  We investigate the normal selfenergy and the mass enhancement factor in the
Hubbard model on the two-dimensional square lattice. Our purpose in this paper
is to evaluate the mass enhancement factor more quantitatively than the
conventional third order perturbation theory. We calculate it by expanding
perturbatively up to the fourth order with respect to the on-site repulsion
$U$. We consider the cases that the system is near the half-filling, which are
similar situations to high-$T_c$ cuprates. As results of the calculations, we
obtain the large mass enhancement on the Fermi surface by introducing the
fourth order terms. This is mainly originated from the fourth order
particle-hole and particle-particle diagrams. Although the other fourth order
terms have effect of reducing the effective mass, this effect does not cancel
out the former mass enhancement completely and there remains still a large mass
enhancement effect. In addition, we find that the mass enhancement factor
becomes large with increasing the on-site repulsion $U$ and the density of
state (DOS) at the Fermi energy $\rho(0)$. According to many current reseaches,
such large $U$ and $\rho(0)$ enhance the effective interaction between
quasiparticles, therefore the superconducting transition temperature $T_c$
increases. On the other hand, the large mass enhancement leads the reduction of
the energy scale of quasiparticles, as a result, $T_c$ is reduced. When we
discuss $T_c$, we have to estimate these two competitive effects.
