Fluctuation effect in relativistic BCS-BEC Crossover Jian Deng, Department of Modern Physics, USTC 2008, 7, 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: J. Deng et al., in preparation

BCS-BEC Crossover

Relativistic case I Color superconductivity in neutron stars

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

Recent works by other group: Nishida & Abuki, PRD 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: v1 – NJL+phase diagram Brauner, arXiv: – Collective excitations Relativistic BCS-BEC crossover for quark-quark pairing

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

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

Beyond MFA

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

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

2PI diagrams and DS equations

Pseudo-gap

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)

gap and density equations

At small T The results are similar to the MFA results

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

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

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.

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 ).

Thanks a lot

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

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

Pairing with imbalance population

Fermi surface topologies

Approximation

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

But still first-order phase transition