The two-pion contribution from low energies to the muon magnetic moment anomaly, although small, has a large relative uncertainty since in this region the experimental data on the cross sections are neither sufficient nor precise enough. It is theref
ore of interest to see whether the precision can be improved by means of additional theoretical information on the pion electromagnetic form factor, which controls the leading order contribution. In the present paper we address this problem by exploiting analyticity and unitarity of the form factor in a parametrization-free approach that uses the phase in the elastic region, known with high precision from the Fermi-Watson theorem and Roy equations for $pipi$ elastic scattering as input. The formalism also includes experimental measurements on the modulus in the region 0.65-0.70 GeV, taken from the most recent $e^+e^-to pi^+pi^-$ experiments, and recent measurements of the form factor on the spacelike axis. By combining the results obtained with inputs from CMD2, SND, BABAR and KLOE, we make the predictions $a_mu^{pipi, LO},[2 m_pi,, 0.30 gev]=(0.553 pm 0.004) times 10^{-10}$ and $a_mu^{pipi, LO},[0.30 gev,, 0.63 gev]=(133. 083 pm 0.837)times 10^{-10}$. These are consistent with the other recent determinations, and have slightly smaller errors.
The moments of the hadronic spectral functions are of interest for the extraction of the strong coupling $alpha_s$ and other QCD parameters from the hadronic decays of the $tau$ lepton. Motivated by the recent analyses of a large class of moments in
the standard fixed-order and contour-improved perturbation theories, we consider the perturbative behavior of these moments in the framework of a QCD nonpower perturbation theory, defined by the technique of series acceleration by conformal mappings, which simultaneously implements renormalization-group summation and has a tame large-order behavior. Two recently proposed models of the Adler function are employed to generate the higher order coefficients of the perturbation series and to predict the exact values of the moments, required for testing the properties of the perturbative expansions. We show that the contour-improved nonpower perturbation theories and the renormalization-group-summed nonpower perturbation theories have very good convergence properties for a large class of moments of the so-called reference model, including moments that are poorly described by the standard expansions. The results provide additional support for the plausibility of the description of the Adler function in terms of a small number of dominant renormalons.
