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The two-pion low-energy contribution to the anomalous magnetic moment of themuon, $a_\mu\equiv(g-2)_\mu/2$, expres sed as an integral over the modulussquared of the pion electromagnetic form fac tor, brings a relatively largecontribution to the theoretical error, since the low accuracy of experimentalmeasurements in this region is amplified by the drastic increase of theintegration kernel. We derive stringent constraints on the two-pioncontribution by exploiting analyticity and unitarity of the pionelectromagnetic form factor. To avoid the poor knowledge of the modulus of thisfunction, we use instead its phase, known with high precision in the elasticregion from Roy equations for pion-pion scattering via the Fermi-Watsontheorem. Above the inelastic threshold we adopt a conservative integralcondition on the modulus, determined from data and perturbative QCD. Additionalhigh precision data on the modulus in the range $0.65-0.71$ GeV, obtained from$e^+e^-$ annihilation and $\tau$-decay experiments, are used to improve thepredictions on the modulus at lower energies by means of a parametrization-freeanalytic extrapolation. The results are optimal for a given input and do notdepend on the unknown phase of the form factor above the inelastic threshold.The present work improves a previous analysis based on the same technique,including more experimental data and employing better statistical tools fortheir treatment. We obtain for the contribution to $a_\mu$ from below 0.63 GeVthe value $(133.258 \pm 0.723)\times 10^{-10}$, which amounts to a reduction ofthe theoretical error by about $6 \times 10^{-11}$.

Precise determination of the low-energy hadronic contribution to the muon $g-2$ from analyticity and unitarity: An improved analysis