  In this work, the dressed molecules theory is used to describe the
two-dimensional quantum anomaly of breathing mode in the recent experimental
system\cite{Holten2018,Peppler2018}. With the aid of a beyond mean-field,
Gaussian pair fluctuation theory, we employ the dressed molecules states to
characterize the axial excited states and the Feshbach molecular states, and
propose a complete low energy effective theory. We show that, in the whole
crossover from a Bardeen-Cooper-Schrieffer (BCS) superfluid to a Bose-Einstein
condensate (BEC), our theory can describe the two-dimensional experimental
systems precisely in low energe region. We explain the the puzzling
experimental observations of the smaller than expected breathing mode frequency
perfectly. Our establishment of the dressed molecules theory for 2D fermions is
crucial to understand the conformal anomaly in the quasi low-dimensional
quantum systems.
