Renormalized $\epsilon$-finite master integrals and their virtues: the three-loop self energy case

Loop diagram calculations typically rely on reduction to a finite set of master integrals in $4 - 2\epsilon$ dimensions. It has been shown that for any problem, the masters can be chosen so that their coefficients are finite as $\epsilon \rightarrow 0$. I propose a definition of renormalized $\epsilon$-finite master integrals, which incorporate ultraviolet divergence subtractions in a specific way. A key advantage of this choice is that in expressions for physical observables, expansions to positive powers in $\epsilon$ are never needed. As an example, I provide the subtractions for general three-loop self-energy integrals. The differential equations method is used to compute numerically the renormalized $\epsilon$-finite master integrals for arbitrary external momentum invariant, in special cases with internal masses equal to a single scale or zero. These include the ones necessary for the three-loop QCD corrections to the self-energies of the W, Z, and Higgs bosons. In principle, the same method should provide for numerical computation of general three-loop self energies with any masses.