Non-factorisable contributions to $t$-channel single-top production at the LHC and FCC

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.

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