Subtraction of the $t\bar{t}$ contribution in $tW\bar{b}$ production at the one-loop level

The $tW\bar{b}$ production contributes to the real corrections to the $tW$ cross section. It would interfere with the top quark pair production, causing difficulties in a clear definition of the $tW{\bar b}$ events. The subtraction of the $t\bar{t}$ contributions has been performed in the diagram removal or diagram subtraction schemes for the tree-level processes. However, these schemes rely on the ability to identify the double resonant diagrams and thus can not be extended to loop diagrams. We propose a new scheme to subtract the $t\bar{t}$ contributions by power expansion of the squared amplitude in the resonant region. In order to cancel the infra-red divergences of the loop amplitudes, a widely used method is to introduce the dipole counter-terms, an ingredient in calculations of the full next-to-leading order QCD corrections. In our scheme, these counter-terms are also power-expanded. As a proof of principle, we calculate the one-loop correction to the $d\bar{d}\to \bar{b}Wt$ process, and present the invariant mass distribution of the $W\bar{b}$ system.