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The electromagnetic coupling constant, $\alpha$, is one of the fundamentalparameters of the Standard Model (SM). Its value at the Z boson mass,$\alpha(M_Z)$, is of particular interest as it enters electroweak precisiontests. When running $\alpha$ from low energies up to the Z mass, five orders ofmagnitude in precision are lost. This makes it one of the least well determinedparameters of the SM at that scale. The largest source of error comes fromnon-perturbative hadronic effects in the low energy region. Thesenon-perturbative effects can be determined from ab-initio calculations inlattice QCD. At higher energies, needed to match onto QCD perturbation theory,discretization errors become large. In addition, the hadronic vacuumpolarization receives logarithmically-enhanced cutoff effects which render thecontinuum extrapolation more difficult. To better control this extrapolation athigher energies, we test a number of improvement procedures based on latticeperturbation theory. To illustrate their effect, we present a preliminaryanalysis of the light quark, connected contribution to the Adler function atEuclidean $Q^2=5$ GeV$^2$. The lattice results are obtained using simulationswith $2+1+1$ flavors of staggered fermions at physical values of the quarkmasses.

Hadronic contribution to the running of the electromagnetic coupling constant from lattice QCD: challenges at short distances