Relativistic BCS-BEC Crossover at Quark Level

Abstract. The non-relativistic G0G formalism of BCS-BEC crossover at finite temperature is extended to relativistic fermion systems. The theory recovers the BCS mean field approximation at zero temperature and the non-relativistic results in a proper limit. For massive fermions, when the coupling strength increases, there exist two crossovers from the weak coupling BCS superfluid to the non-relativistic BEC state and then to the relativistic BEC state. For color superconductivity at moderate baryon density, the matter is in the BCS-BEC crossover region, and the behavior of the pseudogap is quite similar to that found in high temperature superconductors.


3) Fluctuations at T 3) Fluctuations at T 0 0 the the Landau mean field theory
Landau mean field theory is a good approximation only at T=0 is a good approximation only at T=0 where there is no thermal excitation, where there is no thermal excitation, one has to go beyond the mean one has to go beyond the mean field at finite temperature.field at finite temperature.mf pg mf pg t q t q t q t q i q T ig t q g q k i q T rGk G q k 2 full pair propagator full pair propagator

G G mean field theory in scheme mean field theory in scheme introducing a condensed introducing a condensed--pair propagator pair propagator
going beyond mean field in scheme going beyond mean field in scheme  thermodynamics thermodynamics

number fractions at number fractions at T T c c , , in RBEC, in RBEC, T T c c is large enough and is large enough and there is a strong competition there is a strong competition between condensation and between condensation and dissociation. dissociation. discussion on
can the symmetry be restored in the can the symmetry be restored in the pseudogap pseudogap phase?phase?

Kandanoff Kandanoff and Martin:
and Martin: the scheme can not give a correct symmetry restorat the scheme can not give a correct symmetry restoration ion picture and the specific heat is wrong.picture and the specific heat is wrong.

color superconductivity color superconductivity color breaking from SU(3) to SU(2) diquark diquark & meson polarizations & meson polarizations D diquark diquark & meson propagators at RPA & meson propagators at RPA leading order of 1/N leading order of 1/N c c for quarks, and next to leading order for mesons & for quarks, and next to leading order for mesons & diquarks diquarks gap equations for gap equations for chiral chiral and and diquark diquark condensates at T=0 condensates at T=0
strong coupling, small pair size, r strong coupling, small pair size, r--space pairing, ideal gas of bosons, space pairing, ideal gas of bosons, first realization in dilute atomic gas with bosons in 1995.first realization in dilute atomic gas with bosons in 1995.
can not be completely removed, and a the ultraviolet divergence can not be completely removed, and a momentum cutoff still exists in the theory.momentum cutoff still exists in the theory.broken universality: broken universality: explicit density dependence !explicit density dependence !July, 2008 Summer School on Dense Matter and HI July, 2008 Summer School on Dense Matter and HI DubnaIn non-relativistic case, only one dimensionless variable, , changing the density of the system can not induce a BCS-BEC crossover.However, in relativistic case, the extra density dependence may induce a BCS-BEC crossover.

2 t
gap equation in condensed phase is determined by uncondensed pai gap equation in condensed phase is determined by uncondensed pairs rs non happen in non--perturbative perturbative temperature and density region, the coupling is strong.temperature and density region, the coupling is strong.* relativistic BCS * relativistic BCS--BEC crossover is controlled by , the BEC BEC crossover is controlled by , the BEC--BCS BCS crossover would happen when the light quark mass changes in the crossover would happen when the light quark mass changes in the QCD medium.QCD medium.
2008 Summer School on Dense Matter and HI July, 2008 Summer School on Dense Matter and HI Dubna the magnetic instability but the magnetic instability is cured in BEC region.is cured in BEC region.

order parameters of spontaneous order parameters of spontaneous chiral chiral and color symmetry breaking and color symmetry breaking quark propagator in 12D quark propagator in 12D Nambu Nambu--Gorkov Gorkov space space
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