On-shell matrix elements of the EMT trace in gauge theories and heavy quark masses

We present a novel diagrammatic proof of the identity between the forward matrix element of the energy-momentum-tensor (EMT) trace operator over a single particle's on-shell state and its perturbative pole mass to any loop orders in perturbative gauge theories (with gauge bosons and fermions), without appealing to any pre-laid operator renormalization conditions or Ward identities. The proof is based on the equation of mass-dimensional analysis in dimensional regularization, the topological properties of contributing Feynman diagrams and the on-shell renormalization condition. Considering for definiteness perturbative QED and QCD with at most one fermion kept massive, we have verified the aforementioned identity, up to three loops, for all elementary particles through direct computation of dimensionally-regularized matrix elements of the relevant bare operators. We observe interestingly that the trace-anomaly contribution seemingly contains all leading-renormalon effects (explicitly verified up to three loops), and we propose accordingly a new scheme- and scale-independent trace-anomaly-subtracted mass definition for heavy $t$-,$\,b$-,$\,c$-quarks and electrons. A list of amusing numbers is subsequently presented for the composition of their perturbative pole masses.