Bose-Einstein condensation
in relativistic quasi-chemical equilibrium system
--- from color superconductivity to diquark BEC---- QCD phase diagram
Introduction to Color superconductivity (CSC)
Pair fluctuation above Tc in strong coupling region (=low density) Contents: ・ Cooper instability in quark matter and BCS theory
・ Patterns of symmetry breaking in CSC ・ Effects of quark-pair fluctuation above Tc ---Pseudo Gap, specific heat,…
・ Diquark formation and its Bose-Einstein Condensation (BEC) Nakano, Eiji (NTU) 4) Summary and outlook

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1) QCD phase diagram 150~170MeV Color Superconductivity(CSC) Hadrons T Chiral Symm.
Broken 0 Tc~100MeV Hadronic excitations in QGP phase Soft mode of chiral transition - Hatsuda, Kunihiro.
qq quasi bound state - Shuryak, Zahed; Brown, Lee, Rho
Lattice simulations – Asakawa, Hatsuda; etc. r

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2) Introduction to Color superconductivity (CSC) Basic concept of CSC is quite similar to BCS theory : Electron-phonon system Quark-gluon system phonon Attractive interaction comes from
background lattice vibration, phonon,
Not from the gauge field. Attractive interaction exists
in the elementary level,
exchange of gauge bosons. Attractive interaction causes an instability of fermion many-body system.

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Cooper instability (T=0) T-matrix (two-particle collective mode near F.S.) The existence of the pole does not imply the bound state of two fermions,
but instability of normal phase against the two-particle collective excitation
with zero energy ----- a condensation of pair field. BSC state : superposition of two-particle occupied and absent states,
there is no singularity (pole) any more.
Break down of U(1) symmetry (Phase transition to superconductor!) 2-body problem in medium

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Attractive channels in quark matter Quark-quark interaction is mediated by gluons,
which has attractive channels for color anti-symmetric quark pairs. flavor and spin are determined
so as to antisymmetrize the two-quark state: Flavor anti-symmetric, r Hadron 2SC phase CFL phase T Order parameter of CSC: 2nd order 1st order Similar to
standard model
Higgs diquark

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・ Symmetry breaking pattern and RG analysis shows IR stable fixed point.
・ GL analysis shows Type-II superconductivity (fluctuation of gluon is negligible) Critical phenomena of 2SC = 2nd order phase transition,
because Determination of Tc : Thouless Criterion 2nd order Thus, one can employ the Thouless criterion for 2nd order phase transition:
Singularity of T-matrix at finite T gives Tc. at Tc Thermodynamic
Potential D W(D) Dominance of pair fluctuation

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strong coupling! Mean field approx.
works well. Nature of CSC There exists large quantum fluctuation of pair field above Tc. Large quantum fluctuation ( to be Diquark composite) Large coherence of pair field weak coupling

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3) Pair fluctuation above Tc
in Strong coupling region (= low density region) Study on pair fluctuation above Tc (by Kitazawa, Kunihiro)
1) Appearance of Pseudo-Gap
2) Precursory phenomena---
heat capacity, electric conductivity

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e.g, T-matrix Approximation in NJL model In Random Phase Approximation,
(Kitazawa, Kunihiro, PRD2003) T-matrix (pair collective mode) : Gc

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Quarks in BCS Theory (below Tc) Characterized by finite O.P. : Gap opens around the Fermi surface! Gap function =

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Pair fluctuation effect (above Tc) Quasi-particle energy: Dispersion relation: Density of State: Free quark Pseudo Gap

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(Quasi) Level repulsion of spectrum GC=4.67GeV-2 Fluctuation causes a virtual mixing
between quarks and holes w k w nf (w) kF hole paritcle

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The pseudogap survives up to e =0.05~0.1 ( 5~10% above TC ). Numerical Result : Density of State

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Enhancement of cV ~e -1/2 above Tc. Fluctuation effect on Specific heat Abrupt delay of cooling in compact-star evolutions. Quark matter core

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Summarizing the points so far, T Weak Coupling
(High density) Strong Coupling
(Low density) CSC(= CFL) Pair Fluctuation develops No Pair Fluctuation CSC(= 2SC) Pseudo Gap (pair fluc.)
vanishes ?

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T baryon Confinement
Phase Quark Fermi-degeneracy. Attractive channel. + (color-3 , flavor-1 , total J=0) SU(3) * SU(2) c c 2-flavor Color
Superconductivity (2SC) Large quark-pair fluctuation
with asymptotic freedom. Bose statistics of diquark. Diquark Bose-Einstein
Condensation loosely-bound
Cooper pair tightly-bound
diquark cluster BEC-BCS crossover QGP Quasi-Chemical Equilibrium Theory. Properties of diquark-BEC

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Contribution from Pair fluctuation (Diquark propagator) Free quark part We obtain the equation for the Baryon number density: : Bose distribution : Fermi distribution

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Derivative of the phase shift in dilute limit: For sufficiently large coupling,
there appear resonant or bound states below the Fermi Sea
in addition to scattering states near the Fermi Sea. Thus the Baryon number density becomes, :measured from which shows a chemical equilibrium between two quark and diquark composite. (= spectral function of T-matrix)

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From the above argument, we reached an ancient approach to superconductivity:
Quasi Chemical Equilibrium Theory (QCET) ( Schafroth, Butler, Blatt, 1956)
which is revived as a strong coupling theory of CSC. The number conservation: Chemical equilibrium between quark and diquark: We have only two parameters,
constituent quark mass :
and
diquark-composite mass : : Diquark as resonant state : Diquark as bound state For a fixed Baryon number N_B, gives Tc for Diquark BEC.
(This is nothing but Thouless criterion.) (These masses are originally determined from QCD.)

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Application for QCD with (u, d) quark matter Diquark molecules with 2SC-type paring state (color-3 , flavor-1 , total J=0) : * q + q (qq) = D chemical eq. 2SC Other less attractive quark-channels (color-3 , flavor-3 , total J=1)
has been recently suggested. * E.Nakano, et.al.,PRD 68,105001(2003) D.H.Rischke, et.al.,PRD 69,094017(2004) One-BEC theorem Multi-component fermionic matter ; (color, flavor, spin, etc) Composite-boson molecules with various channels ; BEC-singularity occurs only on the ground state
of the most stable channel ( ) : F1, F2 , F3 , B1, B2 , B3 , F+F B ( mB1 < mB2 < mB3 < ) B1 B mB1 Diquark-BEC is ‘homogeneous’ (= no-coexisting state). c.f. Color-Superconductivity * * Anti-diquark cannot be condensed into BEC with positive baryon number density ( ). d

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One-BEC Theorem Multi-component fermionic matter ; (color, flavor, spin, etc) Composite-boson molecules with various channels ; F1, F2 , F3 , B1, B2 , B3 , F+F B ( mB1 < mB2 < mB3 < ) Total fermion number conservation. A composite boson is constructed by 2 fermions ( 2-body correlations are included in the theory ). Helmholtz free-energy density with above constraint ; Minimum condition of free-energy ( , ) gives ; Chemical equilibrium condition. 2 2 is free-energy for one particle. If , system
loses free-energy from fermionic degrees of freedom and
gains free-energy from bosonic degrees of freedom. Chemical eq. means the balance between
these lose and gain of free-energy. (one chemical potential control the whole system). * *

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positive norm condition. One-BEC Theorem Shared chemical potential ( ) must be smaller than any ground state of boson spectra ; Constraint given by B1 is most severe ! B1 bosons B bosons i ( ) If T is lowered, will increase to maintain the conserved number density
and finally saturate at . : BEC-singularity At thermodynamical limit ( V ), gives the macroscopic contributions. [B1-BEC] : no BEC-singularity The lightest composite bosons can only be condensed
to the BEC states (one-BEC theorem). Bose-Einstein condensation occurs only on a ground state of whole boson spectra in the system.

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Diquark Bose-Einstein Condensation Diquark molecules with 2SC-type paring state (color-3 , flavor-1 , total J=0) : * q + q (qq) = D chemical eq. 2SC 1) Total baryon number conservations law : 2) Chemical equilibrium condition : c f s color-3 , flavor-1 , J=0 Composite-factor Environmental parameters : Mass parameters :

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Critical Temperature with Various Mass Values ; Tc * will increase with No-BEC phase will decrease with * “many-body effect” of BEC. Mass phase diagram determines the occurrence of BEC
at a certain temperature for various mass values ; . *

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Mass Phase Diagram T > 0 T = 0 0 0 0 finite finite 0 Single bose gas case. region 1 ( bound state case ; ) BEC-phase With the manifest advantage of binding energy,
all quarks are combined into diquarks at T 0
and condensed into the ground state. loss of kinetic energy
with Pauli-blocking. loss of resonance energy ; Small - ( loss of resonance energy is small ). Large - ( loss of resonance energy is large ). BEC-phase no BEC-phase region 2 ( resonant state case ; )

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Critical Temperature with Various Mass Values ; * with fixed corresponds to strong coupling limit. (Strong interaction may change the mass of composites with relativistic-energy scale.) (strong coupling limit) gives . 『 』 * If : no-thermal part. All the conserved baryon number density are bound into diquarks
and condensed into the ground state. 1) Non-relativistic RPA gives the saturation of Tc at the strong coupling limit. c.f. P.Nozieres and S.Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985) 2) Photon has no BEC.

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Deconfinement phase transition and Chiral phase transition
occurs at the same point. Chiral symmetry restoration cannot precede deconfinement. < Instanton-induced interaction : E.V.Shuryak, Nucl. Phys. B203, 140 (1982) > < QCD sum rules : A.I.Bochkarev, et.al., Nucl. Phys. B268, 220 (1986) > These 2 are same phase transition. < glueball-sigma mixing : Y.Hatta, et.al., Phys. Rev. D69, 097502 (2004) > * 150 MeV 150 MeV 0 0 Case 1 : remainig Chiral sym. breaking Case 2 : Chiral sym. restored Deconfinement phase transition * * Chiral
symmetry breaking Chiral
symmetry breaking Deconfinement phase transition

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QCD Phase Diagram * Tc ~ 100MeV : comparative with Tc of Color Superconductivity. c.f. K.Rajagopal and F. Wilczek, hep-ph/0011333 (2000) * Tc < Tc case1 case2 with light diquark mass . case2 Case 1 : remainig Chiral sym. breaking Case 2 : Chiral sym. restored diquark-BEC diquark-BEC * Current quark mass in case 2 is too small relative to the energy scale of diquark-BEC.

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Density Profile * Diquarks will condense into the ground state ( D )
below Tc (2nd-order phase transition). (0) No anti-particle case.

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High-T Region of Density Profile 2 1 1 T * Quantum Statistics (Fermi or Bose) gets more important
for high-T region with pair creation. Boltzmann statistics only appears
around moderate temperature region. There is no dissociation for both meaning of baryon number density
and particle (anti-particle) number density, without following effects in QCET, * 1) Asymptotic freedom 2) Medium effect (Pauli-blocking). = Compositeness : Symmetry : , Statistics : At least, we might have to introduce a energy cut-off of O(B.E.) in diquark density.

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Effect of Diquark Interactions Diquarks are colored objects (color-3 ), not singlet. Diquarks can scatter into different states through the residual interaction (gluon-exchange). Strong-coupling limit may not correspond to free bose gas,
but (strongly) correlated bose gas system in QCD (?) * * T is very sensitive for the residual interactions between bosons
in general BEC study. * c BEC Effect of diquark interactions is not included
in Gaussian-type approximation like RPA. * c.f. P.Nozieres and S.Schmitt-Rink, J. Low Temp. Phys. 59, 195 (1985). Effect of m lowers Tc, up to freeze-out. -transition of liquid He, =2.17K (c.f. =3.1K). * Effect of ‘density homogenization’ rises up Tc by ~10%. P.Gruter D.Ceperley, and F.Laloe, PRL 79,3549(1997). H.T.C.Stoof, PRA 45, 8398(1992). G.Baym, J.P.Blaizot, PRL 83,1703(1999). Effect of μ does not change Tc at all in single bose gas case. * A.L.Fetter and J.D.Walecka, Quantum Theory of
Many-Particle System (McGraw-Hill, New York, 1971). 4

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Phase diagram obtained from QCET (This effect never appears
in T-matrix appr. (=RPA) ) Tc~100 MeV for CSC
Tc~ 30 MeV for BEC

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Effect of Diquark Interactions 3-component vector field ; color-3 diquak * * Contact -term describes the diquark-diquark scattering effect. * Gross-Pitaevski approach Higher-order scattering terms ( ) are renormalized into
two-body interaction, as usual in nucleon case. J.D.Jackson, Annu. Rev. Part. Sci. 33, 105 (1983) Effective Lagrangian Interaction energy : HI MF approximation , Single particle energy spectra of diquark :

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-Renormalization * * * -renormalization does not change Tc of BEC at all
in single Bose gas case. A.L.Fetter and J.D.Walecka, Quantum Theory of Many-Particle Systerms All the information about interaction
is fully lost in BEC condition. * * -renormalization gives the leading order of interaction effect
in equilibrium system. chemical equilibrium q+q D Isolated quarks feel the effect of O( ) . Diquark number density : * * Quasi-chemical eq. theory (free-q and free-D) * + -renormalization ; free

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Tc with Diquark Interaction Residual interaction between color-3 diquarks is estimated
from the mass-difference of nucleon and with the assumption of quark-hadron continuity : * : repulsive J.F.Donoghue and K.S.Sateesh, Phys. Rev. D38, 360 (1988) The positiveness of is also suggested by using P-matrix method. R.L.Jaffe and F.E.Low, Phys. Rev. D19. 2105 (1979) Residual diquark-diquark interaction will lower the Tc of diquark-BEC
by ~50% from that in non-interacting case. * Gaussian-type approximation like Nozieres-Schmitt-Rink approach
may not be able to describe the strong-coupling region in QCD ; diquark-BEC (?) * ` ~

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Summary Diquark Bose-Einstein condensation is investigated with
Quasi-Chemical Equilibrium theory. Future Work The effect of 3-body correlations (q-D, q-q-q)
for the phenomena of 2-body clustering matter.

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Summary We viewed the quark-pair correlation (fluctuation) at finite density
from weak (high density) to strong (low density) regimes. Outlook weak strong 1) Color superconductivity
2) Pair fluctuation develops above Tc
・Pseudo gap phenomena
・Enhance of specific heat
3) Formation of Quasistable diquarks (= quantum fluctuation)
・Crossover to Diquark BEC Observable consequences in experiments or in astrophysical observations,
e.g., effects on dilepton or neutrino production rate,
and response to external magnetic field. I thank Mr. Nawa (Dept. of Phys. in Kyoto Univ.) for his close collaborations.

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High Tc ! Feshback resonance scattering Observation of di-fermion BEC Interaction strength can be controlled artificially! Fermion Atoms in trapping potential.

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Softening of Pair Fluctuations Dynamical Structure Factor e =0.05 The peak grows from e ~ 0.2
electric SC：e ~ 0.005 m= 400 MeV m= 400 MeV Pole of Collective Mode pole: The pole approaches the origin
as T is lowered toward Tc. (the soft-mode of the CSC)

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Diquark Coupling Dependence GC m= 400 MeV
e=0.01

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Numerical results in QCETa The explicit form of the equation Dispersion of quark and diquark are given by : where Upper bound of :

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