Hadronic vacuum polarization to three loops in chiral perturbation theory

Hadronic vacuum polarization at low virtualities limits the precision of experimental tests of the standard model via important physical observables. Here we compute that effect in two-flavor chiral perturbation theory to three loops. Among the master integrals that describe the amplitude, six are elliptic functions of the momentum. Of these five are new to this work, although all can be related to the three-loop sunset integral. The renormalizability of the amplitude hinges on relations between the master integrals that were not previously known and that are not consequences of the integration-by-parts reduction. Our result is intended to serve as a starting point for phenomenological calculations, as well as the computation of finite-volume corrections in lattice QCD.