BEC-BCS crossover in NJL model; towards QCD phase diagram ... · BCS-BEC crossover zChiral field...
Transcript of BEC-BCS crossover in NJL model; towards QCD phase diagram ... · BCS-BEC crossover zChiral field...
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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
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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)
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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)
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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)
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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
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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
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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!
.
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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
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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
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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)
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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±→ → −
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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
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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)
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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)
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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)
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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
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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
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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
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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
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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
+ +
+ +
− − −
+ + +
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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
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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)
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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)
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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
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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)
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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
=∑∼
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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+
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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)
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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!
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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+ ∂
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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
⎧⎪⎪ = −⎪⎪⎪⎪⎨⎪⎪ = −⎪⎪⎪⎪⎩
∆
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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
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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
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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
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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
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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!
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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”
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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 += ∂ + − + ∑
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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 + ∂ =
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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
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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
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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
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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)
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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.
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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)