Based on

1
BCS to BEC transitionin Quark Matter
?
Hiroaki Abuki INFN, Bari

• Based on
• Y. Nishida and H. Abuki, PRD72, 096004 (05)
• and
• H. Abuki, arXivhep-ph/0605081
• (publication in process, NPA)

2
Plan

• Basic concepts of BCS/BEC crossover
• Can it be realized in relativistic superfluid?
• How can we describe it?
• Thermodynamics of relativistic BCS/BEC
  transition
• How does the dynamics of soft mode change?
• Implication to Quark matter and QGP
• Summary and Outlook

3
IntroductionWhats the BCS/BEC transition? Can
it be realized in Quark Matter?Relativistic
BCS/BEC crossover?
4
Whats the BCS/BEC transition
A question how large the critical
temperature can be in the strong coupling limit?
Eagles (1969), Leggett (1980) Nozieres
Schmitt-Rink (1985)
1957
1924
5
Whats the BCS/BEC transition
How to describe the smooth change from BCS to BEC?
T 0 Eagles (1969), Leggett (1980) T ? 0
Nozieres Schmitt-Rink (1985) NSR theory

• Thouless criterion (divergent fluctuation)
• Fixed number condition with Gaussian fluctuation

g
fluctuation loop
response function to pair fluctuation at one loop
(Cooperon)
6
Whats the BCS/BEC transition
??
??
??
7
Whats the BCS/BEC transition
Where it might be realized? Is it merely a
theoretical prediction?
Even in nuclear matter?
Isospin Mott transition from BCS n-p pairing
at high density to insulator BEC of deuteron
gas when density is decreased
Alm, Roepke, Schnell, Stein, Z. Phys. A 351,295
95 Lombardo, Nozieres, Schuck, Schulze and
Sedrakian, PRC 64, 01
deuteron EB 2.22 MeV in 3SD1 channel
as(3S1) 5.4 fm
8
In relativistic quark matter?
BCS-like
What is relativistic matter?

• kF/m
• 10-9 in atom system
• ?10-7 in 3He
• ?10-2 nuclear matter
• relativity parameter

BEC-like?
Color superconductivity
Hadrons
stronger / diluter BEC-like?? BCS
Natural question to be addressed is Whats the
BCS/BEC crossover in relativistic system? in the
relativistic limit m ? 0, fermions cannot
form bound states at least via a contact
attraction
9
PART IRelativistic NSR framework
Refs Nishida-Abuki, PRD72, 096004 (05) H. Abuki,
arXivhep-ph/0605081
10
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 to NSR, we write this in terms of phase
shift
11
Our approach (2) Nishida-Abuki, RRD72, 096004

• Thouless criterion (Gap equation)
• Fixed number condition in the gaussian
  approximation

We obtain (m, T ) as functions of (G, kF)
12
Our approach (3) Nishida-Abuki, RRD72, 096004
spectral function r (w, p m, T )
When G is strong enough, the bound state poles
appear in r
13
Note vacuum fluctuation
arXivhep-ph/0605081
?
Sum of boson zero point energy measured from
vacuum one
Quantum Casimir pressure due to in-medium
spectral shift! This is another source of
fluctuation specific in relativistic case!
14
Note vacuum fluctuation (2)
arXivhep-ph/0605081
Thermodynamic potential for charged relativistic
boson
Vacuum fluctuation zero point energy which is
usually thrown away because it has no (m, T)
dependence
15
Including Nc colors
arXivhep-ph/0605081
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 nothing to TC
in the BCS, while it lowers the BEC temperature
by factor 1/NC2/3 (normal BEC)
16
PART IIThermodynamics of Crossoverin a
relativistic superfluid
Refs Nishida-Abuki, PRD72, 096004 (05) H. Abuki,
arXivhep-ph/0605081
17
Results
(Nc3, NF2, fixed m/L 0.2, kF /m 0.2)
arXivhep-ph/0605081
??
BCS BEC
Tc/EF
mc/EF
g g

18
Whats the relativistic BEC?
Kapusta, PRD24, (81) Haber-Weldon, PRL46, (81)
boson
anti-boson
Net charge becomes maximum at mBmB
is decreasing function of T
TBEC determined by
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RBEC phase
1. Tc rapidly increases to the order of
EF Fairly approximated by relativistic TBEC

• Tc is well above
• boson mass (mB2m)
• Anti-bosons are
• excited

• Superdense system
• with antiparticles!

Recently confirmed also at T0 He, L and Zhuang,
P, hep-ph/0703042
20
Entropy vs. Coupling
BCS
BEC
RBEC
Entropy density/Ntot

• Mass and precursory
• pairing effects in BCS
• 2. Plateau in the BEC regime
• (increasing coupling can
• only affect internal structure)
• 3. Large for RBEC
• (a lot of degrees of freedom!)

g
21
Cooper pair wave functions
arXivhep-ph/0605081
BCS g -0.35
22
Pair size Phase coherence
arXivhep-ph/0605081
BCS
BEC
RBEC
Pair size (coherence length)
Lengths/Ntot1/3
Ginzburg-Landau healing length
Scattering length
Sa de Melo et al., PRL 71 (1993)
g
23
Spectral function in BEC
arXivhep-ph/0605081
TTc Stable boson pole and anti-boson pole
BEC g 0.5
r(w, 0)m2
TTdiss gtTc stable boson pole disappears! (anti-b
oson pole survives)
(w2m)/m
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Dissociation of bound state
BCS
BEC
RBEC
arXivhep-ph/0605081
pair dissociation
cf. J/y survival above Tdec. Asakawa-Hatsuda,
PRL92 (04) Datta et al., PRD69 (04)
Umeda-Nomura-Matsufuru, EPJC39S1 (05)
Gunji-Hamagaki-Hatusda-Hirano
hep-ph/0703061 Precursory modes above Tc
Hatsuda-Kunihiro, PRL55 (85)
Castorina-Nardulli-Zappala, PRD72 (05)
Temperature/EF
superfluid
g
25
Quantum fluctuation
arXivhep-ph/0605081
BCS
BEC
RBEC
Quantum fluctuation In BCS region is large!
Quantum fluctuation gives a negative contribution
to boson density
Tc/EF
So as to make the total density fixed, fermion
density, and Tc should increase
g
26
Density (kF/m) dependence
arXivhep-ph/0605081
BCS
BEC
RBEC
BCS/BEC boundary shifted to higher value of
chemical pot.
Crossover characteristics of Tc somewhat
smeared In high densities
Tc/Ntot
Pauli-blocking prevents the bound state formation
in medium!
Universal feature at the unitarity is lost by
kF/m!
g
27
Some speculative remarksMapping to QCD phase
diagram
cf. diquark (quasi) bound states above Tc is
relevant to small viscosity in sQGP by Shuryak
and Zahed PRC70, 021901, PRD70, 054507,04 ,etc
Plasmino mass
?
BEC criterion
preformed boson phase
Probable realization of BEC in
non-perturbativeregion of CSC
BCS
(R)BEC(ggt1)
Crossover regime from BCS to BEC is liquid
like? Gelman, Shuryak Zahed, arXivnucl-th/04100
67 T. Schafer, arXivcond-mat/0701251 Low
viscosity (h/s ? 1/3 small !)
28
Summary and Outlook

• Summary
• Developed the NSR formalism to relativistic model
• New source of fluctuation quantum fluctuation
• 2-step crossover BCS/BEC/RBEC in relativistic
  superfluid!
• In RBEC, all the degrees of freedom
  participate
• in thermodynamics. (q, qbar, b, bbar)
• In QCD phase diagram?
• Definite crossover observed in the dynamic
  equation
• TDGL eq. for diffusive mode in weak BCS
• Gross-Pitaevskii eq. for repulsive boson gas in
  BEC
• Relativistic Gross-Pitaevskii eq. in RBEC

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Summary and Outlook

• Future perspectives
• How can we go beyond the Gaussian approximation?
• Self-consistent T-matrix approach?
• (Baym-Kadanof approximation to Luttinger-Ward
  potential)
• Haussmann, PRB49 (94) Haussmann et al.,PRA75
  (07)
• Crossover in a gas with density imbalance
  (stress)
• Mannarelli-Nardulli-Ruggieri,
  arXivcond-mat/0604579
• How can we describe unitary regime in a
  systematic way?
• Nishida-Sons epsilon expansion
• Nicolic-Sachdevs 1/N-expansion of fermion
  system with many species
• How do fermions behave throughout the crossover?
• Kitazawa et al., PLB631,157, 05 pseudo gap?

30
PART IIICrossover in excitations?Dynamic
equations for soft mode?
Ref H. Abuki, arXivhep-ph/0605081
31
Beyond gauss approximation
Sa de Melo et al., PRL 71 (1993) 3202
Beyond the gauss approximation
Interaction between bosons
Thouless criterion tachyonic mode below Tc
Low energy expansion near Tc
d wave function renormalization (d is real in
the BEC region) c diffusion constant b boson
interaction parameter
32
Beyond gauss approximation
soft boson amplitude gives stronger fermion
coupling gets, Weaker repulsion between
bosons becomes (Duality)
4-body Schrodinger equation
(Petrov et al, Strinati et al.) Also similar
value obtained in renormalization group (Y.Ohashi)
33
Extension to relativistic case
arXivhep-ph/0605081
Effective action for soft mode up to quartic term
Gauss app.
Boson int.
Dynamic susceptibility Propagation of boson
(Kinetic part)
34
Change in soft mode property
arXivhep-ph/0605081
Low energy and long wavelength expansion We
find, for example
Simple expansion is allowed only for D (T ) ? w
? min(Tc, m-m)
Abrahams-Tsuneto, Phys. Rev. 152 (1966) Sa de
Melo et al., PRL 71 (1993)
35
Change in soft mode dynamics
Analytic formulas in the leading order in m/Tc
expansion Find In this limit, fluctuation
is purely diffusive as in Nonrela
Imaginary part can be neglected to this order due
to approximate particle-hole symmetry
36
Change in soft mode dynamics
Complex effective mass

• I diffusive mode
• I damped oscillation
• III propagating

Fluctuation decays with
37
Interacting soft modes
arXivhep-ph/0605081
Low energy and long wavelength expansion
Beyond Gauss app.
38
Interacting soft modes
arXivhep-ph/0605081
BCS
BEC
RBEC
Fermion scattering length monotonically decreases
boson-boson repulsive force becomes large when
the RBEC sets in
Boson scattering Length first decreases along
with as, and then increases
2
Its ratio becomes larger beyond the
nonrelativistic limit value 2
g
39
Dynamic equation in BCS
arXivhep-ph/0605081
Low energy coefficients in BCS
d is imaginary dominant, d2 is negligible
Rescaling the boson field by
TDGL equation for diffusive fluctuating pair mode
under Tc , long wavelength fluctuation about Y
0 becomes tachyonic
40
Dynamic equation in BEC
arXivhep-ph/0605081
Low energy coefficients in BEC
d is pure real, d2 is negligible
Then rescaling D by
The Gross-Pitaevskii (GP) equation
Sa de Melo et al., PRL 71 (1993)
chemical pot. mb goes to 0- as T?? Tc0 This
signals an instability to BEC formation!
41
Dynamic equation in RBEC
arXivhep-ph/0605081
Low energy coefficients in RBEC
d, d2 are pure real and and Tcd2 ? d
Then rescaling D by d2
Find the relativistic Gross-Pitaevskii (RGP)
equation
under Tc , boson mass squared becomes negative
42
Application of effective theory
arXivhep-ph/0605081
How soft modes contribute to the transport
property?
Within two-fluid model of superfluid, we have
shear viscosity h second viscosities (z1, z2,
z3) heat conductivity k etc..
X X X
bulk viscosity
Kinast-Turlapov-Thomas, PRA70 (04)
Damping rate of radial breathing mode of 6Li gas
in the optical trap seems to reach minimum near
the unitary
Son, arXivhep-ph/0511721
General coordinate invariance and conformal
invariance imply
z1 z2 0 a perfect liquid?
T. Schaefer, arXivcond-mat/0701251
h/s ? 1/3 small !
43
Shear viscosity from soft modes
arXivhep-ph/0605081
In weak BCS regime fluctuation is mostly
diffusive
We employ the classical Kubo formula (w/Tc ltlt 1)
Stress tensor arising from soft mode may be
defined by
under
44
Shear viscosity from soft modes
arXivhep-ph/0605081
Aslamasov-Larkin type diagram
Assuming the time dependence We find the
formula (Gauss app.) Where relaxation time
Aslamazov and Larkin, Soviet Solid State 10, 875
(68) Phys. Lett. 26A, 238 (68)
c.f. for some topics regarding this graph in 2CS,
see, Kitazawa et al., PTP114 (05)
45
Shear viscosity from soft modes
arXivhep-ph/0605081
In weak BEC regime fluctuation is propagating
mode (real boson) interacting via short range
repulsive force
Assuming soft mode distribution is dilute in
phase space, we evaluate it using the Boltzmann
equation
Thermal de-Broglie length
Quantum radius
The boson-boson s-wave cross section
Our assumption means
46
Shear viscosity from soft modes
arXivhep-ph/0605081
BCS
BEC
Fermion contribution
Fermion contribution
X s-wave unitary bound Pauli-blocking
Viewed as lower limit.
Soft modes
X Important near unitary X KSS bound is nearly
saturated at unitary
Kovtun, Son, Starinets, PRL 94, 111601 (05).
g
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BACKUP SLIDES
48
Tc , mc and number contents
arXivhep-ph/0605081
Tc/EF
N/Ntot
g g

49
Boson mass in medium
arXivhep-ph/0605081
BCS
BEC
RBEC
Anti boson are lighter than boson
Anti boson forms before the formation of boson
MB/2
Pauli-blocking effects Is less significant In
anti boson sector
g
50
GL parameters
arXivhep-ph/0605081
BCS
BEC
RBEC
GL parameters
g
51
Kinematical region
arXivhep-ph/0605081
BCS
BEC
RBEC
Limit frequency
GP/RGP boundary
w/m
g
52
Boltzmann gas or liquid?
arXivhep-ph/0605081
BCS
BEC
RBEC
Calculating the viscosity in the
Boltzmann equation is not good
Inter-particle distance is smaller than quantum
radius
Worse in RBEC because it is strongly coupled
dense system!
g
53
Gap vs. Coupling
54
GL coherence length
arXivhep-ph/0605081
Static fluctuation heals at a length
Ginzburg-Landau healing length
Also, anomalous heat capacity is described by
c.f. Kitazawa et al., PTP114 (05)
Voskresensky, PRC69 (04)
Moreover, this GL length scale determine dynamic
response to electric field
Aslamasov-Larkin, Sov. Phys. Solid State 10 (68)
55
Modified coupling
arXivhep-ph/0605081
Introducing the modified coupling
by Expanding on-shell T-matrix for 2-body
scattering in momentum, So at Low energy, GR
is related to scattering length Define
dimensionless coupling by
g dimensionless coupling with GC being the
critical coupling for zero binding
56
BEC critical temperature
Why is Tc in BEC constant?
Once 1/as crosses zero, there appears boson
with binding energy
These bosons with mB2m, NBN/2 condense into
zero mode below
The condensation has rather kinetic origin
thus Tc in BEC regime is independent
of coupling!
57
Kinematical dissociation
Whats the meaning of naïve Tc in BEC regime?
It can be re-written in terms of boson binding
energy (in vacuum) as
This is order of binding energy
Can it be interpreted as the pair dissociation
temperature? Yes. The chemical equilibrium
condition b ? f? f?
yields the same expression up to logarithmic
(entropic) correction