The two-mass contributions to the three-loop massive operator matrix elements $${\widetilde{A}}_{Qg}^{(3)}$$ and $${\Delta \widetilde{A}}_{Qg}^{(3)}$$

A bstract We calculate the two-mass three-loop contributions to the unpolarized and polarized massive operator matrix elements $${\widetilde{A}}_{Qg}^{(3)}$$ and $${\Delta \widetilde{A}}_{Qg}^{(3)}$$ in x -space for a general mass ratio by using a semi-analytic approach. We also compute Mellin moments up to N = 2000(3000) by an independent method, to which we compare the results in x -space. In the polarized case, we work in the Larin scheme. We present numerical results. The two-mass contributions amount to about 50% of the full $$O({T}_{F}^{2})$$ and $$O({T}_{F}^{3})$$ terms contributing to the operator matrix elements. The present result completes the calculation of all unpolarized and polarized massive three-loop operator matrix elements.