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The single-top production is an important process at the LHC to test theStandard Model (SM) and search for the new physics beyond the SM. Although thecomplete next-to-next-to-leading order (NNLO) QCD correction to the single-topproduction is crucial, this calculation is still challenging at present. Inorder to efficiently reduce the NNLO single-top amplitude, we improve theauxiliary mass flow (AMF) method by introducing the $\epsilon$ truncation. Fordemonstration we choose one typical planar double-box diagram for the $tW$production. It is shown that one coefficient of the form factors on itsamplitude can be systematically reduced into the linear combination of 198scalar integrals.

Reduction of planar double-box diagram for single-top production via auxiliary mass flow