Hadronic vacuum polarization for the muon g-2 from lattice QCD: Analyzing long-distance effects at high precision

The Hadronic Vacuum Polarization (HVP) contribution continues to be the precision bottleneck to the Standard Model calculation of muon g-2. As of this year, lattice QCD results have replaced their data-driven counterparts in the consensus estimate of the HVP contribution.  This achievement is in-part due to significant precision gains in lattice determinations of the long-distance region of the HVP. In this talk, we discuss results from the Fermilab Lattice, HPQCD, and MILC collaborations for this contribution. We give special attention to the finite volume correction, which connects our finite-volume simulations to the infinite volume limit. We include results from our recent large volume analysis, and discuss noise reduction techniques used in our datasets to control statistical errors at long distance.  All calculations are performed on N_f=2+1+1 highly-improved staggered-quark (HISQ) ensembles with a physical pion mass.  This is a companion talk to Jake Sitison's presentation on our collaboration's results for the full HVP contribution to g-2.