  We study the 2D coupled wave-Klein-Gordon systems with semi-linear null
nonlinearities $Q_0$ and $Q_{\alpha\beta}$. The main result states that the
solution to the 2D coupled systems exists globally provided that the initial
data are small in some weighted Sobolev space, which do not necessarily have
compact support, and we also show the optimal time decay of the solution.
  The major difficulties lie in the slow decay nature of the wave and the
Klein-Gordon components in two space dimensions, in addition, extra
difficulties arise due to the presence of the null form $Q_0$ which is not of
divergence form and is not compatible with the Klein-Gordon equations. To
overcome the difficulties, a new observation for the structure of the null form
$Q_0$ is required.
