  Axial breathing modes are studied within the nuclear
energy--density--functional theory to discuss the modification of the nucleon
effective mass produced beyond the mean--field approximation. This analysis is
peformed with the subtracted second random--phase--approximation (SSRPA) model
applied to two nuclei, $^{48}$Ca and $^{90}$Zr. Analyzing the centroid energies
of axial breathing modes obtained with the mean--field--based random--phase
approximation and with the beyond--mean--field SSRPA model, we estimate the
modification (enhancement) of the effective mass which is induced beyond the
mean field. This is done by employing a relation, obtained with the Landau's
Fermi liquid theory, between the excitation frequency of axial modes to
$\sqrt{m/m^*}$, where $m$ ($m^*$) is the bare (effective) mass. Such an
enhancement of the effective mass is discussed in connection with the
renormalization of single--particle excitation energies generated by the
energy--dependent SSRPA self-energy correction. We find that the effective
beyond--mean--field compression of the single--particle spectrum produced by
the self--energy correction is coherent with the increase of the effective mass
estimated from the analysis of axial breathing modes.
