Chiral extrapolation of the pion scattering amplitudes and the hadronic vacuum polarization

Chiral extrapolations are essential to connect lattice QCD simulations, often performed at unphysically large pion masses, to experimental data. In these proceedings, we summarize how the combination of dispersion relations and Chiral Perturbation Theory (ChPT) provides a controlled framework for these extrapolations in low-energy hadronic observables. We start by presenting compact analytic expressions for the one- and two-loop pion-pion partial-wave amplitudes in ChPT. We then analyze lattice-QCD data for the $\pi\pi$ $P$-wave scattering amplitude using the inverse amplitude method at subleading order and develop a fit strategy that allows for the first time for a stable two-loop fit. Finally, we use these results to study the pion-mass dependence of the two-pion channel in the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon. Our results provide valuable insight into the functional form of chiral extrapolations and even allow interpolations around the physical pion mass, providing guidance for future lattice-QCD studies.