Hadronic light-by-light contribution to $(g-2)_\mu$ from lattice QCD: a complete calculation

The tension between the theoretical prediction and the experimental result of the anomalous magnetic moment of the muon, $a_\mu\equiv(g-2)_\mu/2$, is one of the long-standing puzzles of modern particle physics. After the update of the Fermilab E989 experiment in April 2021, the discrepancy between theory and experiment lies at the 4.2-sigma level. Further possible reduction of the error on the theory side relies solely on the control over hadronic processes. With recent developments, it has become possible for lattice QCD to provide competitive predictions on some of the most important hadronic contributions to $a_\mu$. In this talk, the Mainz determination of the hadronic light-by-light (hlbl) contribution to $a_\mu$ computed with $N_f=2+1$ lattice ensembles will be presented. Although at subleading order in $\alpha_{\textrm{QED}}$, $a_\mu^{\textrm{hlbl}}$ was not sufficiently precisely determined in the past and represented a non-negligible source of uncertainty for the total error budget of $a_\mu$. With our continuum and infinite-volume QED setup, we obtain a value of $a_\mu^{\textrm{hlbl}}=106.8(15.9)\times 10^{-11}$ after an infinite-volume, continuum and chiral extrapolation, with estimates for the contributions of all five Wick-contraction topologies.