Renormalized <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>ε</mml:mi></mml:math> -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 their coefficients are as $\epsilon \rightarrow 0$. I propose definition renormalized $\epsilon$-finite integrals, which incorporate ultraviolet divergence subtractions specific way. A key advantage this choice is expressions physical observables, expansions positive powers $\epsilon$ never needed. As an example, provide general three-loop self-energy integrals. The differential equations method used compute numerically arbitrary external momentum invariant, special cases with internal masses equal single scale or zero. These include ones necessary QCD corrections self-energies W, Z, and Higgs bosons. In principle, same should numerical computation self energies masses.