi He, PZ, 2007

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BCS-BEC in Pion Superfluidity

( , ) 2 Im ( , )k D kρ ω ω= −

meson spectra functionmeson mass, Goldstone mode

BEC

BCS

Gaofeng Sun, Lianyi He, PZ, 2007

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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 the QCD key problems: vacuum, color symmetry, chiral symmetry, isospin symmetry ……

* BCS-BEC crossover in color superconductivity and pion superfluidity 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, NICA/JINR and RHIC/BNL) and compact stars.

thanks for your patience

May, 2009 CSQCDII Beijing 15

backups

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vector meson coupling and magnetic instability

vector-meson coupling ( ) ( )2 2

5V VL G µ µψγ ψ ψγ γ τψ = − +

gap equation

02V VGρ ψγ ψ=

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

vector condensate

( )3

3

/ 3 / 38 / 3(2 )

k B k BV V k B

E Ed kG EE Eµ µρ µ

π + −∆ ∆

+ −= − +Θ − +

∫

1η =

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

r mµ −

May, 2009 CSQCDII Beijing 17

beyond mean field

0 100 200 MeV 300 500 MeVqµ∆ = − ↔ = −0 ( 0)T∆ = ∆ = 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

May, 2009 CSQCDII Beijing 18

NJL 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 SV

q q

σ π

σ σ σ σ π π

−Ω = + −

∂Ω ∂ Ω ∂Ω ∂ Ω ∂Ω ∂ Ω ∂Ω ∂ Ω= ≥ = ≥ = ≥ = ≥

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

( ) ( ) ( )( )2 20 0 5NJL iL i m G iµ

µψ γ µγ ψ ψψ ψ τ γ ψ= ∂ − + + +

, , u d u duu ddσ ψψ σ σ σ σ= = + = =chiral and pion condensates with finite pair momentum

quark propagator in MF

0 51

5 0

2( , )

2u

d

p q m iGS p q

iG k q m

µµ

µµ

γ γ µ γ πγπγ γ γ µ γ

− − ⋅ + −= + ⋅ + −

0 2m m Gσ= −

2 25 52 , 2

2 2iq x iq xui d e di u eπ ππ γ π γ⋅ − ⋅

+ −= = = =

quark chemical potentials0 / 3 / 2 0

0 0 / 3 / 2u B I

d B I

µ µ µµ

µ µ µ+

= = −

thermodynamic potential and gap equations:

pion superfluid

May, 2009 CSQCDII Beijing 19

5

5

3 5 0

1, ,,,

m

mi mi mi m

στ γ πτ γ πτ γ π

+ +

− −

= =Γ = = =

( )4

*4( ) Tr ( ) ( )

(2 )mn m nd pk i S p k S pπ

Π = Γ + Γ∫

meson polarization functions

meson propagator at RPA

pole of the propagator determines meson masses

mixing among normal in pion superfluid phase, the new eigen modes are linear combinations of

considering 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

σσ σπ σπ σπ

π σ π π π π π π

π σ π π π π π π

π σ π π π π π π

+ −

+ + + + − +

− − + − − −

+ − = =

− Π − Π − Π − Π − Π − Π − Π − Π

= − Π − Π − Π − Π − Π − Π − Π − Π

0

, ,σ π π+ −

mM

, ,σ π π+ −

D

mesons in RPA

, ,σ π π+ −

May, 2009 CSQCDII Beijing 20

I Bµ µ−

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:

0BT qµ= = =

cI mπµ =

( )B BnµIµ

pion superfluidity phase diagram in plane at T=0

: 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

0q ≠

0q =

phase diagram of pion superfluid