  By far the biggest contribution to hadronic vacuum polarization (HVP) arises
from the two-pion channel. Its quark-mass dependence can be evaluated by
combining dispersion relations with chiral perturbation theory, providing
guidance on the functional form of chiral extrapolations, or even
interpolations around the physical point. In addition, the approach allows one
to estimate in a controlled way the isospin-breaking (IB) corrections that
arise from the pion mass difference. As an application, we present an updated
estimate of phenomenological expectations for electromagnetic and strong IB
corrections to the HVP contribution to the anomalous magnetic moment of the
muon. In particular, we include IB effects in the $\bar K K$ channel, which are
enhanced due to the proximity of the $\bar K K$ threshold and the $\phi$
resonance. The resulting estimates make it unlikely that the current tension
between lattice-QCD and data-driven evaluations of the HVP contribution is
caused by IB corrections.
