  The physics of the pseudogap state is intimately linked with the pairing
mechanism that gives rise to superfluidity in quantum gases and to
superconductivity in high-Tc cuprates, and therefore, both in quantum gases and
superconductors, the pseudogap state and preformed pairs have been under
intensive experimental scrutiny. Here, we develop a path integral treatment
that provides a divergence-free description of the paired state in
two-dimensional Fermi gases. Within this formalism, we derive the pseudogap
temperature and the pair fluctuation spectral function, and compare these
results with the recent experimental measument of the pairing in the
two-dimensional Fermi gas. The removal of the infrared divergence in the number
equations is shown both numerically and analytically, through a study of the
long-wavelength and low-energy limit of the pair fluctuation density. Besides
the pseudogap temperature, also the pair formation temperature and the critical
temperature for superfluidity are derived. The latter corresponds to the
Berezinski-Kosterlitz-Thouless (BKT) temperature. The pseudogap temperature,
which coincides with the pair formation temperature in mean field, is found to
be suppressed with respect to the pair formation temperature by fluctuations.
This suppression is strongest for large binding energies of the pairs. Finally,
we investigate how the pair formation temperature, the pseudogap temperature
and the BKT temperature behave as a function of both binding energy and
imbalance between the pairing partners in the Fermi gas. This allows to set up
phase diagrams for the two-dimensional Fermi gas, in which the superfluid
phase, the phase-fluctuating quasicondensate, and the normal state can be
identified.
