Three-loop charge renormalization effects due to quartic scalar self-interactions

Dimensionally regularized dispersion theory is used to compute the $O\left({\hslash }^3{g}^3{f}^2\right)$ contribution to the charge renormalization function ${\beta }_g$, where $g$ is a gauge field coupling and $f$ is a quartic (pseudo) scalar selfcoupling. Some motivations for and systematics of the calculation are discussed. Special attention is given to an $N=4\mathrm{}$ globally supersymmetric gauge theory.