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Bose-Einstein condensation in relativistic quasi-chemical equilibrium system --- from color superconductivity to diqua

Bose-Einstein condensation in relativistic quasi-chemical equilibrium system --- from color superconductivity to diquark BEC ----. Nakano, Eiji (NTU). Contents: . QCD phase diagram Introduction to Color superconductivity (CSC)

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Bose-Einstein condensation in relativistic quasi-chemical equilibrium system --- from color superconductivity to diqua

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  1. Bose-Einstein condensation in relativistic quasi-chemical equilibrium system --- from color superconductivity to diquark BEC---- Nakano, Eiji (NTU) Contents: • QCD phase diagram • Introduction to Color superconductivity (CSC) • Pair fluctuation above Tc in strong coupling region (=low density) ・ 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) 4) Summary and outlook

  2. Pre-critical region of CSC • large pair fluctuations – Kitazawa, Kunihiro • Crossover from BCS to BEC – Nishida, Abuki 1) QCD phase diagram Hadronic excitations in QGP phase T • Soft mode of chiral transition - Hatsuda, Kunihiro. • qq quasi bound state - Shuryak, Zahed; Brown, Lee, Rho • Lattice simulations – Asakawa, Hatsuda; etc. 150~170MeV Chiral Symm. Broken Tc~100MeV Color Superconductivity(CSC) Hadrons r 0

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

  4. has a pole for arbitrary weak attractive interaction at T=0. = + O.P. : Cooper instability (T=0) 2-body problem in medium 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. (Phase transition to superconductor!) BSC state : superposition of two-particle occupied and absent states, there is no singularity (pole) any more. Break down of U(1) symmetry

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

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

  7. Nature of CSC Short coherence length x -1/3 N Large coherence of pair field Large quantum fluctuation ( to be Diquark composite) weak coupling strong coupling! Mean field approx. works well. There exists large quantum fluctuation of pair field above Tc.

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

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

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

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

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

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

  14. Fluctuation effect on Specific heat CV /107 free (BCS approx.) Tc from collecitve mode e Quark matter core Enhancement of cV ~e -1/2 above Tc. Abrupt delay of cooling in compact-star evolutions.

  15. Crossover to BEC T_pair fluc. Dissociation temp. Diquark-quark mixture Tc Diquark BEC Thouless Criterion Summarizing the points so far, T Pseudo Gap (pair fluc.) vanishes No Pair Fluctuation Pair Fluctuation develops ? CSC(= 2SC) CSC(= CFL) Weak Coupling (High density) Strong Coupling (Low density)

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

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

  18. Derivative of the phase shift in dilute limit: (= spectral function of T-matrix) :measured from 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, which shows a chemical equilibrium between two quark and diquark composite.

  19. 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 (These masses are originally determined from QCD.) For a fixed Baryon number N_B, gives Tc for DiquarkBEC. (This is nothing but Thouless criterion.)

  20. Application for QCD with (u, d) quark matter * Diquark molecules with 2SC-type paring state (color-3 , flavor-1 , total J=0) : chemical eq. q + q (qq) = D 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) F1, F2 , F3 , Composite-boson molecules with various channels ; B1, B2 , B3 , F+F B ( mB1 <mB2 <mB3 < ) BEC-singularity occurs onlyon the ground state of the most stable channel ( ) : mB1 B1 B * Diquark-BEC is ‘homogeneous’ (= no-coexisting state). c.f.Color-Superconductivity * Anti-diquark cannot be condensed into BEC with positive baryon number density ( ). d

  21. ( ) for B F+F One-BEC Theorem Multi-component fermionic matter ; (color, flavor, spin, etc) F1, F2 , F3 , Composite-boson molecules with various channels ; B1, B2 , B3 , ( mB1 <mB2 <mB3 < ) F+F B 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 ; * is free-energy for one particle. Chemical equilibrium condition. * If , system loses free-energy from fermionic degrees of freedom and gains free-energy from bosonic degrees of freedom. 2 Chemical eq. means the balancebetween these lose and gain of free-energy. (one chemical potential control the whole system).

  22. i can always be neglected for V . One-BEC Theorem positive norm condition. Shared chemical potential ( ) must be smaller than any ground state of boson spectra ; Constraint given byB1 is most severe ! If Tis lowered, will increase to maintain the conserved number density and finally saturate at . B1 bosons : BEC-singularity At thermodynamical limit ( V ), gives the macroscopic contributions. B bosons [B1-BEC] ( ) i : 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.

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

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

  25. (0) q + q D Mass Phase Diagram T > 0 0 finite finite T = 0 0 0 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 atT0 and condensed into the ground state. region 2( resonant state case ; ) Small - ( loss of resonance energy is small ). BEC-phase Large - ( loss of resonance energy is large ). loss of kinetic energy with Pauli-blocking. loss of resonance energy ; no BEC-phase

  26. 『』 『』 Tc 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 . : no-thermal part. If Allthe conserved baryon number density are bound into diquarks and condensed into the ground state. 1) Non-relativistic RPA gives the saturation ofTc 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.

  27. m Quark Mass & Diquark Mass q m d < Lattice-QCD simulation for = 0 : T.Celik, et.al., Nucl. Phys. B256. 670 (1985) > 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) > Case 1 : remainig Chiral sym. breaking Case 2 : Chiral sym. restored Deconfinement phase transition Deconfinement phase transition 150 MeV 150 MeV Chiral symmetry breaking Chiral symmetry breaking 0 0

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

  29. ‘ Saturation of μ’ * ‘Dissociation’ of diquark-molecules is strongly suppressed. Density Profile No anti-particle case. (0) Diquarks will condense into the ground state ( D ) below Tc (2nd-order phase transition). *

  30. 2 High-T Region of Density Profile 2 1 1 Compositeness : T , Symmetry : = Statistics : 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). At least, we might have to introduce a energy cut-off of O(B.E.) in diquark density.

  31. T T 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. BEC c Effect of mlowersTc, up to freeze-out. * 4 -transition of liquid He, =2.17K (c.f. =3.1K). P.Gruter D.Ceperley, and F.Laloe, PRL 79,3549(1997). Effect of ‘density homogenization’ rises upTc by ~10%. H.T.C.Stoof, PRA 45, 8398(1992). G.Baym, J.P.Blaizot, PRL 83,1703(1999). * Effect of μ does not changeTc 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). * 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).

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

  33. L eff n d n d Effect of Diquark Interactions Effective Lagrangian * ; color-3 diquak 3-component vector field * 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) Interaction energy : HI MF approximation , Single particle energy spectra of diquark :

  34. * Diquark number density : * -Renormalization * free * Quasi-chemical eq. theory (free-q and free-D) * + -renormalization ; * * -renormalization does not changeTc 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. Isolated quarks feel the effect ofO( ) . chemical equilibrium q+q D

  35. 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 (?) *

  36. Diquark-BEC is ‘homogeneous’ (= no-coexisting state). Anti-diquark cannot condense into BEC with positive baryon number density. 『』 Tc (strong coupling limit) gives ; relativistic effect . Tc ~ 100MeV ; comparative with Tc of Color Superconductivity. BEC ‘Dissociation’ is strongly suppressed with pair creation. Quantum statistics still remains for T with pair creation. Residual diquark-diquark interaction lowersTc by ~50%. ( less applicability of Gaussian-type approximation ?) 『』 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.

  37. Summary We viewed the quark-pair correlation (fluctuation) at finite density from weak (high density) to strong (low density) regimes. weak 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 strong Outlook 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.

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

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

  40. Diquark Coupling Dependence stronger diquark coupling GC ×1.3 ×1.5 m= 400 MeV e=0.01 GC

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

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