Dissecting the hadronic contributions to $(g-2)_\mu $ by Schwinger's sum rule

The theoretical uncertainty of $(g-2)_\mu $ is currently dominated by hadronic contributions. In order to express those in terms of directly measurable quantities, we consider a sum rule relating $g-2$ to an integral of a photo-absorption cross section. The sum rule, attributed to Schwinger, can be viewed as a combination of two older sum rules: Gerasimov-Drell-Hearn and Burkhardt-Cottingham. The Schwinger sum rule has an important feature, distinguishing it from the other two: the relation between the anomalous magnetic moment and the integral of a photo-absorption cross section is linear, rather than quadratic. The linear property makes it suitable for a straightforward assessment of the hadronic contributions to $(g-2)_\mu $. From the sum rule we rederive the Schwinger $\alpha/2\pi$ correction, as well as the formula for the hadronic vacuum-polarization contribution. As an example of the light-by-light contribution we consider the single-meson exchange.