Fluctuation effect in relativistic BCS - BEC Crossover

1. Introduction • Boson-fermion model for BCS-BEC crossover • beyond MFA, fluctuation effect • Discussions and outlooks Fluctuation effect in relativistic BCS-BEC Crossover Jian Deng, Department of Modern Physics, USTC 2008, 7, 12 @ QCD workshop, Hefei • 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. 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 Possible strong coupling Quark Gluon PlasmaIn Relativistic Heavy Ion Collisions

5. Relativistic BCS-BEC crossover for quark-quark pairing • 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

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 phaseWilfgang Ketterle (MIT) arXiv:0805.0623 Realization of a strongly interacting Bose-Fermi mixture from a two-component Fermi gas

8. BeyondMFA

9. The fluctuation of condensate sets in Higgs andNambu-Goldstonefields:

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

27. But still first-order phase transition