Finite-Size Effects of the HVP Contribution to the Muon $g-2$ with C$^{\star}$ Boundary Conditions

The muon $g-2$ is a compelling quantity due to the current standing tensions among the experimental average, data-driven theoretical results, and lattice results. Matching the final target accuracy of the experiments at Fermilab and J-PARC will constitute a major challenge for the lattice community in the coming years. For this reason, it is worthwhile to consider different options to keep the systematic errors under control. In this proceedings, we discuss finite-volume effects of the leading Hadron Vacuum Polarization (HVP) contribution to the muon $g-2$ in the presence of C$^{\star}$ boundary conditions. When considering isospin-breaking corrections to the HVP, C$^{\star}$ boundary conditions provide a possible consistent formulation of $\mathrm{QCD+QED}$ in finite volume. Even though these boundary conditions can be avoided in the calculation of the leading HVP contribution, we find the interesting result that they remove the leading exponential finite-volume correction. In practice, compared to the periodic case, C$^{\star}$ boundary conditions cut the finite-size effects in half on a lattice of physical size $M_{\pi}L=4$ and by a factor of almost ten for $M_{\pi}L=8$. We discuss the origin of this reduction and implications for computational efficiency.