  We extend a previously proposed field-theoretic self-consistent perturbation
approach for the equilibrium dynamics of the Dean-Kawasaki equation presented
in [J. Stat. Mech. 2008 P02004]. By taking terms missing in the latter analysis
into account we arrive at a set of three new equations for correlation
functions of the system. These correlations involve the density and its
logarithm as local observables. Our new one-loop equations, which must
carefully deal with the noninteracting Brownian gas theory, are more general
than the historic Mode-Coupling one in that a further and well-defined
approximation leads back to the original mode-coupling equation for the density
correlations alone. However, without performing any further approximation step,
our set of three equations does not feature any ergodic-non ergodic transition,
as opposed to the historical mode- coupling approach.
