  We analytically investigate the effective-diffusivity tensor of a tracer
particle in a fluid flow endowed with a short correlation time. By means of
functional calculus and a multiscale expansion, we write down the main
contributions to the eddy diffusivity due to each single physical effect and to
their interplays. Namely, besides molecular diffusivity and a constant uniform
mean streaming, we take into account the possibility for the (incompressible,
Gaussian, stationary, homogeneous, isotropic) turbulent fluctuations to break
parity invariance. With respect to the classical turbulence-driven diffusivity
amplification for delta-correlated flows, we find that the presence of a short
temporal correlation induces a diminution even when coupled with such effects,
with two principal exceptions. Notably, the diffusivity is --- perturbatively
--- enlarged not only by the helical contribution itself, but also by the
interference between molecular diffusion and mean flow.
