The muon anomalous magnetic moment is one of the most precisely measured quantities in particle physics. In a recent experiment at Brookhaven it has been measured with a remarkable 14-fold improvement of the previous CERN experiment reaching a precision of 0.54ppm. Since the first results were published, a persisting "discrepancy" between theory and experiment of about 3 standard deviations is observed. It is the largest "established" deviation from the Standard Model seen in a "clean" electroweak observable and thus could be a hint for New Physics to be around the corner. This deviation triggered numerous speculations about the possible origin of the "missing piece" and the increased experimental precision animated a multitude of new theoretical efforts which lead to a substantial improvement of the prediction of the muon anomaly aµ = (gµ − 2)/2. The dominating uncertainty of the prediction, caused by strong interaction effects, could be reduced substantially, due to new hadronic cross section measurements in electron-positron annihilation at low energies. Also the recent electron g − 2 measurement at Harvard contributes substantially to the progress in this field, as it allows for a much more precise determination of the fine structure constant α as well as a cross check of the status of our theoretical understanding.In this report we review the theory of the anomalous magnetic moments of the electron and the muon. After an introduction and a brief description of the principle of the muon g − 2 experiment, we present a review of the status of the theoretical prediction and in particular discuss the role of the hadronic vacuum polarization effects and the hadronic light-by-light scattering correction, including a new evaluation of the dominant pion-exchange contribution. In the end, we find a 3.2 standard deviation discrepancy between experiment and Standard Model prediction. We also present a number of examples of how extensions of the electroweak Standard Model would change the theoretical prediction of the muon anomaly aµ. Perspectives for future developments in experiment and theory are briefly discussed and critically assessed. The muon g − 2 will remain one of the hot topics for further investigations.

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The correction to the muon anomalous magnetic moment from the pion-pole contribution to the hadronic light-by-light scattering is considered using a description of the π 0 − γ * − γ * transition form factor based on the large-N C and short-distance properties of QCD. The resulting twoloop integrals are treated by first performing the angular integration analytically, using the method of Gegenbauer polynomials, followed by a numerical evaluation of the remaining twodimensional integration over the moduli of the Euclidean loop momenta. The value obtained, a LbyL;π 0 µ = +5.8 (1.0) × 10 −10 , disagrees with other recent calculations. In the case of the vector meson dominance form factor, the result obtained by following the same procedure reads a LbyL;π 0 µ | V M D = +5.6 × 10 −10 , and differs only by its overall sign from the value obtained by previous authors. The inclusion of the η and η ′ poles gives a total value a LbyL;PS µ = +8.3 (1.2) × 10 −10 for the three pseudoscalar states. This result substantially reduces the difference between the experimental value of a µ and its theoretical counterpart in the standard model.

Starting from the study of the low-energy and high-energy behaviours of the QCD three-point functions VAP , V VP and AAP , several O(p 6 ) low-energy constants of the chiral Lagrangian are evaluated within the framework of the lowest meson dominance (LMD) approximation to the large-N C limit of QCD. In certain cases, values that differ substantially from estimates based on a resonance Lagrangian are obtained. It is pointed out that the differences arise through the fact that QCD short-distance constraints are in general not correctly taken into account in the approaches using resonance Lagrangians. We discuss the implications of our results for the O(p 6 ) counterterm contributions to the vector form factor of the pion and to the decay π → eν e γ, and for the pion-photon-photon transition form factor.

The hadronic light-by-light contribution to a(mu), the anomalous magnetic moment of the muon, is discussed from the point of view of an effective low-energy theory. As an application, the coefficient of the leading logarithm arising from the two-loop graphs involving two anomalous vertices is computed, and found to be positive. This corresponds to a positive sign for the pion-pole contribution to the hadronic light-by-light correction to a(mu), and to a sizable reduction of the discrepancy between the present experimental value of a(mu) and its theoretical counterpart in the standard model.

We present a lattice QCD calculation of the double-virtual neutral pion transition form factor, with the goal to cover the kinematic range relevant to hadronic light-by-light scattering in the muon g − 2. Several improvements have been made compared to our previous work. First, we take into account the effects of the strange quark by using the N f = 2 + 1 CLS gauge ensembles. Second, we have implemented the on-shell O(a)-improvement of the vector current to reduce the discretization effects associated with Wilson quarks. Finally, in order to have access to a wider range of photon virtualities, we have computed the transition form factor in a moving frame as well as in the pion rest frame. After extrapolating the form factor to the continuum and to physical quark masses, we compare our results with phenomenology. We extract the normalization of the form factor with a precision of 3.5% and confirm within our uncertainty previous somewhat conflicting estimates for a low-energy constant that appears in chiral perturbation theory for the decay π 0 → γγ at NLO. With additional input from experiment and theory, we reproduce recent estimates for the decay width Γ(π 0 → γγ). We also study the asymptotic large-Q 2 behavior of the transition form factor in the double-virtual case. Finally, we provide as our main result a more precise model-independent lattice estimate of the pion-pole contribution to hadronic light-by-light scattering in the muon g − 2: a HLbL;π 0 µ = (59.7 ± 3.6) × 10 −11 . Using in addition the normalization of the form factor obtained by the PrimEx experiment, we get the lattice and data-driven estimate a HLbL;π 0 µ = (62.3 ± 2.3) × 10 −11 .

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We calculate the π 0 → γ * γ * transition form factor F π 0 γ * γ * (q 2 1 , q 2 2 ) in lattice QCD with two flavors of quarks. Our main motivation is to provide the input to calculate the π 0 -pole contribution to hadronic light-by-light scattering in the muon (g − 2), a HLbL;π 0 µ . We therefore focus on the region where both photons are spacelike up to virtualities of about 1.5 GeV 2 , which has so far not been experimentally accessible. Results are obtained in the continuum at the physical pion mass by a combined extrapolation. We reproduce the prediction of the chiral anomaly for real photons with an accuracy of about 8 − 9%. We also compare to various recently proposed models and find reasonable agreement for the parameters of some of these models with their phenomenological values. Finally, we use the parametrization of our lattice data by these models to calculate a HLbL;π 0 µ .