BCS to BEC crossovers in
dense Quark Matter Physics
Hiroaki Abuki
Frankfurt University
thanks a lot for fruitful collaborations!!
Y. Nishida and H. A., PRD72, 096004 (05)
H. A., Nuclear Physics A791, 117 (07)
H. A., M. Ciminale, R. Gatto, N.D. Ippolito,
G. Nardulli, M. Ruggieri, PRD78, 014002 (08)
H. A., R. Anglani, R. Gatto, M. Pellicoro, M. Ruggieri,
arXiv:0809.2658
H. A., and T. Brauner, work in progress
26 Sep. 2008
QGP meets cold atoms @ GSI
Where to find BCS/BEC
crossover in QCD?
Fig from: Xuguang Huang
quark-gluon plasma
u
d*
1. QCD at large isospin density (Son, Stephanov, etc...)
Charged pion condensate, to (ud*)
pairs
hadron
gas
J P = 0 - p+
Fermi
kF
u
QI = +1
pion BCS
u
surface
pion
BEC
u
*
*
d
d
d*
x  r-1/3 x-space pairing
putting isospin charge,
p+’s spatially overlaps
k-space pairing x  r-1/3
2. QCD at large baryon density (Bailin-Love, Iwasaki, etc...)
Color superconductivity, (qq) Cooper pairs
kF
JP = 0 + d
u
?
B = +2/3
u
CSC
d
reducing baryon charge,
d
coupling becomes stronger,
Fermi surfaces disappear
2
I
PART I
QCD at finite isospin density;
From a pion condensate
to a pionic superfluid
What is the role of baryon chemical potential?
What is the role of electric charge neutrality?
Ref: H. A., R. Anglani, R. Gatto, M. Pellicoro, M. Ruggieri,
arXiv:0809.2658; H. A., M. Chiminale, R. Gatto,
N.D. Ippolito, G. Nardulli, M. Ruggieri, PRD78, 014002 (08)
Large Isospin matter?
SU(2) symmetry is satisfied in hadron spectroscopy
æu ÷
ö up (+1/2)
q = ççç ÷
÷
÷ down (-1/2)
çèd ø
æe i a / 2u ÷
ö
çç
by U(1)  SU(2)
÷
q ® ç - ia / 2 ÷
÷
ççe
d÷
÷
è
ø
u
Large isospin U(1) charge density means
0 = n I = u + u - d + d = n u - nd Þ
n u = nd =
kF
nI
d*
2
Pairs in pion channel due to gluonic attraction on Fermi surfaces
ìï p x = D cos a
D ia
with p i = qi g 5 t i q
d (i g 5 )u =
e or ïí
ïï p y = D sin a
2
î
spontaneous U(1), P symmetry breaking! = “pionic superfluid”
4
Crossover in pionic superfluid
Crossover to Bose-Einstein condensate of p+ meson?
X Son, Stephanov, PRL86, 592 (01)
(chiral perturbation at low mI , and perturbative QCD at high mI )
X Kogut, Sinclair, PRD70, 094501 (04);
X de Forcrand, Stephanov, Wenger, PoS LAT2007, 237 (07)
(Lattice QCD at finite mI is free from the sign problem)
How to study the smooth crossover analytically?
Effective models at hadron level can only describe the BEC side
L = q (i g m¶ m - m )q + Lint
+ dL
NJL
ìï Lint = G é(qq)2 + (qi g tr q)2 ù : simulates gluons
5
ïï NJL
ë
û
í
ïï dL = m q + t q : isospin chemical potential
3
I
ïî
X L. He, P. Zhuang, PRD71, 116001 (05)
5
Crossover in pionic superfluid
L. He, P. Zhuang, PRD71, 116001, ’05
ìï
ïï
ïï
1. í ìïï
ïï ïí
ïï ï
ïïî ïïî
r
{f , p } =
r
{ qq , qi g5t q
}
M - m 0 = - 2G f
{N , N z } = {2G p x2 + p y2 , - 2G p z }
2. det ij éêdij - 2G P i , j (w, p; M , N )ùú= 0
ë
û
1: Chiral & Pion
condensates
In Mean field app.
2: Mesonic excitations
In the RPA
Oj
pj
w, p
q
Oi
pi
ìï i P (w, p) = F .T . q(t )[qO q(x ), qO q(0)]
ïï i , j
i
j
q
í
ïï O = {1, i g t , i g t , i g t } for {s , p , p , p } mesons
+
5 1
5 2
5 3
0
ïî i
Note1: N is the gap at Fermi surface: E u = E d = (E - mI / 2)2 + N 2
Note2: once N becomes nonzero, there is mixing among {s,p+,p-}
6
Crossover in pionic superfluid
M p + = mI > 2M Q ( mI ,T )
dissoc.
M0
N0
Goldstone boson
The same critical point
mp+ =mp- mI
Figures from:
L. He, P. Zhuang, PRD71, 116001, ’05
G. Sun, L. He, P. Zhuang, PRD75, 096004, ’07
7
Pion BEC in neutral quark matter
at finite baryon density?
Ebert, Klimenko, J. Phys. G32, 599, ’06;
dL = Le + q + ( m - Qˆ me )q;
(Qˆ u = (2/ 3)u,
Qˆ d = (- 1 / 3)d )
: role of isospin chemical potential
ìï mI = - me
ïí
ïï m = m - (1/ 6)me : Quark mean chemical potential
î
The effect of m is to break the balance b/w quark and antiquark.
ìï E =
ïï u
í
ïï E =
ïïî d
(E - mI / 2)2 + N 2 - m
(E - mI / 2)2 + N 2 + m
: mismatched u-dbar
Fermi surfaces?
Pion condensate tries to keep the Fermi surfaces matched
n u = n d ( for N > m)
Fermi surfaces persist for small m !
8
The baryon density induced
gapless pion condensate?
Type II
Type III
Type I
gapless
N
mgapless
> N 2 + pion
(mI / 2 + M )2
(1) m >pion
(2) m >gapless
(3)
N 2 + éë(pion
mI / 2)2 - M 2 ù
û
condensate
condensate
condensate
negative isospin decouples
antiquarks
decouple
m = 0.5M
p0/M
mI = 4.0M
N = 0.25M
m = 3.2M
ubar(-)
d(-)
dbar(+)
pF
m = 1.2M
d
dbar
u(+)
u
u
p/M
gapless u quark with
two effective Fermi surfaces
gapless u quark with
one effective Fermi surface
Even d quark becomes
gapless
ìï E = (E - m / 2)2 + N 2 - m; E = (E + m / 2)2 + N 2 + m
ïï Type
u
I: Huang,
I Shovkovy (03) u
I
í
2
2
2
2
ïï Type
II:(EKitazawa,
Rischke,
Shovkovy,
PLB637,
367
(06)
E
=
+
m
/
2)
+
N
m
;
E
=
(
E
m
/
2)
+
N
+ m
d
I
I
d
îï
9
Result from NJL model analysis
in the chiral limit (m=0)
N = 2G p x2 + p y2
Continuous
transition
from cSB to
charged p
condensate
Ebert, Klimenko, J. Phys. G32, 599, ’06;
Andersen, Kyllingstad, hep-ph/0701033
Neutral ground state is built
dW
= 0 ® me ( m)
d me
cSB
M0
N0
typeIII
gapless
p cond.
mc1
M=N=0 (normal QM)
mc2
1st order trnsition
m (quark)
chiral condensate:
S = f + p x i g consequences:
t x + p yig t y
Phenomenological
5
2
Q. Wang,2P.
x
X. Huang,
+S] f
Its magnitude Tr[S
5
2
Zhuang,
PRD76 094008
y is continuous
+ p + p
(07)
10
cSB/pcon
1st orde
transition
Normal QM
What is the role of quark mass?
H.A., R. Anglani, R. Gatto,
M. Pellicoro, M. Ruggieri, arXiv:0809.2658
Now take the finite quark mass into account
1st order
r 2ù
0
2
ˆ
é
L = Le + q (i g ¶ m + ( m - Q me )g + m )q + G ë(qq) + (qi g 5 t q) û
Normal QM
m
Pion condensate
in the
gapless
p neutral ground state?
condensate
¶W
¶W
¶W
= 0,
= 0,
= 0
¶N
¶M
¶ me
m=10keV, mp=6MeV
1.
p condensate exists but only
in the vicinity of the chiral limit
2.
p condensate appears as soon
as p- mode becomes gapless
Electrically neutral
ground states and
meson spectra in RPA
11
The role of current quark mass;
a conclusion
When m = mc1 is reached from N=0 phase at low density,
mec hits Mp- from below; the Bose-Einstein condensation
mec  Mp- - 0 (at critical point)
Gapless pion condensate disappears for quark mass larger
than order of 10keV. (vacuum pion mass order of 10MeV)
Gell-Mann-Oakes-Renner relation, m p µ
m
Goldstone mass is drastically increased
even by a tiny explicit symmetry breaking term
The effect of m can so easily wash out gapless pionic supperfluid
me  mp  Mp- (by quark mass, no BEC)
12
PART II
QCD at finite baryon density;
From color superconductivity
to BEC of tightly bound diquarks
What is the relativistic BCS/BEC crossover?
What is the role of preformed quark pair fluctuation?
Ref: Y. Nishida, H. A., PRD72, 096004 (05)
H. A, Nucl. Phys. A791, 117, (’07)
Relativistic BCS-BEC crossover?
Role of fluctuations beyond mean field:
RBEC
here: X Nishida, Abuki, 2005, 2007, Nozieres-Schmitt-Rink, RBEC
X He, Zhuang, 2006, 2007, G0G scheme to relativistic systems
X Deng, Wang, Wang, 2008, Bose-fermion model
with Baym-Kandanoff-Martin self consistent approach
Crossover in collective excitations, phase diagram
X Nawa, Nakano, Yabu, 2006, Statistical analysis
X Deng, Schmitt, Wang, 2007, Bose-fermion model
X Kitazawa, Rischke, Shovkovy, 2007, Phase diagram, NJL
X Brauner, 2008, collective excitations, NJL
X Chatterjee, Mishra, Mishra, 2008, variational approach
14
10
abuki-hatsuda-itakura, PRD 65 07014 ’02
SD eq. with improved ladder approx.
5
c / dq
BCS-like
In relativistic quark matter?
c/d q
10
What is “relativistic” matter?
kF/mc
10
10
10
T
4
3
2
1
BEC-like?
10
0
10
10-9 in atom system
10-7 in 3He
10-3 deuteron gas
“relativity parameter”
or “density parameter”
3
10
4
10
5
μ
[
MeV
[MeV]]
10
6
Color superconducting phase
Hadrons
stronger / diluter
BEC-like??
BCS
Natural question to be addressed is:
What’s the BCS/BEC crossover in relativistic system?
What is the dependence on relativity parameter?
In the limit m  0, fermions cannot form bound states.
m
15
Our approach
Nishida-Abuki, RRD72, 096004
Thouless
(Gap equation)
Apply thecriterion
NSR formalism
to a relativistic fermion model
m: Dirac mass
attractive 4-point (contact) interaction
m: chemicalin
pot.
in J P = 0+ channel
Fixed number condition
the gaussian approximation
denoted by
dWNSR =
G
N F  N F   N MF 
denoted by
N fluc .
We obtain (m, T ) as functions of (G, kF)
N fluc  N B  N B  N unstable
16
Tc/EF
BCS
(Nc=3, NF=2, fixed m/L = 0.2, kF /m = 0.2)
RBEC
??
BEC
mc/EF
BCS
RBEC
??
BEC
1.0
1
m
0.8
0.1
Tc
MF
0.0001
-0.5
NR
NR
0.01
0.001
TBEC
Tc
 NB 2 



3
/
2




with mB  2 
BEC
TNR

0.0
2
mB
0.5
g
c/ EF
(Temperatures, Chemical potential)
EF
Results
mc = m
Relativistic criterion
for BEC

c
0.4
2/3
1.0
0.6
mc =
M B ( mc ,T c )
0.2
mcNR = 1.5
0.0
-0.5
0.0
2
EB
: Nishida, Abuki (05)
: NSR (80)
2
0.5
g
1.0
1.5
17
What’s the RelativisticBEC?
Kapusta, PRD24, (’81); Haber-Weldon, PRL46, (’81)
dp 
N B   B , mB , T   
f
3  B
 2  

 
mB2  p 2   B  f B
boson

mB2  p 2   B 

anti-boson
Net charge reaches maximum N B  mB , mB , T  at mB = mB;
TBEC by  nB  N B  mB , mB , TBEC   TBEC  mB , nB   ,
NB(m
B,T) n 1/3
For
mB,m
Bc/
B
(dilute, non-relativistic
nB2 / 3 limit)
2  nB 
BEC
TNR 

  mB
mB    3 / 2 T
BEC
only bosons T
contribute
to NB
BEC
For mBc/  nB1/3
(dense, relativistic limit)
TRLBEC 
3nB
 mB
mB
anti-bosons contribute to NB
18
Tc/EF
BCS
(Nc=3, NF=2, fixed m/L = 0.2, kF /m = 0.2)
N/Ntot
RBEC!
BEC
1.5
Superdense system
with antiparticles!
1
0.1
BEC
TRL

3 NB  NB 
2mB
MF
T
B c
0.001
TBEC
NB Nq
1.0
TBEC
with m  2 
0.01
 q, q , d , d 
RL
Number fractions
(Temperatures, Chemical potential)
EF
Results
NR
Tc
Tc rapidly increases
to the order of EF
mB = 
Nfluc
0.5
NMF
0.0
c
Tc is above boson mass
Anti-bosons are beexcited
0.0001
-0.5
0.0
0.5
g
1.0
NBbar Nqbar
1.5
-0.5
-0.5
0.0
0.5
g
1.0
1.5
19
Density (kF/mc) dependence
BCS
BEC
RBEC
1
high density
BCS/BEC boundary
shifted to higher
value of chemical pot.
Tc(kF=2.4m)
Tc/EF
T c ( kF)
EF
0.1
2mc=Tc
0.01
Pauli-blocking
prevents the bound
state formation in high
density medium!
Tc(kF=0.2m)
MB=2mc2m
k =0.2m x n
0.001
0.0001
-0.5
F
with n=1,2,...,10
low density
bottom
to top)
Universal feature at the(from
unitarity
is lost
by a relativity parameter kF/mc !!
0.0
0.5
g
Gc
GR
1.0
Crossover
characteristics of Tc
somewhat smeared
at high densities
1.5
20
Summary and Outlook
Mean field analyses for isospin matter with imbalance
Examined the roles of electric neutrality and current quark mass
Developed the NSR formalism to a relativistic pairing model
2-step crossover “BCS/BEC/RBEC” in a relativistic superfluid!
In RBEC, all the degrees of freedom participate
in thermodynamics. (q, qbar, d, dbar)
How do fluctuations affect the Thouless criterion and/or
gap equation? 1/N expansion might be useful
Nikolic and Sachdev, PRA75 (07);
Veillette, Sheehy, Radzihovsky, PRA75 (07)
See next talk by Tomas Brauner
21
Backup slides
22
Naïve application of BCS
leads to power law blow up
1
What’s the BCS/BEC
T transition
/E ~
naive
c
F
k F2 as2 log  2 / k F2 as2 
c
A question: howTlarge
the critical temperature
BEC
F
can be in the E
strong
coupling limit?
Tc / EF ~ 0.2314..
“universal value”
not depending
on coupling
1957
BCS
Tc / EF ~ e  / 2 kF |as |
1924
depends exponentially
on coupling
1
0
Weak (dense)
Unitarity limit
c.f Eagles (’69), Leggett (’80), Nozieres&Schimitt-Rink (’80)
k F as
Strong (dilute)
23
Why Mp-= me at the critical
point?
H.A., R. Anglani, R. Gatto,
M. Pellicoro, M. Ruggieri, arXiv:0809.2658
The inverse propagator of p- mode in the RPA is
m
pGp + p - (Q 0 ; me ) =
- Q 02N c c (Q 0 ; me )
2GM
w, p
with Q0 = w + me , c being the polarization
u
d
ig5 / 2
Gp + p - (Q 0 = ± M p m ; me ) = 0 : determines charged pion masses
Assuming 2nd order transition, we expand W in N 2 = 4G 2 (px2+py2)
1
a( me ) 2 b( me ) 4
a
(
m
)
=
Gp + p - (Q 0 = me ; me )
W=
N +
N + K it is shown
e
4
2
4
ìï M - = mec ( mec > 0)
p
a( mec ) = 0 : determines the critical point  ïí
ïï M + = - mec ( mec < 0)
ïî p
24
phase diagram in the (m, me)-plane
for a physical quark mass
mec 2 ( m);
mec ( m); a( me ( m)) = 0
Wc SB = Wc SR
or equivallently,
M p - ( mec ( m)) = mec
f0
p 0
f 0
f0
p= 0
pion condensation impossible
in neutral quark matter with
a physical quark mass!
Q <0
Q =0
Q >0
meneut ( m);
dW
= 0
d me
25
Our approach (1) Nishida-Abuki, RRD72, 096004
Apply the NSR formalism to the relativistic systems
m: Dirac mass
m: chemical pot.
attractive 4-point (contact) interaction
in J P = 0+ channel
Gaussian fluctuations to TDP is
Following NSR, we write this in terms of phase shift
26
Including Nc colors H. Abuki, NPA791, 117 (’07)
Including N colors and 2 flavors is straightforward..
If pairing is in the color anti-symmetric and flavor singlet channel,
Thouless criterion will not be affected
Fixed number condition in the gaussian approximation
If kF is fixed, increase of NC does not affect TC in the BCS,
while it lowers the BEC temperature by factor 1/NC2/3 (normal BEC)
27
Cooper pair wave functions
 (r )  oscillation
4 kF r  (0, rin
) (0,0)
Friedel
BCS
like S-wave bound state w.f.
2x10
0.2
-3
0.1
BEC
g = 0.5
10 x 
c
BCS
g = -0.35

0.1
1
0.0



0
10
20
30

0
0.0
r
sin(kF r )  c
BCS
 (r ) 
e
( k F r )3
BEC (r )  e
1
kF r
2
kFr / 2 
2
r
2as
-1
-0.1
0

3
0
1
2
kFr / 2 
3
28
Dissociation of bound state
BCS
1
BEC
RBEC
Pre-formed boson phase
above Tc!
pair dissociation
Temperature/EF
Temperature
EF
Tdiss
cf. J/y survival above Tdec.
Asakawa-Hatsuda, PRL92 (’04)
Datta et al., PRD69 (’04)
Umeda-Nomura-Matsufuru,
EPJC39S1 (’05)
0.1
0.01
Tc
0.001
0.0001
-0.5
Precursory modes above Tc
Hatsuda-Kunihiro, PRL55 (’85)
Castorina-Nardulli-Zappala,
PRD72 (’05)
superfluid
0.0
0.5
g
1.0
1.5
29
Speculative remarks;
Mapping to QCD phase diagram
Plasmino mass
mq  g
cf. diquark (quasi) bound states above Tc
is relevant to small viscosity in sQGP
by Shuryak and Zahed
PRC70, 021901, PRD70, 054507,04 ,…etc
BEC criterion : g  
Probable realization of
BEC in non-perturbative
region of CSC
T
?
BCS (R)BEC
(g>1)

Crossover regime from BCS to BEC is liquid like?
Gelman, Shuryak & Zahed, arXiv:nucl-th/0410067
T. Sch”afer, PRA76, 063618 (h/s  1/2; small !)
30
Really in Quark matter?
Critical analysis
Effective dimensionless coupling
1
g(m ,G ) = +
2
G L L2 + m 2 - p2
éL + L 2 + m 2 ù
2
ú
2Gm log êê
ú
L
êë
ú
û
For any fixed bare coupling G < p2/L2 (corresponding to
extremely large coupling for Majorana mass generation at vacuum)
when m  
when m  0
g(m , G ) ® 1
: BEC like
G L2 - p2
g(m , G ) ®
® - ¥
é
ù
2
L
2Gm 2 log ê ú
êëm ú
û
: BCS like
31
Really in Quark matter?
Critical analysis
Existence of mass, m*, the separation scale b/w BEC and BCS
g(m *(G ),G ) = 0
For usually adopted diquark coupling, one obtains
æ
3 1.8 ÷
ö
( L : 1 GeV)
m * ççG =
» 0.53L
2 ÷
è
4L ø
Huge fermion mass is needed to have BEC-like system
Kitazawa-Rischke-Shovkovy, PLB663, 228, ’08
Even though some lattice calculation seems encouraging
Nakamura, Saito, Prog. Theor. Physics 112, 183 ’04
Petreczky et al., Nucl. Phys. Proc. Suppl. 106, 513 ’02
32
Outlook
How can we go beyond the Gaussian approximation?
Self-consistent T-matrix approach?
(Baym-Kadanoff approximation to Luttinger-Ward potential);
Haussmann, PRB49 (’94); Haussmann et al.,PRA75 (’07)
How do fluctuations affect the Thouless criterion and/or
gap equation? 1/N expansion might be useful
Nikolic and Sachdev, PRA75 (07);
Veillette, Sheehy, Radzihovsky, PRA75 (07)
See next talk by Tomas Brauner
Thanks for your attention
33