Dispersive Treatments of $K_{\ell4}$ Decays and Hadronic Light-by-Light Scattering

In this thesis, I present dispersive treatments of two hadronic processes: the semileptonic kaon decay $K_{\ell4}$ and hadronic light-by-light scattering. The $K_{\ell4}$ decay is one of the best sources of information on some of the parameters of chiral perturbation theory. The dispersion relation for $K_{\ell4}$ provides a resummation of $\pi\pi$- and $K\pi$-rescattering effects. In contrast to a pure chiral treatment, it reproduces the observed curvature of one of the form factors. The matching of the dispersion relation to the chiral representation of the form factors allows the extraction of the values of three low-energy constants. Hadronic light-by-light scattering appears as a virtual process in the calculation of the anomalous magnetic moment of the muon $(g-2)_\mu$. For more than a decade, a discrepancy of about $3\sigma$ has persisted between the experimental determination and the standard-model prediction of the $(g-2)_\mu$. It is expected that within a few years hadronic light-by-light scattering will dominate the uncertainty of the theory prediction of the $(g-2)_\mu$. So far, only model calculations of the hadronic light-by-light contribution are available. However, in view of forthcoming $(g-2)_\mu$ experiments at Fermilab and J-PARC it is crucial that the hadronic light-by-light calculation can be improved systematically. The dispersive description presented here provides a formalism for a data-driven determination of hadronic light-by-light scattering and hence opens up an avenue towards a model-independent evaluation of the $(g-2)_\mu$