  We study Gaussian fluctuations of the zero-temperature attractive Fermi gas
in the 2D BCS-BEC crossover showing that they are crucial to get a reliable
equation of state in the BEC regime of composite bosons, bound states of
fermionic pairs. A low-momentum expansion up to the fourth order of the
quadratic action of the fluctuating pairing field gives an ultraviolent
divergent contribution of the Gaussian fluctuations to the grand potential.
Performing dimensional regularization we evaluate the effective coupling
constant in the beyond-mean-field grand potential. Remarkably, in the BEC
regime our grand potential gives exactly the Popov's equation of state of 2D
interacting bosons, and allows us to identify the scattering length $a_B$ of
the interaction between composite bosons as $a_B=a_F/(2^{1/2}e^{1/4})= 0.551...
a_F$, with $a_F$ is the scattering length of fermions. Remarkably, the value
from our analytical relationship between the two scattering lengths is in full
agreement with that obtained by recent Monte Carlo calculations.
