  We present the analytical results of the two-loop amplitudes for hadronic
$tW$ production, focusing on the leading color and light fermion-loop
contributions. The calculation of the two-loop integrals is performed using the
method of canonical differential equations. The results have been expressed in
terms of multiple polylogarithms and checked by comparing the infra-red
divergences with the predictions from anomalous dimensions. Combined with the
one-loop squared amplitudes we have computed previously, we obtain the hard
function relevant to a NNLO Monte Carlo calculation. We find that the hard
function varies slowly in the region with small top quark velocity but
increases dramatically in the region with very large top quark velocity. After
phase space integration, the leading color hard function gives an about $5.4\%$
correction to the leading order cross section, while the light fermion loop
contributes about $-1.4\%$.
