Bose-Einstein condensation Bose-Einstein condensation in relativistic quasi-chemical equilibrium system --- from color superconductivity to diquark BEC from color superconductivity to diquark BEC )QCD phase diagram 2)Introduction to Color superconductivity (CSC) 3)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

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. Pre-critical region of CSC large pair fluctuations – Kitazawa, Kunihiro Crossover from BCS to BEC – Nishida, Abuki 

2) Introduction to Color superconductivity (CSC) Basic concept of CSC is quite similar to BCS theory : Electron-phonon system Quark-gluon system gluon 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.

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!) O.P. : 2-body problem in medium has a pole for arbitrary weak attractive interaction at T=0.

Attractive channels in quark matter [ ３ ] c ×[ ３ ] c ＝ [ ３ ] c ＋ [ ６ ] c gluon 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,  Hadron 2SC phase CFL phase T Order parameter of CSC: 2-flavor case, 2SC phase 3-flavor case, CFL phase 2 nd order 1 st order Similar to standard model Higgs diquark

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

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 Short coherence length  N -1/3

3) Pair fluctuation above Tc in Strong coupling region (= low density region) 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 Hadron 2SC phase  T Tc This region

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

Quasi-particle energy: Density of State: Quarks in BCS Theory (below Tc) Characterized by finite O.P. : Gap opens around the Fermi surface! Gap function =

Pair fluctuation effect (above Tc) Quasi-particle energy: Dispersion relation: ×1.5 F.S. Density of State: Spectral function: Characterized by zero O.P. : Density of State: Free quark Pseudo Gap

(Quasi) Level repulsion of spectrum G C =4.67GeV -2 Fluctuation causes a virtual mixing between quarks and holes  k  n f (  ) kFkF hole paritcle

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

Enhancement of c V ~  -1/2 above T c.  C V /10 7 TcTc free (BCS approx.) from collecitve mode Fluctuation effect on Specific heat Abrupt delay of cooling in compact-star evolutions. Quark matter core

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 Thouless Criterion Tc Crossover to BEC Diquark BEC Dissociation temp. Diquark-quark mixture T _pair fluc. ?

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. diquark-BEC Properties of diquark-BEC Critical temperature ( T c). Density profile. Residual Interaction between diquaks.

Contribution from Pair fluctuation (Diquark propagator) Free quark part We obtain the equation for the Baryon number density: : Bose distribution: Fermi distribution : scattering phase shift defined by Thermodynamics with pair fluctuation

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)

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.)

Application for QCD with (u, d) quark matter 2SC 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 ; only BEC-singularity occurs only on the ground state of the most stable channel ( ) : F 1, F 2, F 3, B 1, B 2, B 3, F+F B ( m B 1 < m B 2 < m B 3 < ) B1B1 B mB1mB1 Diquark-BEC is ‘homogeneous’ (= no-coexisting state). c.f. Color-Superconductivity * * Anti-diquark cannot be condensed into BEC with positive baryon number density ( ). d

One-BEC Theorem Multi-component fermionic matter ; (color, flavor, spin, etc) Composite-boson molecules with various channels ; F 1, F 2, F 3, B 1, B 2, B 3, F+F B ( m B 1 < m B 2 < m B 3 < ) Total fermion number conservation. 2 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 ; for( ) 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. F+F B Chemical eq. means the balance between these lose and gain of free-energy. (one chemical potential control the whole system). * *

positive norm condition. One-BEC Theorem Shared chemical potential ( ) must be smaller than any ground state of boson spectra ; Constraint given by B 1 is most severe ! B 1 bosons B bosons i i ( ) If T is lowered, will increase to maintain the conserved number density saturate and finally saturate at. BEC-singularity : BEC-singularity At thermodynamical limit ( V ), gives the macroscopic contributions. B 1 -BEC [B 1 -BEC ] BEC-singularity : no BEC-singularity can always be neglected for V. 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.

Diquark Bose-Einstein Condensation 2SC 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 :

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

Mass Phase Diagram T > 0 T = 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. q + q D (0) 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 ; )

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. 『 』 * TcTc 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 T c 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.

Deconfinement phase transition and Chiral phase transition occurs at the same point. Chiral symmetry restoration cannot precede deconfinement. These 2 are same phase transition. * 150 MeV 0 0 Case 1 : remainig Chiral sym. breakingCase 2 : Chiral sym. restored Deconfinement phase transition Quark Mass & Diquark Mass m q m d * * Chiral symmetry breaking Chiral Deconfinement phase transition

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

Density Profile * Diquarks will condense into the ground state ( D ) below T c (2 nd -order phase transition). (0) ‘ Saturation of μ ’ * ‘Dissociation’ of diquark-molecules is strongly suppressed. No anti-particle case.

High-T Region of Density Profile 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.

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 T c, up to freeze-out. -transition of liquid He, =2.17K (c.f. =3.1K). * Effect of ‘density homogenization’ rises up T c 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 T c 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 T T

Phase diagram obtained from QCET ( This effect never appears in T-matrix appr. (=RPA) ) We expect decrease of Tc due to the quark-pair fluctuation and diquark-diquark interaction. (=Diquark composite) 0 90 MeV BEC Hadron Phase Tc~100 MeV for CSC Tc~ 30 MeV for BEC Turn on diquark-diquark int. enhances Diquark mass. (decreases Tc)

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 : HIHI L eff MF approximation d n d n, Single particle energy spectra of diquark :

-Renormalization * * * -renormalization does not change T c 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

T c 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. D (1979) Residual diquark-diquark interaction will lower the T c 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 (?) * ` ~

Summary Diquark Bose-Einstein condensation is investigated with Quasi-Chemical Equilibrium theory. Diquark-BEC is ‘homogeneous’ (= no-coexisting state). Anti-diquark cannot condense into BEC with positive baryon number density. (strong coupling limit) gives ; relativistic effect. 『 』 TcTc T c ~ 100MeV ; comparative with T c of Color Superconductivity. BEC ‘Dissociation’ is strongly suppressed with pair creation. Quantum statistics still remains for T with pair creation. Residual diquark-diquark interaction lowers T c by ~50%. ( less applicability of Gaussian-type approximation ?) Future Work The effect of 3-body correlations ( q-D, q-q-q ) for the phenomena of 2-body clustering matter.

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.

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

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

stronger diquark coupling G C Diquark Coupling Dependence GCGC ×1.3×1.5  = 400 MeV  =0.01

Numerical results in QCETa The explicit form of the equation Dispersion of quark and diquark are given by : where Upper bound of : RHS 0