  This paper presents a theoretical analysis of the closure kinetics of a
polymer with hydrodynamic interactions. This analysis, which takes into account
the non-Markovian dynamics of the end-to-end vector and relies on the
preaveraging of the mobility tensor (Zimm dynamics), is shown to reproduce very
accurately the results of numerical simulations of the complete non linear
dynamics. It is found that Markovian treatments, based on a Wilemski-Fixman
approximation, significantly overestimate cyclization times (up to a factor 2),
showing the importance of memory effects in the dynamics. In addition, this
analysis provides scaling laws of the mean first cyclization time (MFCT) with
the polymer size $N$ and capture radius $b$, which are identical in both
Markovian and non-Markovian approaches. In particular, it is found that the
scaling of the MFCT for large $N$ is given by $T\sim N^{3/2}\ln (N/b^2)$, which
differs from the case of the Rouse dynamics where $T\sim N^{2}$. The extension
to the case of the reaction kinetics of a monomer of a Zimm polymer with an
external target in a confined volume is also presented.
