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QCD-like theories possess a positively definite fermion determinant at finitebaryon chemical potential $\mu_{\text B}$ and the lattice simulation can besuccessfully performed. While the chiral perturbation theories are sufficientto describe the Bose condensate at low density, to describe the crossover fromBose-Einstein condensation (BEC) to BCS superfluidity at moderate density weshould use some fermionic effective model of QCD, such as theNambu-Jona-Lasinio model. In this paper, using two-color two-flavor QCD as anexample, we examine how the Nambu-Jona-Lasinio model describes the weaklyinteracting Bose condensate at low density and the BEC-BCS crossover atmoderate density. Near the quantum phase transition point $\mu_{\text B}=m_\pi$($m_\pi$ is the mass of pion/diquark multiplet), the Ginzburg-Landau freeenergy at the mean-field level can be reduced to the Gross-Pitaevskii freeenergy describing a weakly repulsive Bose condensate with a diquark-diquarkscattering length identical to that predicted by the chiral perturbationtheories. The Goldstone mode recovers the Bogoliubov excitation in weaklyinteracting Bose condensates. The results of in-medium chiral and diquarkcondensates predicted by chiral perturbation theories are analyticallyrecovered. The BEC-BCS crossover and meson Mott transition at moderate baryonchemical potential as well as the beyond-mean-field corrections are studied.Part of our results can also be applied to real QCD at finite baryon or isospinchemical potential.

Nambu-Jona-Lasinio model description of weakly interacting Bose condensate and BEC-BCS crossover in dense QCD-like theories