Leading-renormalon-free Trace-anomaly-subtracted $\sigma$-mass for Heavy Quarks up to Five Loops in QCD

We demonstrate that the leading IR-renormalon divergence in the perturbative pole mass of a massive quark resides entirely in the contribution from the trace anomaly of the energy-momentum tensor in QCD. Consequently, the recently proposed trace-anomaly-subtracted $\sigma$-mass definition for heavy quarks is not only scheme- and scale-invariant, but also free from the leading IR-renormalon ambiguity. We further derive a formula connecting this $\sigma$-mass to the perturbative pole mass, solely in terms of the QCD $\beta$-function, quark-mass anomalous dimension ${\gamma }_m$ and a proper rewritten form of the pole-to-$\overline{\mathrm{M}\mathrm{S}}$ mass conversion factor. Utilizing this formula along with the ingredients available in the literature, we present the explicit five-loop result for the perturbative relationship between the $\sigma$-mass and the perturbative pole mass in QCD under the approximation of keeping only a single quark massive. Given the theoretical merits of this mass definition and the availability of high-precision conversion relations, we encourage its application to high-energy processes with heavy quarks, e.g., $H\to b\bar{b}+X_{\mathrm{Q}\mathrm{C}\mathrm{D}}$, and to current-current correlators used in determining heavy-quark masses and decay widths.