We compute the form factors of the photon-quark–anti-quark vertex and the effective vertex of a Higgs-boson and two gluons to three-loop order within massless perturbative quantum chromodynamics. These results provide building blocks for many third-order cross sections. Furthermore, this is the first calculation of complete three-loop vertex corrections.

We present results for the relation between a heavy quark mass defined in the on-shell and minimal subtraction ($\overline{\mathrm{MS}}$) scheme to four-loop order. The method to compute the four-loop on-shell integral is briefly described and the new results are used to establish relations between various short-distance masses and the $\overline{\mathrm{MS}}$ quark mass to next-to-next-to-next-to-leading order accuracy. These relations play an important role in the accurate determination of the $\overline{\mathrm{MS}}$ heavy quark masses.

We demonstrate that pure $\mathcal{N}=4$ supergravity is ultraviolet divergent at four loops. The form of the divergence suggests that it is due to the rigid $U(1)$ duality-symmetry anomaly of the theory. This is the first known example of an ultraviolet divergence in a pure ungauged supergravity theory in four dimensions. We use the duality between color and kinematics to construct the integrand of the four-loop four-point amplitude, whose ultraviolet divergence is then extracted by standard integration techniques.