Abstract:
The leading-order hadronic contribution to the muon magnetic moment anomaly $a_\mu\equiv (g_\mu-2)/2$, calculated using a dispersion integral of $e^+e^-$ annihilation data and $\tau$ data, is briefly reviewed. This contribution has the largest uncertainty to the predicted value of $a_\mu$, which differs from the direct measurement by $\sim 3.6 (2.4)$ standard deviations for the $e^+e^- (\tau)$ based analysis. Recent results and main open issues on the subject are discussed.

Abstract:
After discussing the relevance of a first principles theory-prediction of the hadronic vacuum polarisation for Standard Model tests, the theoretical challenges for its computation in lattice QCD are reviewed. New ideas that will potentially improve the quality of lattice simulations will be introduced and the status of ongoing simulations will be presented briefly.

Abstract:
We present a calculation of the hadronic vacuum polarization (HVP) tensor within the framework of Dyson--Schwinger equations. To this end we use a well-established phenomenological model for the quark-gluon interaction with parameters fixed to reproduce hadronic observables. From the HVP tensor we compute both the Adler function and the HVP contribution to the anomalous magnetic moment of the muon, $a_\mu$. We find $a_\mu^{HVP}= 6760\times 10^{-11}$ which deviates about two percent from the value extracted from experiment. Additionally, we make comparison with a recent lattice determination of $a_\mu^{HVP}$ and find good agreement within our approach. We also discuss the implications of our result for a corresponding calculation of the hadronic light-by-light scattering contribution to $a_\mu$.

Abstract:
We present a lattice calculation of the hadronic vacuum polarization and the lowest-order hadronic contribution to the muon anomalous magnetic moment, a_\mu = (g-2)/2, using 2+1 flavors of improved staggered fermions. A precise fit to the low-q^2 region of the vacuum polarization is necessary to accurately extract the muon g-2. To obtain this fit, we use staggered chiral perturbation theory, including the vector particles as resonances, and compare these to polynomial fits to the lattice data. We discuss the fit results and associated systematic uncertainties, paying particular attention to the relative contributions of the pions and vector mesons. Using a single lattice spacing ensemble (a=0.086 fm), light quark masses as small as roughly one-tenth the strange quark mass, and volumes as large as (3.4 fm)^3, we find a_\mu^{HLO} = (713 \pm 15) \times 10^{-10} and (748 \pm 21) \times 10^{-10} where the error is statistical only and the two values correspond to linear and quadratic extrapolations in the light quark mass, respectively. Considering systematic uncertainties not eliminated in this study, we view this as agreement with the current best calculations using the experimental cross section for e^+e^- annihilation to hadrons, 692.4 (5.9) (2.4)\times 10^{-10}, and including the experimental decay rate of the tau lepton to hadrons, 711.0 (5.0) (0.8)(2.8)\times 10^{-10}. We discuss several ways to improve the current lattice calculation.

Abstract:
I present a preliminary calculation of the hadronic vacuum polarization for 2+1 flavors of improved Kogut-Susskind quarks by utilizing a set of gauge configurations recently generated by the MILC collaboration. The polarization function $\Pi(q^2)$ is then used to calculate the lowest order (in $\alpha_{QED}$) hadronic contribution to the muon anomalous magnetic moment.

Abstract:
Recent calculations of the hadronic vacuum polarisation contribution are reviewed. The focus is put on the leading-order contribution to the muon magnetic anomaly involving $e^+e^-$ annihilation cross section data as input to a dispersion relation approach. Alternative calculation including tau data is also discussed. The $\tau$ data are corrected for various isospin-breaking sources which are explicitly shown source by source.

Abstract:
Recently, it was shown that insertions of hadronic vacuum polarization at O(alpha^4) generate non-negligible effects in the calculation of the anomalous magnetic moment of the muon. This result raises the question if other hadronic diagrams at this order might become relevant for the next round of g-2 measurements as well. In this note we show that a potentially enhanced such contribution, hadronic light-by-light scattering in combination with electron vacuum polarization, is already sufficiently suppressed.

Abstract:
We present a quenched lattice calculation of the lowest order (alpha^2) hadronic contribution to the anomalous magnetic moment of the muon which arises from the hadronic vacuum polarization. A general method is presented for computing entirely in Euclidean space, obviating the need for the usual dispersive treatment which relies on experimental data for e^+e^- annihilation to hadrons. While the result is not yet of comparable accuracy to those state-of-the-art calculations, systematic improvement of the quenched lattice computation to this level of accuracy is straightforward and well within the reach of present computers. Including the effects of dynamical quarks is conceptually trivial, the computer resources required are not.

Abstract:
Adjusting a previously developed Grotch-type approach to a perturbative calculation of the electronic vacuum-polarization effects in muonic atoms, we find here the two-loop vacuum polarization relativistic recoil correction of order $\alpha^2(Z\alpha)^4m^2/M$ in light muonic atoms. The result is in perfect agreement with the one previously obtained within the Breit-type approach. We also discuss here simple approximations of the irreducible part of the two-loop vacuum-polarization dispersion density, which was applied to test our calculations and can be useful for other evaluations with an uncertainty better than 1%.

Abstract:
The hadronic vacuum polarization contribution to the muon (g-2) value is calculated by considering a known dispersion integral which involves the $R_{e+ e-}(s)$ ratio. The theoretical part stemming from the region below 1.8 GeV is the largest contribution in our approach, and is calculated by using a contour integral involving the associated Adler function $D(Q^2)$. In the resummations, we explicitly take into account the exactly known renormalon singularity of the leading infrared renormalon in the usual and in the modified Borel transform of $D(Q^2)$, and map further away from the origin the other renormalon singularities by employing judiciously chosen conformal transformations. The renormalon effect increases the predicted value of the hadronic vacuum polarization contribution to the muon (g-2), and therefore diminishes the difference between the recently measured and the SM/QCD-predicted value of (g-2). It is also shown that the total QED correction to the hadronic vacuum polarization is very small, about 0.06 percent.