Three-Loop Hadronic Vacuum Polarization in Chiral Perturbation Theory

One of the many hurdles facing lattice computations of quantities such as the hadronic vacuum polarization (HVP), are finite-volume effects. Since these are dominated by low-energy phenomena, they can be predicted and controlled using the low-energy effective field theory of QCD, Chiral Perturbation Theory (ChPT). I present the first three-loop (next-to-next-to-next-to-leading order) computation of this quantity in infinite volume, and show some glimpses into its ongoing generalization to finite volume. As a three-loop amplitude without propagating massless particles, the HVP exhibits surprisingly deep mathematical structures not seen in QED and other typical subjects of high-loop calculations. I will therefore demonstrate several ways in which we have pushed the envelope of current loop integral technology.