Two-loop master integrals for the single top production associated with $W$ boson

The $tW$ associated production has the second-largest cross section among three single top production channels. A complete study of NNLO QCD corrections to $tW$ production is still missing in literature. We present the calculation of part of the requisite two-loop master integrals for the process $b+g \to t+W$ at NNLO QCD. It turns out that the 80 master integrals in two families, involving up to three massive internal lines, can be expressed by Goncharov polylogarithms. The canonical differential equations that these integrals obey contain at most three square roots. By separating the canonical master integrals into groups according to their dependences on roots, we succeed in achieving optimization of final expressions, which are validated by independent numerical checks.