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18 Mar 2010 NFQCD10
BEC-BCS crossover in NJL model; towards QCD phase diagram & role of fluctuationsHiroaki Abuki
(Tokyo Science University)
Many thanks to:
Gordon Baym, Tetsuo Hatsuda, Naoki Yamamoto& Tomas Brauner
Plan
BEC-BCS crossover with axial anomaly
Notion of BEC-BCS crossoverIn general relativistic systems?Axial anomaly induced BEC-BCS transition(1)
How can we deal with fluctuations?
1/N expansion vs. Nozieres-Schmitt-RinkApplied to Nonrelativistic Fermi gas(2)
CFL/2SC models of color superconductor(2)
(1) HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408(2) HA, T. Brauner, PRD78, 125010 (2008)
Plan
BEC-BCS crossover with axial anomaly
Notion of BEC-BCS crossoverIn general relativistic systems?Axial anomaly induced BEC-BCS transition(1)
How can we deal with fluctuations?
1/N expansion vs. Nozieres-Schmitt-RinkApplied to Nonrelativistic Fermi gas(2)
CFL/2SC models of color superconductor(2)
(1) HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408(2) HA, T. Brauner, PRD78, 125010 (2008)
What is the BEC-BCS crossover?
(1) D. M. Eagles, Phys. Rev. 186, 456 (1969) (2) A. J. Leggett, J. Phys. C 41, 7 (1980)(3) Nozières and Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985)
What is the BEC-BCS crossover?
1F sk a
c
F
TE
0Weak Strong
BCS
depends exponentiallyon coupling
/ 2 | |/ ~ π− F sk ac FT E e
1957
1924
Naïve application of BCSleads power law blow up
( )naive
2 2 2 2
1/ ~log 2 /c F F s F s
T Ek a k a
Eagles (1969), Leggett (1
980)
Nozieres & Schmitt-Rink
(1985) …
BEC
“universal value”not depending
on coupling
/ ~ 0.2314..c FT E/ 1dξ
1/3d N −=( ) 1/32 / 2
2T Bd N
mTπλ −= ≈ =
1/3 2 /3,F Fk n E n∝ ∝ Unitarity limit
Crossover observed in UCA
Note: interaction is tunable via Magnetic field!K40: Regal et al., Nature 424, 47 (03), PRL92 (04): JILA groupLi6: Strecker et al., PRL91 (03): Rice group; Zwierlen et al., PRL91 (03): MIT group; Chin et al., Science 305, 1128 (04): Austrian group
K40 (JILA)
Relevant scales:
Tc ∼ 100nKTF ∼ 1µKρ ∼ 1014cm-3λ ∼ 100nm
N0/N
BCSBEC
In general relativistic systems?
What is a relativistic matter?
kF/mc∼λc/d (:=relativity parameter)
∼10−9 in cold atom systems∼10−7 in 3He∼10−2 nuclear matter (deuteron gas;
Lombardo, Nozieres, Schuck, Schulze, Sedrakian,PRC64, 064314 (2001)
In cold quark matter with CSC
kF/mc 100 relativistic enough!
.
A schematic phase diagram of QCD
∼300milion tons/cm3
CP
χSBCSC
µq
T
∼300MeV
mq ≠ 0
100
101
102
103
104
105
ξ c/d
q
103 104 105 106
qc d/ξ
Cooper pair sizevs.
inter-quark spacing
HA, T. Hatsuda, K. Itakura, PRD 65 (02)See also M. Matsuzaki, PRD62 (00) for pair size
]MeV[μ
BEC-like?
BCS-like
NJL model with axial anomaly (1)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
〈φ〉≠0〈d〉=0 〈φ〉≠0
〈d〉≠0
〈φ〉=0〈d〉=0
NJL model with axial anomaly (2)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Model setup (chiral limit)
(6)(4)0( )q i q LLL µγ= + +∂ +
( )( )
(4)
(4)
8 tr
2 tr R Rd L L
G
H d d d d
L
L
χ φ φ+
+ +
⎧⎪ =⎪⎪⎨⎪ = +⎪⎪⎩
Chiral fields: Diquark fields:
L
H
L
L L
L
G
L
R R
( )ij Rj Liq qφ = ( )a abc b c
L ijk Lj Lkid q Cqε ε=
For general construction of GL free energy (→ Talk: T. Hatsuda, 3/10)see also Yamamoto, Tachibana, Baym & T.H., PRL97 (06)
(r,g,b)(u,d,s)
NJL model with axial anomaly (3)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Model setup (chiral limit)
KMT term for U(1)A breaking
K responsible for η’ mass decoupling
(6)(4)0( )q i q LLL µγ= +∂ + +
( )
( )KMT
KMT'
(6)
(6)
8 det h.c.
tr' h.c.R LK
L
L d d
K φ
φ+
⎧⎪ = − +⎪⎪⎨⎪ = +⎪⎪⎩ K
uL uR
sL sR
dL dR
/36 ( ) ( )
(1) ( ( 1) )nA L R L R
U Z q q±→ → −
NJL model with axial anomaly (4)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Favorable mean fields
Chiral: 〈φij〉 = δij (χ/2) (V-singlet, 0+)
CFL: 〈dLaj〉 = −−〈dRaj〉 = δaj (s/2)(for K’ >0 ) (V+C-singlet, 0+)
Spontaneous symmetry breaking in CFLAlford, Rajagopal, Wilczek, NPB99
(3) (3) (3) (1) (1)C L R B ASU SU SU U U× × × ×
2(3)L R CSU Z+ +→ × Z6
NJL model with axial anomaly (5)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Dirac/Mojorana mass formations
.
Lots of work done:
(with variety of CSC phases)Ruester et al, (05)Blaschke et al., PRD72 (05)
(+vector interaction)Zhang, Fukushima, Kunihiro, PRD79 (09)Zhang, Kunihiro, PRD80 (09)
etc…See for a review:Buballa, Phys. Rept. 407 (05)
NJL model with axial anomaly (6)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
〈φ〉=0〈d〉=0
Pisarski-Wilczek (‘84)
〈φ〉≠0〈d〉=0 〈φ〉=0
〈d〉≠0
K’=0 (no interplay b/w chiral and super)
(G,K,H):M. Buballa, Phys. Rept. 407 (05)
NJL model with axial anomaly (7)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
〈φ〉≠0〈d〉=0 〈φ〉≠0
〈d〉≠0
K’ = 4.2K0 (turning on the interplay)
〈φ〉=0〈d〉=0
Talk; T. Hatsuda 3/10,Hatsuda, Tachibana, Yamamoto, Baym (06)
For another possible realization:Talk; T. Kunihiro 3/10,Kitazawa, Kunihiro, Koide, Nemoto (02)Zhang, Kunihiro (09, w Fukushima 08)
NJL model with axial anomaly (8)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
〈φ〉≠0〈d〉=0 〈φ〉≠0
〈d〉≠0
m = 5.5 MeV (turning on quark mass)
〈φ〉≈0〈d〉=0
Critical Point (CEP)Asakawa, Yazaki (89)Chiral sym. is onlyapproximate one
NJL model with axial anomaly (9)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Chiral transition; dependence on (K’,K)
Effect of K’ :• weaken the chiral
transition
Effect of K :• strengthen 1st order
chiral transition
Effect of mq :• helps to weken the
chiral transition
K/K0
K’/
K0
5
4
3
2
1
03210
1st order
crossover
K'c
m q=0
5
4
3
2
1
03210
1st order
crossover
m q=
5 MeV
mq= 1
40 Me
V
NJL model with axial anomaly (10)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
〈d〉≠0
BEC-BCS crossover in the CFL (COE)?
Transitions to theCFL (COE) phase :
• Both are 2nd order
• Both singal U(1)Bbreaking and arecharacterized byThouless condition
• However, differentphysical origins
NJL model with axial anomaly (12)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Development of precursory to CSC:Kitazawa, Kunihiro, Koide, Nemoto (02)
BEC of bound diquarks:Nishida,Abuki (PRD05, 07)Kitazawa,Rischke,Shovkovy (08)
Mass of diquark hitschemical potential!
ω =ΜD−2µ = 0µ=Mq
NJL model with axial anomaly (11)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
Effect of axial anomaly on (qq) attraction
Neglecting fluctuation in chiral field
KMT’ term increases (qq)-attraction in the positive parity channel
' (1/ 4) 'H HH K Hχ→ = + >
( )KMT'(6) t' r R L R LL dK d d dφ φ+ + += +
( )( )( )( )( )( )
KMT'(6) (1/4) ' tr
(1/4) ' tr
R L R L
R L R L
L K d d d d
K d d d d
χ
χ
+ +
+ +
− − −
+ + +
NJL model with axial anomaly (13)HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408
µ [MeV]
K’/
K0
How robust is theBEC-like CFL?
• K’ increases, thenBEC sets in at somecritical K’ (Q)
• This happens beforethe chiral transitiongets crossover (P)
• Between (Q) and (P)transition from BECto BCS is 1st order!
Q
P
Summary of 1st partRich phase diagram due to axial anomalyand its coupling to chiral/diquark fields
Chiral crossover
Diquark field contributes as if it is an external field for chiral transitionLow-T critical point appears!
BCS-BEC crossover
Chiral field behaves as if it is a mediating fieldthat increases (qq)-attraction
GL prediction: Hatsuda, Tachibana, Yamamoto, Baym (06)
c.f. Crossover by increasing H: Nishida, Abuki (05,07)NJL phase diagram: Kitazawa,Rischke,Shovkovy (08)
Plan
BEC-BCS crossover with axial anomaly
Notion of BEC-BCS crossoverIn general relativistic systems?Axial anomaly induced BEC-BCS transition(1)
How can we deal with fluctuations?
1/N expansion vs. Nozieres-Schmitt-RinkApplied to Nonrelativistic Fermi gas(2)
CFL/2SC models of color superconductor(2)
(1) HA, G. Baym, T. Hatsuda, N. Yamamoto, arXiv:1003.0408(2) HA, T. Brauner, PRD78, 125010 (2008)
Introduction of 1/N expansionFluctuations are obviously important in strong coupling (unitary) regime
1/N is a method to take into account fluctuations about MF in a systematic, controlled way
X Nikolic, Sachidev, PRA75 (2007) 033608 (NS)1. TC at unitarity
X Veillette, Sheehy, Radzihovsky, PRA75 (2007) 043614 (VSR)2. TC at unitarity3. T=0 off the unitarity
X Abuki, Brauner, PRD78, 125010 (2008) (AB)4. TC off the unitarity
1/N expansion (1)Euclidian lagrangian for ψ = (ψ↑, ψ↓)T
Introduce N “flavors” of spin doublet:
( )( )2 2
*2 2 ,2 4
TGLmτ
ψ µ ψ ψ σ ψ ψ σ ψ+ +⎛ ⎞∇ ⎟⎜= ∂ − − −⎟⎜ ⎟⎟⎜⎜⎝ ⎠
( )( )2 2
*2 22 4
TN N
GLNmτα α
ψ µ ψ ψ σ ψ ψ σ ψ+ +⎛ ⎞∇ ⎟⎜= ∂ − − −⎟⎜ ⎟⎟⎜⎜⎝ ⎠
( )1 1 2 2( , ) , , , ,..., ,NT
TNαψ ψ ψ ψ ψ ψ ψ ψ ψ ψ↑ ↓ ↑ ↓ ↑ ↓ ↑ ↓= → =
2 2 2, 1T
N N NU U U Uσ σ+= = : Sp(2N,R) ⊂ SU(2N)
1/N expansion (2)
SU(2) singlet Cooper pairSp(2N,R) singlet pairing field
No additional symmetry breaking No unwanted NG bosons There is only physical one, the Anderson-Bogoliubov mode associated with correct U(1) (total number) breaking
21
( ) ( ) ( )2
NTGx x
Nxα α
αφ ψ σ ψ
=∑∼
1/N ordering of correctionsBosonized action
For systematic handling, setting
Thermodynamic potential at NLO
[ ]TrLog2
1/1
0
( , )( , )
TN
xS d dx D x
Gφ τ
τ φ τ−⎡ ⎤⎢ ⎥= −⎢ ⎥⎢ ⎥⎣ ⎦∫ ∫
, 0
( , ) ( , )1 i ipxp
x p eN
ωτ
ωφ τ δφ ω − +
≠
= ∆ + ∑ Bosonic fluctuation
Fermion one loop:Mean Field app.
Boson one loop:quantum effect
LO ∼ O(1) NLO ∼ O(1/N)
(1/(0) )1 ( , , ,1/ ) 1 ...NsF NT k a
Nµ δΩ ∆ = Ω +Ω+
Coupled equations to be solved
Equations that have to be solved:
And similar equations of (∆0 , µ0) at T=0Self-consistent solution? … Dangerous!
Gapless-conserving dichotomy (Violation ofGoldstone theorem or Conservation law)
(1/ )
(1/
(0)
2 20 0
3(0)
2
)
1 0
13
N
NF
N
kNµ
δπ
δ
µ
∆= ∆=
⎧⎪∂Ω ∂⎪ + =⎪⎪ ∂Ω
Ω
∆ ∂∆⎪⎪⎨⎪∂Ω ∂⎪⎪ + = −⎪⎪ ∂ ∂⎪⎩
(TC , µC)
X Haussmann et al, PRA75 (2007) 023610X Strinati and Pieri, Europphys. Lett. 71 359 (2005)X T. Kita, J. Phys. Soc. Jpn. 75, 044603 (2006)
Order by order expansion (NS)
Want to solve:
Expand also
LO equation …
… NLO equation
(0) (1/ )1( ) ( ) ... 0NF x F xN
+ + =
(0(0) )( ) 0F x =
0
(0)(1/ )( )
01/1 1 ( ) 0
x
N
x
N dF xFN dx
xN=
+ =
(1 )(0) /1 ...NxxN
x= + +
no dangerous propagator inside!
Comparison with NSR
Nozieres-Schmitt-Rink theory
Missing 1/N correction to Thouless criterionSo no problem in solving two equations selfconsistently in (µ(kF), T(kF))1/N term in number eq. dominates in the strong coupling and recovers TC in BEC limitThe phase diagram in (µ, T)-plane unaffected: Only the bulk (µ, kF)-relations affected
2
(0)
(0)
03
20
(1/ )
( , ) 0
1( , ) ( , )3
C
FC C
N
T
kT T
Nµ µ
µ
µ µπ
δ
∆ ∆=
∆=
⎧⎪∂ =⎪⎪⎪⎪⎨⎪⎪∂ + ∂ = −⎪⎪⎪Ω
⎩
Ω
Ω
2(1/
0
)1 ( , )N CTNδ µ
∆ ∆=Ω+ ∂
The results: UnitarityNS VSR
Corrections are large! 1/N corrections to (βC, βCµC) and (TC, µC) formally equivalent at large N, but not at N=1!
5.3172.014
2.7851.504CC
F
C
EN
N
T
Tµ
⎧⎪⎪ = +⎪⎪⎪⎪⎨⎪⎪ = +⎪⎪⎪⎪⎩
1.3100.496
0.5801.747
C
C
F
F
E N
E N
T
µ
⎧⎪⎪ = −⎪⎪⎪⎪⎨⎪⎪ = −⎪⎪⎪⎪⎩
T=00
0 0.1630.686
0.3120.591
F
F
E N
E Nµ
⎧⎪⎪ = −⎪⎪⎪⎪⎨⎪⎪ = −⎪⎪⎪⎪⎩
∆
Results: TC, µC off the unitarity
1/N to βC (1/TC)
1/N to TC
NSR LO (MFA)
• Chemical potential in the BCS limit coincides with perturbativeresult: Reproduces second-order analytic formula
c.f. Fetter, Walecka’s textbook
1st
2nd
0.23
BEC BCS
( )224(11 2 log2)
15 Fk a
π−+
413 sFF
k aEµ
π= +
BEC BCS
Summary of 1/N to Nonrelativistic Fermi gas
Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion
TC useless at unitarity, even negative!Only qualitative conclusion: fluctuation lowers TC
On the other hand, β=1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7):E. Burovski et al., PRL96 (2006) 160402
Expansion about MF fails in molecular BEC regime
1/N expansion may still give useful prediction in the BCS region
Summary of 1/N to Nonrelativistic Fermi gas
Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion
TC useless at unitarity, even negative!Only qualitative conclusion: fluctuation lowers TC
On the other hand, β=1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7):E. Burovski et al., PRL96 (2006) 160402
Expansion about MF fails in molecular BEC regime
1/N expansion may still give useful prediction in the BCS region
Summary of 1/N to Nonrelativistic Fermi gas
Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion
TC useless at unitarity, even negative!Only qualitative conclusion: fluctuation lowers TC
On the other hand, β=1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7):E. Burovski et al., PRL96 (2006) 160402
Expansion about MF fails in molecular BEC regime
1/N expansion may still give useful prediction in the BCS region
Summary of 1/N to Nonrelativistic Fermi gas
Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion
TC useless at unitarity, even negative!Only qualitative conclusion: fluctuation lowers TC
On the other hand, β=1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7):E. Burovski et al., PRL96 (2006) 160402
Expansion about MF fails in molecular BEC regime
1/N expansion may still give useful prediction in the BCS region
Summary of 1/N to Nonrelativistic Fermi gas
Extrapolation to N=1 is troublesome: Final predictions depend on which observable is chosen to perform the expansion
TC useless at unitarity, even negative!Only qualitative conclusion: fluctuation lowers TC
On the other hand, β=1/TC-based extrapolation yields TC/EF=0.14, close to MC result 0.152(7):E. Burovski et al., PRL96 (2006) 160402
Expansion about MF fails in molecular BEC regime
1/N expansion may still give useful prediction in the BCS region
1/N expansion in dense, relativistic Fermi system: Color superconductivity
Impact of fluctuation on the (µ, Τ)-plane?Several strong coupling effects discussed:
M. Matsuzaki (00), Abuki-Hatsuda-Itakura (02), Kitazawa, Koide, Kunihiro, Nemoto (02)Kitazawa, Rischke, Shovkovy (08)
NSR scheme also applied: no change in (µ, Τ)-phase diagram:
Nishida-Abuki (05), Abuki (07)
Are fluctuation effects different for several pairing patterns? (2SC, CFL, etc…)
Lets try 1/N expansion!
1/N expansion in dense, relativistic Fermi system: Color superconductivity
take the simplest NJL (4-Fermi) model
Several species but with equal mass, equal chemical potentialqq pairing in total spin zero, positive parityFor color-flavor structure:
2SC:
CFL:
( ) ( )( )5 5014
BNC CGL q i m q q F q q F qα α
αµγ γ γ +
=
= ∂ + − + ∑
[ ] 1 ( , , )2
ij ija bc abcF ra g bε ε= =
1 (( ( , ),...,( , ))2
, )jk ijk
ai abcbcF u r sa i bε ε⎡ ⎤ = =⎣ ⎦
3 diquark “flavors”
9 diquark “flavors”
Introduce a new flavor “taste” of quarks q qA (A=1,2,3,…,N)
SU(3)C×(flavor group)×SO(N)Assume SO(N)-singlet pair field, then
No unwanted NG bosons. We make a systematic expansion in 1/N and set N=1 at the end of calculation
5( , ) 2a AC
AA
aGx q qN
Fφ τ γ= ∑
( ) ( )( )5 50 4 B BA A AC C
a aa
A
GL q i m q q F q q F qN
µγ γ γ += ∂ + − + ∑
1/N expansion to shift of TCInterested in shift of TC in (µ,T)-plane:Not interested in bulk (µ, ρ)-relation since density can not be controlled(µ is fundamental in equilibrium)
Approach from normal phase T TC +0
Then consider Thouless criterion alone
(0) 1 (1/ ) 11( 0) ( 0) 0NB BG P G PNµ µδ− −= + = =
Pair fluctuation becomes massless at TC
2 2
( 1/ )0) (( , ) ( , ) 0NN T Tδ µµ∆ ∆
Ω∂ Ω + ∂ =
1/N expansion to inverse boson propagator, NLO Thouless criterion
Boson propagator at LO:
NLO correction: Boson self energy
(0) 1 (1/ ) 11( 0) ( 0) 0NB BG P G PNµ µδ− −= + = =
Pµ
φaχpair : Cooperon
(1/ ) 11 ( 0)NBG PN µδ −= =
(0) 1( )1 ( )B
GN
G PG Pµ µχ
= ≡− pair
(0)( ) 1/BG Q Nµ ∼
Pµ=0
LO ∼ O(N) NLO ∼ O(1)
∼ O(1)
N∼vertex:
∼ O(1/N)
Color-Flavor structure of vertexes in loop gives
Information of color/flavor structure of pairing pattern condenses in simple algebraic factor NB/NF
NLO correction to boson self energy
0
(1/ ) 1 (0)
2
3 0+,
1 ( 0) ( ) ( )
( ) ( , , , ; )(2 )
NB B
Q nT
e e
B
efe f
F
G P dQNG Q F Q
dF Q I e f Q
NNN µ π
δ
ξ ξπ
−
=
=±
= =
≡ +
∑ ∫
∑ ∫ k k qk k k q
Tr BF
NF F F F
Nα γγδ β δ αβδ δ+ +⎡ ⎤ =⎢ ⎥⎣ ⎦∼γδ
α
(0)( )BG Qµ γδδ
β
Qµ
NLO fluctuation effect in CFL is twice as large as 2SC one
Mean field Tc’s split at NLO
Phase dependent algebraic factor
199CFL
1/2632SC
111“BCS”
NB/NFNFNBpairing
Caveat: NLO integral badly divergentTake the advantage of HDET (µ ∆)
Only degrees of freedom close to Fermi surface are relevant for pairing physicsRenormalize the bare coupling G in favor of mean field Tc(0) : TC(0)/µ parameterize the strength of coupling
In the weak coupling limit TC(0)=0.567∆0(0)
High density approximation
(1)2
(0)11 (0)
2B
F
CNLO
FB
NN N
fk
GTµ
δπ µ
− ⎛ ⎞ ⎛ ⎞⎟⎜ ⎟⎜ ⎟⎜ ⎟⎟⎜⎝ ⎠⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠
(0) 12 (0)(0) log2F
BC
kG N T
Tµπ
− ⎛ ⎞⎟⎜= ⎟⎜ ⎟⎟⎜⎝ ⎠∼O(N): textbook
material
∼ O(1)
f NLO
TC(0) /µ
Numerical result for NLO function(0)
(0)(0) (0) 1
CN
B
FLO
NN B
Tf
CC C C NL
N
FO
TT T e T
Nf
NNµ
µ
⎛ ⎞⎟⎜ ⎟⎜ ⎟− ⎜ ⎟⎜ ⎟⎟⎜⎜ ⎟⎝ ⎠⎛ ⎞⎛ ⎞⎟⎜ ⎟⎜ ⎟⎟⎜= ≈ − ⎜ ⎟⎟⎜ ⎜ ⎟⎟⎟⎜⎜ ⎟⎜ ⎝ ⎠⎝ ⎠
strong
Fluctuation suppressesTC significantly
Phenomenologicallyinteresting couplingregion
15-30% supressionof critical TCweak
Implication to QCD phase diagram
Suppression of TC is phase dependent: CFL TC is more suppressed than 2SC one
There may be afluctuation driven2SC windoweven when Ms=0
Suppression of TC is order of 15-30% :
Non-negligible!
T
µ
mu,d,s = 0
CFLNG
NOR
2SC
Tc(MF)
Summary of 2nd part
1/N expansion to TC at/off the unitarityAvoids problems with self-consistency
Only reliable when the NLO corrections are small (in BCS, not in molecular BEC region)
Color superconducting quark matterFluctuation corrections non-negligible
Suppressions of TC different according to pairing patterns Various phases!
OutlookImprovement: Color neutrality, etc.
Outlook (for 1st part of talk)General Ginzburg-Landau (GL) analysis
What if mu,d ms ? Neutrality condition?
Coupling with the density
Full NJL analysiswith flavor asymmetric matter with vector interaction & KMT term & neutralityPolyakov loop? PNJL
Nucleon (b-f) formation due to diquark-quark residual interaction
c.f. talk: T. Kunihiro 3/10, Zhang, Kunihiro (09)Zhang, Kunihiro, Fukushima (08)
c.f. Fukushima, PLB04, Ratti, Thaler, Weise, PRD06Roessner, Ratti, Weise, PRD07, Abuki, Ruggieri et al., PRD08
c.f. talk: T. Hatsuda 3/10, Maeda,, Baym, Hatsuda, PRL103 (09)