1. Fluctuation effect in relativistic BCS-BEC Crossover Jian Deng, Department of Modern Physics, USTC 2008, 7, 12 @ QCD workshop, Hefei  Introduction  Boson-fermion model for BCS-BEC crossover  beyond MFA, fluctuation effect  Discussions and outlooks J. Deng, A. Schmitt, Q. Wang, Phys.Rev.D76:034013,2007 J. Deng, J.-C. Wang, Q. Wang, arXiv:0803.4360 J. Deng et al., in preparation

2. BCS-BEC Crossover

3. Relativistic case I Color superconductivity in neutron stars

4. Possible strong coupling Quark Gluon Plasma In Relativistic Heavy Ion Collisions Relativistic case II See e.g. Braun-Munzinger, Wambach, 2008 (review) Ruester,Werth,Buballa, Shovkovy,Rischke,2005 Fukushima, Kouvaris, Rajagopal, 2005 Blaschke, Fredriksson, Grigorian, Oztas, Sandin, 2005

5. Recent works by other group: Nishida & Abuki, PRD 2007 -- NJL approach Abuki, NPA 2007 – Static and Dynamic properties Sun, He & Zhuang, PRD 2007 – NJL approach He & Zhuang, PRD 2007 – Beyond mean field Kitazawa, Rischke & Shovkovy, arXiv:0709.2235v1 – NJL+phase diagram Brauner, arXiv:0803.2422 – Collective excitations Relativistic BCS-BEC crossover for quark-quark pairing

6. Boson-fermion model: setting up Global U(1) symmetry: conserved current

7. Phase diagram Non-relativistic Relativistic Shadowed region stand for unstable solutions, which will collapse to LOFF state or separating phase Wilfgang Ketterle (MIT) arXiv:0805.0623 Realization of a strongly interacting Bose-Fermi mixture from a two-component Fermi gas

8. Beyond MFA

9. The fluctuation of condensate sets in Higgs and Nambu-Goldstone fields:

10. The CJT formalism (J. M. Cornwall, R. Jackiw and E. Tomboulis, 1974 ) Full propagator: Tree-level propagator:

11. 2PI diagrams and DS equations

12. Pseudo-gap

13. First order phase transition with fixed chemical potential Introduction of term in : B.I.Halperin, T.C.Lubensky and S. Ma 1974 (magnetic field fluctuations) I. Giannakis, D. f. Hou, H. c. Ren and D. H. Rischke, 2004 ( Gauge Field Fluctuations) Sasaki, Friman, Redlich, 2007 (baryon number fluctuation in 1st chiral phase transition)

14. gap and density equations

15. At small T The results are similar to the MFA results

16. At T=Tc Fluctuations become important in BEC regime. In BEC regime T*>Tc.

17. T-dependence The fluctuation effects become larger. BEC criterion is related to the minimization of the thermodynamics potential.

18. Summary 1. Relativistic boson-fermion model can well describe the BCS- BEC crossover within or beyond MFA. 2.As an fluctuation effect, the pseudo-gap become more important for larger temperature. 3.Fluctuation changes the phase transition to be first-order.

19. Outlook Full self-consistency is needed. BEC criterion for interacting bosons need more close look. Anti-particles and finite size of bosons should be considered carefully Our model can be extended to discuss quarkoynic continuity with finite chemical potential where the confinement and chiral symmetry breaking are not coincide (L. Mclerran and R. D. Pisarski ).

20. Thanks a lot

21. BEC: condensate, number density conservation, critical temperature Distribution function: Density conservation: Thermal bosons at most: Temperature dependence:

22. Boson-fermion model (MFA) With bosonic and fermionic degrees of freedom and their coupling, but neglect the coupling of thermal bosons and fermions as Mean Field Approximation

23. Pairing with imbalance population

24. Fermi surface topologies

25. Approximation

26. Ensure the reliability of gap equation Continuous changing of gap with fixed number density

27. But still first-order phase transition