 Solution

An on-shell renormalization scheme fixes the mass parameter at the physical pole mass and fixes the renormalized field by the chosen pole residue. Its counterterms include finite contributions needed to enforce those conditions. A coupling is likewise defined by a specified physical amplitude or mass-shell condition.

The minimal subtraction scheme removes only poles in the dimensional regulator. The modified minimal subtraction scheme, which is the overbarred scheme in the PDF, removes the accompanying universal combination $\log 4\pi -{\gamma }_E$ as well. In $D=4-ϵ$, the one-loop subtraction is proportional toA subtraction-scheme mass is a running parameter and generally differs from the physical mass. Finite redefinitions relate the schemes while leaving physical quantities unchanged. The overbar is essential for the stated finite formula in part (d).