  Single top quark is mainly produced through the $t$-channel W boson exchange
$q + b \rightarrow q' + t$ at LHC. This process probes Wtb vertex directly and
can be used to measure the CKM matrix element $V_{tb}$ or to constrain the
bottom quark PDF. The non-factorisable contributions are the last missing piece
of the NNLO corrections. In these proceedings, we discuss in a first part the
ostensible importance of such corrections and the obtention of the different
relevant amplitudes. In the second part, recently published results at the
energy of the LHC are compared to new results for proton-proton collision at
$100 \: {\rm TeV}$, the energy of the FCC.
