  We investigate phase transitions in scalar field theories using the
functional renormalization group (RG) equation. We analyze a system with
$U(2)\times U(2)$ symmetry, in which there is a parameter $\lambda_2$ that
controls the strength of the first-order phase transition driven by
fluctuations. In the limit of $\lambda_2\to0$, the $U(2)\times U(2)$ theory is
reduced to an $O(8)$ scalar theory that exhibits a second-order phase
transition in three dimensions. We develop a new insight for the understanding
of the fluctuation-induced first-order phase transition as a smooth
continuation from the standard RG flow in the $O(8)$ system. In our view from
the RG flow diagram on coupling parameter space, the region that favors the
first-order transition emerges from the unphysical region to the physical one
as $\lambda_2$ increases from zero. We give this interpretation based on the
Taylor expansion of the functional RG equations up to the fourth order in terms
of the field, which encompasses the $\epsilon$-expansion results. We compare
results from the expansion and from the full numerical calculation and find
that the fourth-order expansion is only of qualitative use and that the
sixth-order expansion improves the quantitative agreement.
