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We present the analytic results of one-loop squared amplitudes for $tW$production at a hadron collider. The calculation is performed using the methodof differential equations. After renormalization, we have checked that theinfrared divergences agree with the general structure predicted by anomalousdimensions. The finite remainder contributes to the next-to-next-to-leadingorder hard function, one of the essential ingredients in the factorizationformula of the cross section near the infrared region, which can be used inresummation of all-order soft gluon effects or a differentialnext-to-next-to-leading order calculation based on the phase space slicingmethod.

One-loop squared amplitudes for hadronic $tW$ production at next-to-next-to-leading order in QCD