Chiral symmetry and dispersion relations: from $\pi \pi$ scattering to hadronic light-by-light.

<!--HTML--><p>Chiral symmetry provides strong constraints on hadronic matrix elements at low energy, which are most efficiently derived with chiral perturbation theory. As an effective quantum field theory the latter also accounts for rescattering or unitarity effects, albeit only perturbatively, via the loop expansion. In cases where rescattering effects are important it becomes necessary to go beyond the perturbative expansion, e.g. by using dispersion relations. A matching between the chiral and the dispersive representation provides in several cases results of high precision. I will discuss this approach with the help of a few examples, like $\pi \pi$ scattering (which has been tested successfully by CERN experiments like NA48/2 and DIRAC), $\eta \to 3 \pi$ and the hadronic light-by-light contribution to $(g-2)_\mu$. For the latter quantity the implementation of the dispersive approach has opened up the way to a model-independent calculation and the concrete possibility to significantly reduce the theoretical uncertainty.</p>