  On approaching the dynamical transition temperature, supercooled liquids show
heterogeneity over space and time. Static replica theory investigates the
dynamical crossover in terms of the free energy landscape (FEL). Two kinds of
static approaches have provided a self-consistent equation for determining this
crossover, similar to the mode coupling theory for glassy dynamics. One uses
the Morita-Hiroike formalism of the liquid state theory, whereas the other
relies on the density functional theory (DFT). Each of the two approaches has
advantages in terms of perturbative field theory. Here, we develop a replica
field theory that has the benefits from both formulations. We introduce the
generalized Franz-Parisi potential to formulate a correlation functional.
Considering fluctuations around an inhomogeneous density determined by the
Ramakrishnan--Yussouf DFT, we find a new closure as the stability condition of
the correlation functional. The closure leads to the self-consistent equation
involving the triplet direct correlation function. The present field theory
further helps us study the FEL beyond the mean-field approximation.
