Determination of $overline{m}_b/overline{m}_c$ and $overline{m}_b$ from $n_f=4$ lattice QCD$+$QED

We extend HPQCDs earlier $n_f=4$ lattice-QCD analysis of the ratio of $overline{mathrm{MSB}}$ masses of the $b$ and $c$ quark to include results from finer lattices (down to 0.03fm) and a new calculation of QED contributions to the mass ratio. We find that $overline{m}_b(mu)/overline{m}_c(mu)=4.586(12)$ at renormalization scale $mu=3$,GeV. This result is nonperturbative. Combining it with HPQCDs recent lattice QCD$+$QED determination of $overline{m}_c(3mathrm{GeV})$ gives a new value for the $b$-quark mass: $overline{m}_b(3mathrm{GeV}) = 4.513(26)$GeV. The $b$-mass corresponds to $overline{m}_b(overline{m}_b, n_f=5) = 4.202(21)$GeV. These results are the first based on simulations that include QED.