The induced interaction in a Fermi gas with a BEC-BCS crossover

We study the effect of the induced interaction on the superfluid transition temperature of a Fermi gas with a BEC-BCS crossover. The Gorkov-Melik-Barkhudarov theory about the induced interaction is extended from the BCS side to the entire crossover, and the pairing fluctuation is treated in the approach by Nozi\`{e}res and Schmitt-Rink. At unitarity, the induced interaction reduces the transition temperature by about twenty percent. In the BCS limit, the transition temperature is reduced by a factor about 2.22, as found by Gorkov and Melik-Barkhudarov. Our result shows that the effect of the induced interaction is important both on the BCS side and in the unitary region.