The charm/bottom quark mass from heavy quarkonium at N$^3$LO

Understand

We determine the charm and bottom quark masses using the N$^3$LO perturbative expression of the ground state (pseudoscalar) energy of the bottomonium, charmonium and $B_c$ systems: the $\eta_b$, $\eta_c$ and $B_c$ masses.

• We work in the renormalon subtracted scheme, which allows us to control the divergence of the perturbation series due to the pole mass renormalon.
• Our result for the $\overline{\rm MS}$ masses reads ${\overline m}_{c}({\overline m}_{c})=1223(33)$ MeV and ${\overline m}_{b}({\overline m}_{b})=4186(37)$ MeV.
• We also extract a value of $\alpha_s$ from a renormalon-free combination of the $\eta_b$, $\eta_c$ and $B_c$ masses: $\alpha_s(M_z)=0.1195(53)$.