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The comparison of the theoretical and experimental determinations of theanomalous magnetic moment of the muon $(g-2)_\mu$ constitutes one of thestrongest tests of the Standard Model at low energies. In this article, wecompute the leading hadronic contribution to $(g-2)_\mu$ using lattice QCDsimulations employing Wilson quarks. Gauge field ensembles at four differentlattice spacings and several values of the pion mass down to its physical valueare used. We apply the O($a$) improvement programme with two discretizations ofthe vector current to better constrain the approach to the continuum limit. Theelectromagnetic current correlators are computed in the time-momentumrepresentation. In addition, we perform auxiliary calculations of the pion formfactor at timelike momenta in order to better constrain the tail of theisovector correlator and to correct its dominant finite-size effect. For thenumerically dominant light-quark contribution, we have rescaled the lepton massby the pion decay constant computed on each lattice ensemble. We perform acombined chiral and continuum extrapolation to the physical point, and ourfinal result is $ a_\mu^{\rm hvp}=(720.0\pm12.4_{\rm stat}\,\pm9.9_{\rmsyst})\cdot10^{-10}$. It contains the contributions of quark-disconnecteddiagrams, and the systematic error has been enlarged to account for the missingisospin-breaking effects.

The leading hadronic contribution to $(g-2)_\mu$ from lattice QCD with $N_{\rm f}=2+1$ flavours of O($a$) improved Wilson quarks