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Real-Virtual contributions to the inclusive Higgs cross-section at N3LO

We compute the contributions to the N3LO inclusive Higgs boson cross-section from the square of one-loop amplitudes with a Higgs boson and three QCD partons as external states. Our result is a Taylor expansion in the dimensional regulator epsilon, where the coefficients of the expansion are analytic functions of the ratio of the Higgs boson mass and the partonic center of mass energy and they are valid for arbitrary values of this ratio. We also perform a threshold expansion around the limit of soft-parton radiation in the final state. The expressions for the coefficients of the threshold expansion are valid for arbitrary values of the dimension. As a by-product of the threshold expansion calculation, we have developed a soft expansion method at the integrand level by identifying the relevant soft and collinear regions for the loop-momentum.